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+arXiv:2301.00268v1  [math-ph]  31 Dec 2022
+AUTOCORRELATIONS OF CHARACTERISTIC POLYNOMIALS
+FOR THE ALTERNATIVE CIRCULAR UNITARY ENSEMBLE
+BRAD RODGERS, HARSHITH SAI VALLABHANENI
+Abstract. We find closed formulas for arbitrarily high mixed moments of
+characteristic polynomials of the Alternative Circular Unitary Ensemble (ACUE),
+as well as closed formulas for the averages of ratios of characteristic polynomials
+in this ensemble. A comparison is made to analogous results for the Circular
+Unitary Ensemble (CUE). Both moments and ratios are studied via symmetric
+function theory and a general formula of Borodin-Olshanski-Strahov.
+1. Introduction
+In this short note we examine mixed moments and averages of ratios of char-
+acteristic polynomials associated with the Alternative Circular Unitary Ensemble
+(ACUE). Our main results are a closed formula for arbitrarily high mixed moments
+in Theorem 2 and a closed formula for averages of ratios in Theorem 6. The ACUE
+refers to a certain random collection of points on the unit circle of the complex plane
+whose distribution is meant to mimic the points of the Circular Unitary Ensemble
+(CUE) of random matrix theory. Let us use the notation
+∆(x1, ..., xN) :=
+�
+1≤j<k≤N
+(xj − xk),
+for a Vandermonde determinant, an anti-symmetric function in the variables
+x1, ..., xN. For an integer N ≥ 1, we use the label ACUE(N) to denote the ran-
+dom collection of N points {e(ϑ1), ..., e(ϑN)} on the unit circle S1 which have the
+following joint density: for an arbitrary function f : (S1)N → C,
+EACUE(N)
+�
+f(e(ϑ1), ..., e(ϑN))
+�
+= 1
+N!
+1
+(2N)N
+�
+t1,...,tN
+f(e(t1), ..., e(tN)) · |∆(e(t1), ..., e(tN))|2,
+(1)
+where each index ti is summed over the set {0,
+1
+2N ,
+2
+2N , ..., 2N−1
+2N } (so that the sum
+consists of (2N)N terms in total). Likewise we use the label CUE(N) to denote
+the random collection of N points {e(θ1), ..., e(θN)} on the unit circle with joint
+density given by:
+ECUE(N)
+�
+f(e(θ1), ..., e(θN))
+�
+= 1
+N!
+�
+[0,1]N f(e(t1), ..., e(tN)) · |∆(e(t1), ..., e(tN))|2 dNt.
+(2)
+It is known that EACUE(N)[1] = ECUE(N)[1] = 1, so both these expressions indeed
+implicitly define joint probability densities. These joint densities are each sym-
+metric in all variables, so the ACUE(N) and the CUE(N) may be seen as point
+
+2
+BRAD RODGERS, HARSHITH SAI VALLABHANENI
+processes supported on the 2N-th roots of unity or the unit circle of the complex
+plane respectively. We use the notation EACUE(N) or ECUE(N) for the purpose of
+reminding the reader over which ensemble an expectation is being taken. (These
+could be replaced by the more traditional notation E with no change in meaning.)
+The ACUE was put forward in a blog post of T. Tao [28] in order to investigate
+the limitations of certain methods in analytic number theory. Of particular interest
+was a comparison of the k-level correlation functions of the ACUE and the CUE.
+The CUE can be seen as a finite model of how zeros of the Riemann zeta function
+and other L-functions are conjectured to be spaced, while the ACUE can be seen
+as a finite model of how zeros are very unlikely to be spaced but which cannot
+be ruled out by current methods. A similar construction (replacing the CUE and
+ACUE with limiting point processes) was independently studied by J. Lagarias and
+the first author of this paper [21] around the same time. Related point processes
+have also been studied for reasons unrelated to number theory in the past; see e.g.
+[3, 4].
+It is therefore of interest to investigate similarities and differences between the
+CUE and ACUE. In this paper we examine the statistics induced by characteristic
+polynomials associated to the CUE and ACUE. The CUE is naturally associated
+to the eigenvalues of a random Haar distributed unitary matrix, but there is not
+an especially natural matrix interpretation for the ACUE (though see Remark 6 of
+[28]). In order to easily speak of the characteristic polynomial associated to these
+ensembles, define the diagonal matrices
+g := diag(e(ϑ1), ..., e(ϑN))
+(associated to ACUE(N))
+G := diag(e(θ1), ..., e(θN))
+(associated to CUE(N)).
+We refer to the random functions det(1 − zg) and det(1 − zG) in the complex
+variable z as the characteristic polynomials associated with the ACUE and CUE
+respectively. Note that det(1 − zG) will have the same distribution as if G were a
+random unitary matrix chosen according to Haar measure.
+A purpose in this paper is to examine mixed moments of characteristic polyno-
+mials from the ACUE. In his blog post (see Remark 7), Tao made the remarkable
+observation that for quite large powers, moments of characteristic polynomials as-
+sociated to ACUE(N) and CUE(N) agree:
+Theorem 1 (Tao). For positive integers K, L ≤ N,
+EACUE(N)
+� K
+�
+k=1
+det(1 − ukg)
+L
+�
+ℓ=1
+det(1 − vkg)
+�
+= ECUE(N)
+� K
+�
+k=1
+det(1 − ukG)
+L
+�
+ℓ=1
+det(1 − vkG)
+�
+.
+This allows one to compute a large range of moments for the ACUE using known
+results for the CUE. Nonetheless it is interesting to ask if a closed formula can be
+found that allows for the computation of all moments, and this is a main result of
+this paper.
+In order to state it, for an integer ℓ and positive integer m, we introduce the
+notation [ℓ]m ∈ {0, 1, ..., m − 1} to be the reduction of ℓ modulo m, and define the
+
+AUTOCORRELATIONS FOR THE ACUE
+3
+function
+HN,ℓ(v) :=
+�
+0
+if 0 ≤ [ℓ]2N ≤ N − 1
+v[ℓ]2N−N
+if N ≤ [ℓ]2N ≤ 2N − 1.
+(3)
+Theorem 2. For N, K, L ≥ 1, and v1, ..., vK+L ∈ C,
+EACUE(N)
+�
+det(g)−K
+K+L
+�
+k=1
+det(1 + vkg)
+�
+=
+det
+�
+φi(vj)
+�K+L
+i,j=1
+∆K+L(v1, ..., vK+L),
+where
+φi(v) = φ(K,L;N)
+i
+(v) :=
+�
+vN+L+K−i − vL HN, K−i(v)
+for 1 ≤ i ≤ K
+vK+L−i − vL+N−1 HN, i−K−1(1/v)
+for K + 1 ≤ i ≤ K + L.
+Note that for g associated to ACUE(N) we have
+det(g)−1 det(1 + vg) = vN det(1 + v−1g),
+(4)
+so that this formula indeed allows for the computation of mixed moments of char-
+acteristic polynomials and their conjugates.
+This should be compared to the analogous result for the CUE; we state this
+result in the formalism of Bump-Gamburd [8].
+Theorem 3 (Prop. 4 of [8]). For N, K, L ≥ 1,
+ECUE(N)
+�
+det(G)−K
+K+L
+�
+k=1
+det(1 + vkG)
+�
+= s⟨N k⟩(v1, ..., vK+L)
+=
+det
+�
+ψi(vj)
+�K+L
+i,j=1
+∆K+L(v1, ..., vK+L),
+where
+ψi(v) = ψ(K,L;N)
+i
+(v) :=
+�
+vN+L+K−i
+for 1 ≤ i ≤ K
+vK+L−i
+for K + 1 ≤ i ≤ K + L .
+The determinantal ratio here is just a definition of the Schur polynomial
+s⟨N K⟩(v1, ..., vK+L) associated to the partition ⟨N K⟩ = (N, ..., N), with K parts.
+An example makes the pattern of the matrices in the numerators of the right
+hand sides of Theorems 2 and 3 easier to see. If K = 5, L = 4, N = 2, columns in
+the variable v will be:
+For ACUE:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+φ1(v)
+φ2(v)
+φ3(v)
+φ4(v)
+φ5(v)
+−−
+φ6(v)
+φ7(v)
+φ8(v)
+φ9(v)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+v10
+v9 − v5
+v8 − v4
+v7
+v6
+−−
+v3
+v2
+v − v5
+1 − v4
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+,
+For CUE:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ψ1(v)
+ψ2(v)
+ψ3(v)
+ψ4(v)
+ψ5(v)
+−−
+ψ6(v)
+ψ7(v)
+ψ8(v)
+ψ9(v)
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+v10
+v9
+v8
+v7
+v6
+−−
+v3
+v2
+v
+1
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+,
+where the line serves only to visually separate the block with indices i ≤ K from
+the block with indices i ≥ K + 1.
+
+4
+BRAD RODGERS, HARSHITH SAI VALLABHANENI
+Note that if K, L ≤ N then we have
+K − i ≤ N − 1
+for 1 ≤ i ≤ K
+i − K − 1 ≤ N − 1
+for K + 1 ≤ i ≤ K + L,
+so it follows by examining the definition of HN,ℓ that φ(K,L;N) = ψ(K,L;N) in the
+above determinantal formulas. Thus these formulas recover the observation of Tao
+in Theorem 1. By contrast if K > N or L > N these formulas show the moments
+for these models differ, despite having a closely related structure.
+In fact it is by specializing the following formula for averages of ratios of char-
+acteristic polynomials that we derive Theorem 2.
+Theorem 4. For N and J positive integers, and v1, ..., vJ complex numbers and
+u1, ..., uJ complex numbers which are not 2N-th roots of unity,
+EACUE(N)
+� �J
+j=1 det(1 + vjg)
+�J
+j=1 det(1 + ujg)
+�
+=
+1
+det
+�
+1
+ui−vj
+� det
+�
+1
+ui − vj
+eN(ui, vj)
+�
+,
+(5)
+where the determinants on the right hand side are of J×J matrices, over the indices
+1 ≤ i, j ≤ J, and
+eN(u, v) := 1 − uNvN
+1 − u2N .
+This formula in turn is a consequence of a general formula introduced by Borodin-
+Olshanski-Strahov in [5] for computing the average of ratios of characteristic poly-
+nomials associated to what they call Giambelli-compatible point processes. We will
+show the ACUE falls into this class of point processes and then specialize their
+result; see Theorem 6 below.
+Theorem 4 may be compared to an analogous formula for the CUE (see e.g. [24,
+Thm. 4.2], [13, Thm. 5.4], or [19, (4.35)]):
+Theorem 5. For N and J positive integers, and v1, ..., vJ complex numbers and
+u1, ..., uJ complex numbers which do not lie on the unit circle,
+ECUE(N)
+� �J
+j=1 det(1 + vjG)
+�J
+j=1 det(1 + ujG)
+�
+=
+1
+det
+�
+1
+ui−vj
+� det
+�
+1
+ui − vj
+eN(ui, vj)
+�
+,
+where the determinants on the right hand side are of J×J matrices, over the indices
+1 ≤ i, j ≤ J, and
+eN(u, v) :=
+�
+1
+if |u| < 1,
+vN/uN
+if |u| > 1.
+From Theorem 4, a possible strategy for proving Theorem 2 is evident: we take
+appropriately scaled limits, with each ui tending either to 0 or ∞ in order to
+recover the average appearing in Theorem 2. Doing so nonetheless involves several
+nontrivial determinantal manipulations.
+There is at least one alternative strategy for proving Theorems 2 and 4, and
+this is to rely on the theory of orthogonal polynomials. This method has been
+used to derive similar formulas for moments and averages of ratios of characteristic
+polynomials in several random matrix ensembles; see for instance [7, 1, 17] for
+moments and [27, 6] for ratios. One difficulty in the orthogonal polynomial method
+is that the finitely supported weights which define the ACUE allow for at most a
+
+AUTOCORRELATIONS FOR THE ACUE
+5
+finite collection of monic orthogonal polynomials. It would be interesting to see if
+this difficulty can be overcome to give alterative proofs of Theorems 2 or 4.
+It is perhaps a little surprising that moments of characteristic polynomials from
+the ACUE have a structure related to those from the CUE even for very large
+powers. This may ultimately be seen as a consequence of the similarity between
+Theorems 4 and 5 for ratios; another purpose of this paper is to provide an expla-
+nation of how ratio formulas like Theorem 4 can be used to derive moment formulas
+like Theorem 2. It will be evident that the same method could be used to deduce
+Theorem 3 from Theorem 5 as well.
+We note that formulas for the averages of ratios of characteristic polynomials in
+the CUE usually are written in a form involving a sum over ‘swaps’, involving a
+slightly different formalism than Theorem 5, – see for instance [10, Prop 2.1], [9,
+Cor. 1.2], or [8, Thm. 3]. By use of the functional equation, these formulas can be
+deduced from Theorem 5. For instance, the J = 2 case of Theorem 5 entails the
+following: for complex numbers α, β, γ, δ with |γ|, |δ| < 1,
+ECUE(N)
+�
+det(1 − α G) det(1 − β G)
+det(1 − γ G) det(1 − δ G)
+�
+= βN
+δN ECUE(N)
+�
+det(1 − α G) det(1 − β−1 G)
+det(1 − γ G) det(1 − δ−1 G)
+�
+= (1 − βγ)(1 − αδ)
+(1 − δγ)(1 − αβ) + (αβ)N (1 − γα−1)(1 − δβ−1)
+(1 − α−1β−1)(1 − γδ).
+Note that this formula is valid only for |γ|, |δ| < 1. If instead for instance |γ| < 1
+and |δ| > 1, the left hand side would just work out to 1.
+By using the functional equation (4) for det(1 + vg) one can derive expressions
+of this sort for the ACUE as well. For instance, for complex numbers α, β, γ, δ with
+neither γ nor δ equal to 2N-th roots of unity, Theorem 4 reveals,
+EACUE(N)
+�
+det(1 − α g) det(1 − β g)
+det(1 − γ g) det(1 − δ g)
+�
+= βN
+δN EACUE(N)
+�
+det(1 − α g) det(1 − β−1 g)
+det(1 − γ g) det(1 − δ−1 g)
+�
+= (1 − βγ)(1 − αδ)
+(1 − δγ)(1 − αβ)
+�1 − αNγN
+1 − γ2N
+��1 − βNδN
+1 − δ2N
+�
++ (αβ)N (1 − γα−1)(1 − δβ−1)
+(1 − α−1β−1)(1 − γδ)
+�1 − β−NγN
+1 − γ2N
+��1 − α−NδN
+1 − δ2N
+�
+.
+Note that in this case there is no need to assume that |γ|, |δ| < 1. Indeed, the right
+and left hand sides are meromorphic in the variables γ and δ, with singularities
+only at 2N-th roots of unity.
+This procedure can be used to obtain formulas for J > 2 as well. But for mixed
+ratios of more than two characteristic polynomials, expansions like this for the
+ACUE seem to become increasingly more complicated than those for the CUE; by
+contrast the determinantal formula of Theorem 4 remains relatively simple for all
+J.
+It is natural to ask whether Theorems 2 or 6 shed light on any number theoretic
+phenomena. A typical question in number theory involves moments of the Riemann
+zeta-function in which powers K and L are fixed or grow slowly. Theorem 1 of Tao
+is certainly of interest in this regard, but because K and L must be of size at least
+N before Theorem 2 sees a difference between the CUE and ACUE prediction, it
+does not seem that the new information in this theorem will shed light on these
+
+6
+BRAD RODGERS, HARSHITH SAI VALLABHANENI
+sorts of questions. On the other hand, uniform estimates for moments can be of
+some interest in determining extreme values of L-functions (see e.g. [25, Sec. 7]),
+and Theorem 2 may be of some use in examining alternative possibilities here.
+Furthermore Theorem 4 suggests a hypothetical ‘alternative ratio formula’ for the
+Riemann zeta-function – a formula which one would like to rule out but cannot at
+present. This is discussed further in Section 4.
+Acknowledgements: We thank David Farmer and Ofir Gorodetsky for very use-
+ful references, comments, and corrections. B.R. received partial support from an
+NSERC grant and US NSF FRG grant 1854398.
+2. The ratio formula: Theorem 4
+In this section we prove Theorem 4. Our starting point is an application of a
+general formula of Borodin-Olshanski-Strahov to the ACUE.
+Theorem 6 (A Borodin-Olshanski-Strahov Formula for ACUE). For N and J
+positive integers, v1, ..., vJ complex numbers, and u1, ..., uJ complex numbers which
+are not 2N-th roots of unity,
+EACUE(N)
+� �J
+j=1 det(1 + vjg)
+�J
+j=1 det(1 + ujg)
+�
+=
+1
+det
+�
+1
+ui−vj
+� det
+�
+1
+ui − vj
+EACUE(N)
+�det(1 + vjg)
+det(1 + uig)
+��
+,
+(6)
+where the determinants on the right hand side are of J×J matrices, over the indices
+1 ≤ i, j ≤ J.
+Proof. This requires only minor modifications of formulas in [5]. Claims I and II
+of that paper show that if α is a measure on C with finite moments and if a point
+process consisting of N points {z1, ..., zN} in C has a joint density given by
+(const.)
+�
+1≤i<j≤N
+|zi − zj|2
+N
+�
+i=1
+α(dzi),
+then as a formal powers series
+E
+�
+H(α1) · · · H(αJ)E(β1) · · · E(βJ)
+�
+=
+1
+det
+�
+1
+αi+βj
+� det
+�
+1
+αi + βj
+E
+�
+H(αi)E(βj)
+��
+,
+where
+H(α) :=
+1
+�N
+j=1(1 − zjα−1)
+,
+E(β) :=
+N
+�
+j=1
+(1 + zjβ−1).
+This is only claimed for a measure α supported on R, but the proof applies with
+no change to measures supported on C, except that in the proof of Theorem 3.1 the
+moments An =
+�
+R xnα(dx) must be replaced by An,m =
+�
+C znzm α(dz) and later
+in the proof Aλi+N−i+N−j must be replaced by Aλi+N−i , N−j.
+The point process ACUE is induced by such a joint density where α is a proba-
+bility measure uniform on the 2N-th roots of unity in C. This identity may be seen
+
+AUTOCORRELATIONS FOR THE ACUE
+7
+to be true not just for formal powers series but for functions H(α), E(β) by con-
+sidering the case |α1|, ..., |αJ| > 1 (where all power series will converge absolutely)
+and then meromorphically continuing to all α1, ..., αJ.
+Finally, we arrive at (6) simply by setting αj = −u−1
+j , βj = v−1
+j
+and simplifying
+the resulting determinants.
+□
+The remainder of this section is therefore devoted to understanding the expec-
+tation which occurs on the right hand side of (6), accomplished in Proposition 8
+below.
+Lemma 7. Consider a hook partition (a, 1b) with a ≥ 1 and b ≥ 0 of length
+b + 1 ≤ N.
+For the Schur polynomial s(a,1b) associated to this partition in the
+variables e(ϑ1), ..., e(ϑN) of the ACUE(N), we have
+EACUE(N)s(a,1b) =
+�
+(−1)b
+if
+a + b ≡ 0 (mod 2N)
+0
+otherwise.
+Proof. Label ωj = e(ϑj) so that for a partition λ of length ℓ(λ) ≤ N,
+sλ = det(ωλj+N−j
+i
+)
+det(ωN−j
+i
+)
+,
+where if ℓ(λ) < N we adopt the convention λℓ(λ)+1 = · · · = λN = 0, and the
+determinants above are N × N.
+Note that det(ωN−j
+i
+) = ∆(ω1, ..., ωN).
+Hence from the definition (1) of the
+ACUE,
+EACUE(N)s(a,1b)
+=
+1
+N! (2N)N
+�
+t1,...,tN
+det
+�
+e
+�
+(λj + N − j)ti
+��
+det
+�
+e
+�
+(N − j)ti
+��
+,
+(7)
+where each index ti is summed over the set {0,
+1
+2N , ..., 2N−1
+2N }.
+Expanding each
+determinant into a sum over permutations mapping {1, ..., N} to {1, ..., N} one
+sees
+det
+�
+e
+�
+λj + N − j)ti
+��
+det
+�
+e
+�
+N − j)ti
+��
+=
+�
+σ,π∈SN
+(−1)σ(−1)π
+N
+�
+i=1
+e
+�
+(λσ(i) + N − σ(i))ti − (N − π(i))ti
+�
+.
+Thus (7) is
+= 1
+N!
+�
+σ,π∈SN
+(−1)σ(−1)π1
+�
+(λσ(i) − σ(i)) + π(i) ≡ 0 (mod 2N)
+for all i
+�
+=
+�
+π∈SN
+(−1)π1
+�
+λi − i + π(i) ≡ 0 (mod 2N)
+for all i
+�
+,
+(8)
+where 1[ · ] denotes an indicator function, taking the value 1 or 0 depending on
+whether the proposition inside is true or false.
+
+8
+BRAD RODGERS, HARSHITH SAI VALLABHANENI
+In the special case that λ = (a, 1b) this sum has a simple evaluation. In that
+case any nonvanishing summand will have π satisfying
+a − 1 + π(1) ≡ 0 (mod 2N)
+and
+1 − 2 + π(2) ≡ 0 (mod 2N)
+...
+1 − (b + 1) + π(b + 1) ≡ 0 (mod 2N)
+and
+0 − (b + 2) + π(b + 2) ≡ 0 (mod 2N)
+...
+0 − N + π(N) ≡ 0 (mod 2N).
+Since 1 ≤ π(i) ≤ N, the last N − 1 of these equations force
+π(b + 2) = b + 2, ...., π(N) = N,
+π(2) = 1, ...., π(b + 1) = b.
+This forces
+π(1) = b + 1,
+and so at most one permutation π makes a nonzero contribution to (8), and that
+contribution is nonzero if and only if a+b ≡ 0 (mod 2N), since a+b = a−1+π(1).
+Since in cycle notation this permutation is π = (b + 1, b, ..., 2, 1) we have (−1)π =
+(−1)b and this verifies the lemma.
+□
+Proposition 8. For v any complex number and u any complex number which is
+not a 2N-th root of unity,
+EACUE(N)
+det(1 + vg)
+det(1 + ug) = 1 − uNvN
+1 − u2N .
+Proof. We first consider |u| < 1. From a series expansion we have
+det(1 + vg)
+det(1 + ug) =
+N
+�
+j=0
+∞
+�
+k=0
+(−1)kejhkvjuk,
+(9)
+where ej and hk are respectively elementary symmetric polynomials of degree j and
+homogeneous symmetric polynomials of degree k in the variables e(ϑ1), ..., e(ϑN)
+associated to ACUE(N). Note that
+e0 = h0 = 1,
+while other terms can be expression in terms of Schur polynomials in the variables
+e(ϑ1), ..., e(ϑN):
+ejh0 = s(1j)
+for 1 ≤ j ≤ N,
+e0hk = s(k)
+for k ≥ 1,
+ejhk = s(k+1,1j−1) + s(k,1j)
+for 1 ≤ j ≤ N − 1, k ≥ 1,
+eNhk = s(k+1,1N−1)
+for k ≥ 1,
+
+AUTOCORRELATIONS FOR THE ACUE
+9
+with the first two identities following from the combinatorial definition of Schur
+functions [26, Sec. 7.10], and the last two from the Pieri rule [26, Thm. 7.15.7].
+From Lemma 7 it thus follows
+EACUE(N)ejhk =
+
+
+
+
+
+1
+if j = 0, k ≡ 0 (mod 2N),
+(−1)N−1
+if j = N, k ≡ N (mod 2N),
+0
+otherwise.
+Hence from (9),
+EACUE(N)
+det(1 + vg)
+det(1 + ug) = (1 + u2N + u4N + · · · ) − (vNuN + vNu3N + vNu5N + · · · )
+= 1 − uNvN
+1 − u2N ,
+for |u| < 1 and all v. The result then follows by analytic continuation.
+□
+Thus we have:
+Proof of Theorem 4. Apply Proposition 8 to Theorem 6.
+□
+3. The moment formula: Theorem 2
+Our technique in proving Theorem 2 will be to condense the determinants in
+(5) by letting all ui → 0. We begin with several lemmas that are useful for that
+purpose.
+The following is a slight generalization of Lemma 1 of [22].
+Lemma 9 (Determinantal Condensation Identity). Take q ≤ J. For f1, f2, ..., fJ
+functions (mapping R to C) that are at least q times continuously differentiable at
+the point a,
+lim
+u1,...,uq→a
+1
+∆(uq, ..., u1) det
+�
+fj(ui)
+�J
+i,j=1 = det
+
+
+�
+1
+(i−1)!f (i−1)
+j
+(a)
+�
+i≤q, j≤J
+�
+fj(ui)
+�
+q+1≤i≤J, j≤J
+
+ ,
+(10)
+where on the left hand side the limit is taken in the order that first u1 → u2, then
+u2 → u3, ... , uq−1 → uq, and finally uq → a.
+Proof. We prove this identity by induction, viewing ∆(u1) = 1 for the q = 1 case,
+which then becomes trivial. Suppose then that (10) has been proved for a limit in
+q − 1 variables. This implies for a limit in q variables,
+lim
+u1,...,uq→a
+1
+∆(uq, ..., u1) det
+�
+fj(ui)
+�J
+i,j=1
+= lim
+uq→a
+lim
+uq−1→uq
+1
+(uq − qq−1)q−1 det
+
+
+�
+1
+(i−1)!f (i−1)
+j
+(uq−1)
+�
+i≤q−1,j≤J
+�
+fj(ui)
+�
+i≥q,j≤J
+
+ .
+But Taylor expanding the entries of row q as
+fj(uq) =
+q
+�
+i=1
+f (i−1)
+j
+(uq−1)
+(i − 1)!
+(uq − uq−1)i−1 + O((uq − uq−1)q),
+and using multilinearity of the determinant to cancel out the first q − 1 terms of
+the above sum in row q, the claimed result quickly follows.
+□
+
+10
+BRAD RODGERS, HARSHITH SAI VALLABHANENI
+Remark 10. It is likely that a result of this sort remains true no matter the
+path along which a limit is taken (perhaps with further analytic conditions on the
+functions fj), but we won’t require that in what follows.
+Remark 11. It is easy to see by permuting rows of the determinant that this result
+also implies
+lim
+u1,...,uq→a
+1
+∆(u1, ..., uq) det
+�
+fj(ui)
+�J
+i,j=1 = det
+
+
+�
+1
+(q−i)!f (q−i)
+j
+(a)
+�
+i≤q,j≤J
+�
+fj(ui)
+�
+i≥q+1,j≤J
+
+
+(11)
+and
+lim
+uq+1,...,uK→a
+1
+∆(uK, ..., uq+1) det
+�
+fj(ui)
+�J
+i,j=1 = det
+
+
+�
+fj(ui)
+�
+i≤q,j≤J
+�
+1
+(i−q−1)!f (i−q−1)
+j
+(a)
+�
+i≥q+1,j≤J
+
+ ,
+(12)
+where in this last equation the limit is taken in the order uq+1 → uq+2,..., uK−1 →
+uK, uK → a.
+In applying this lemma we need the following computation.
+Lemma 12. For integers ℓ ≥ 0 and N ≥ 1,
+lim
+u→0
+1
+ℓ!
+dℓ
+duℓ
+�
+1
+u − v
+1 − uNvN
+1 − u2N
+�
+= −pN,ℓ(v),
+for pN,ℓ defined by
+pN,ℓ(v) :=
+1
+vℓ+1 −vN−1 HN,ℓ(1/v) =
+1
+vℓ+1 −
+�
+0
+if 0 ≤ [ℓ]2N ≤ N − 1
+v2N−1−[ℓ]2N
+if N ≤ [ℓ]2N ≤ 2N − 1.
+(13)
+Proof. Note that we have
+1
+u − v
+1 − uNvN
+1 − u2N
+= −1
+v
+1
+1 − u/v + uN − vN
+u − v
+uN
+1 − u2N
+= −
+�1
+v + u
+v2 + u2
+v3 + · · ·
+�
++ (vN−1 + vN−2u + · · · + uN−1)(uN + u3N + · · · ),
+taking a series expansion around u = 0. Since the quantity on the left hand side of
+the Lemma is exactly the coefficient of uℓ in this expansion, the claim follows by
+inspection.
+□
+Lemma 13. (Cauchy Determinant Formula) For u1, ..., uJ and v1, ..., vJ collections
+of complex numbers with no elements in common,
+det
+�
+1
+ui − vj
+�J
+i,j=1 = ∆(uJ, ..., u1)∆(v1, ..., vJ)
+□(u; v)
+where
+□(u; v) :=
+J
+�
+i=1
+J
+�
+j=1
+(ui − vj).
+Proof. See for instance [23, Part 7, §1, Ex. 3].
+□
+We can now give a proof of the moment formula for ACUE.
+
+AUTOCORRELATIONS FOR THE ACUE
+11
+Proof of Theorem 2. We set u = (u1, ..., uK) and u′ = (u′
+1, ..., u′
+L) as abbreviations
+for ordered lists, and let u ∪ u′ := (u1, ..., uK, u′
+1, ..., u′
+L) be an (ordered) concatena-
+tion of these lists. We abbreviate ∆(u) = ∆(u1, ..., uK) and also use the notation
+�∆(u) = ∆(uK, ..., u1) = (−1)K(K−1)/2∆(u).
+Our starting point is the identity
+EACUE(N)
+�
+det(g)−K
+K+L
+�
+k=1
+det(1 + vkg)
+�
+= lim
+u→∞ uN
+1 · · · uN
+K lim
+u′→0 EN(u ∪ u′ ; v),
+(14)
+where we define
+EN(u ∪ u′ ; v) := EACUE(N)
+�
+�K+L
+k=1 det(1 + vkg)
+�K
+k=1 det(1 + ukg) �L
+ℓ=1 det(1 + u′
+ℓg)
+�
+.
+The limits u → ∞ and u′ → 0 mean u1, ..., uK → ∞ and u′
+1, ..., u′
+L → 0. In what
+follows we will take these in the order u′
+1 → u′
+2, ..., u′
+L−1 → u′
+L, u′
+L → 0 and
+u1 → u2, ..., uK−1 → uK, uK → ∞ so that Lemma 9 can easily be applied.
+For notational reasons we write
+FN(u, v) :=
+1
+u − v
+1 − uNvN
+1 − u2N .
+We use Theorem 4 and Lemma 13 to see,
+EN(u ∪ u′ ; v) =
+□(u ∪ u′ ; v)
+�∆(u ∪ u′)∆(v)
+det
+
+
+�
+fN(ui, vj)
+�
+i≤K,j≤K+L
+�
+fN(u′
+i−K, vj)
+�
+i≥K+1,j≤K+L
+
+
+=
+□(u ; v)□(u′ ; v)
+�∆(u)�∆(u′)□(u′; u)∆(v)
+det
+
+
+�
+fN(ui, vj)
+�
+i≤K,j≤K+L
+�
+fN(u′
+i−K, vj)
+�
+i≥K+1,j≤K+L
+
+ .
+Taking a limit u′ → 0 and using Lemma 9 – in particular its consequence (12) –
+and Lemma 12,
+lim
+u′→0 EN(u∪u′ ; v) =
+□(u; v) �K+L
+k=1 (−vk)L
+�∆(u) �K
+k=1(−uk)L∆(v)
+det
+
+
+�
+fN(ui, vj)
+�
+i≤K,j≤K+L
+�
+− pN,i−K−1(vj)
+�
+i≥K+1,j≤K+L
+
+ .
+But note the easily verified functional equation
+fN(u, v) = −fN(u−1, v−1)vN−1u−(N+1).
+Thus
+uN
+1 · · · uN
+K lim
+u′→0 EN(u ∪ u′ ; v)
+= (−1)L □(u; v) �K+L
+k=1 vL
+k
+�∆(u)∆(v)
+K
+�
+k=1
+u−L−1
+k
+·det
+
+
+�
+− vN−1
+j
+fN(u−1
+i , v−1
+j )
+�
+i≤K,j≤K+L
+�
+− pN,i−K−1(vj)
+�
+i≥K+1,j≤K+L
+
+ .
+(15)
+
+12
+BRAD RODGERS, HARSHITH SAI VALLABHANENI
+For fixed v, we have as u → ∞,
+□(u; v)
+�∆(u)
+K
+�
+k=1
+u−L−1
+k
+=
+□(u; v) �K
+k=1 u−L−1
+k
+�K
+k=1 uK−1
+k
+∆
+� 1
+u1 , ...,
+1
+uK
+� ∼
+1
+∆
+� 1
+u1 , ...,
+1
+uK
+�.
+Applying Lemma (9) – with its consequence (11) this time – and Lemma 12, the
+limit of (15) as u → ∞ is
+= (−1)L
+�K+L
+k=1 vL
+k
+∆(v)
+det
+
+
+�
+vN−1
+j
+pN,K−i(v−1
+j )
+�
+i≤K,j≤K+L
+�
+− pN,i−K−1(vj)
+�
+i≥K+1,j≤K+L
+
+
+=
+1
+∆(v) det
+
+
+�
+vN+L−1
+j
+pN,K−i(v−1
+j )
+�
+i≤K,j≤K+L
+�
+vL
+j pN,i−K−1(vj)
+�
+i≥K+1,j≤K+L
+
+ .
+By inspection of matrix entries, the above is
+=
+det
+�
+φi(vj)
+�K+L
+i,j=1
+∆(v)
+.
+Recalling (14), this is exactly what we sought to prove.
+□
+4. Hypothetical implications for ratios of ζ(s)
+Let us briefly and somewhat informally discuss these results in the context of the
+distribution of the Riemann zeta-function. For the sake of this discussion, suppose
+the Riemann Hypothesis is true, and label the nontrivial zeros of the zeta-function
+by {1/2 + iγj}j∈Z, so that γj ∈ R for all j. What is widely believed about the local
+distribution of zeros concerns two point processes, the first point process (associated
+to a large parameter T ) given by
+�log T
+2π (γj − t)
+�
+j∈Z
+(16)
+where t ∈ [T, 2T ] is chosen randomly and uniformly, and the second point process
+(associated to a large parameter N) given by
+{Nθi}i=1,...,N
+(17)
+where θ1, ..., θN ∈ [−1/2, 1/2) are identified with the points e(θ1), ..., e(θN) of
+CUE(N). The widely believed GUE (Gaussian Unitary Ensemble) Hypothesis states
+that as T → ∞ and N → ∞ both point processes (16) and (17) tend to the same
+limiting point process.
+(This means that randomly generated configurations of
+points from these two processes will look similar near the origin of the real line.)
+The ACUE was first investigated as one alternative model of how zeros of the
+Riemann zeta-function might be spaced.
+In particular, one considers the point
+process (associated to a large parameter N) given by
+{Nϑi + r
+2}i=1,...,N
+(18)
+where ϑ1, ..., ϑN
+∈ {− 1
+2, −N+1
+2N
+, −N+2
+2N
+, ..., N−1
+2N } are identified with the points
+e(ϑ1), ..., e(ϑN) of ACUE(N), and r ∈ [0, 1) is chosen independently, and uniformly
+at random. As N → ∞ the point process (18) tends to a limiting process, called the
+AH point process in [21]. The AH point process has correlation functions which
+mimic the limiting process for CUE (see [20] for further discussion), but it also
+
+AUTOCORRELATIONS FOR THE ACUE
+13
+has gaps between points which are always half-integers. In this way it is one pos-
+sible – though likely not a unique – candidate for a limiting distribution of the
+zeta-function point process 16 which is compatible with what is currently known
+about the local distribution of zeros of the zeta-function and also with the so-called
+Alternative Hypothesis, a (widely disbelieved) conjecture that gaps between zeros
+always occur close to half-integer multiples of the mean spacing.
+For this reason [28] gave the name AGUE (Alternative Gaussian Unitary En-
+semble) Hypothesis to the hypothetical claim that as T → ∞ the zeta zero point
+process (16) tends to the AH point process. As one would like to rule out the
+Alternative Hypothesis, one would like to rule out the stronger AGUE Hypothesis.
+More details on the AH point process can be found in the references [21, 28],
+while further information on the Alternative Hypothesis in general can be found in
+[2].
+A major impetus for studying mixed moments of characteristic polynomials
+det(1 + uG) for the CUE came from the work of Keating-Snaith, who used in-
+formation about CUE moments to make a conjecture regarding moments of the
+Riemann zeta-function [18, Eq. (19)]. As first observed by Tao and as discussed in
+the introduction, the consequence of Theorem 1 that for sufficiently large N mixed
+moments in the CUE and ACUE agree suggests that even should the zeros of the
+Riemann zeta-function be spaced according to the pattern of the ACUE, this could
+still be consistent with the Keating-Snaith moment conjecture.
+The local spacing of zeros of the Riemann zeta-function is also closely related to
+the averages of ratios of shifts of the Riemann zeta function near the critical line.
+This perspective was first pursued by Farmer [14, 15] and has subsequently been
+investigated by others [9, 11, 12]. In particular note that from Theorem 5,
+lim
+N→∞ ECUE(N)
+�
+J
+�
+j=1
+det(1 − e−νj/NG)
+det(1 − e−µj/NG)
+�
+=
+1
+det
+�
+1
+νj−µi
+� det
+�
+1
+νj − µi
+e(µi, νj)
+�
+,
+for Re µj ̸= 0 for all j, where
+e(µ, ν) :=
+�
+1
+if Re µ > 0
+eµ−ν
+if Re µ < 0.
+From the results proved in [12, 24] it can be seen that the claim
+lim
+T →∞
+1
+T
+� 2T
+T
+J
+�
+j=1
+ζ(1/2 + νj/ log T + it)
+ζ(1/2 + µj/ log T + it) dt =
+1
+det
+�
+1
+νj−µi
+� det
+�
+1
+νj − µi
+e(µi, νj)
+�
+(19)
+for Re µj ̸= 0 for all j, is equivalent to the GUE Hypothesis. (In fact [24] treats
+only real µj, νj, but the method can be adapted to complex values. There is a
+notational difference in [24]; the function E used there satisfies E(ν, µ) = e(µ, ν)
+for the function e used here.)
+A belief in the AGUE Hypothesis would suggest that we replace characteristic
+polynomials det(1 − uG) as they appear above by det(1 − uei2πr/2Ng), where r ∈
+[0, 1) is independent of g and uniformly chosen. For the ACUE, from Theorem 4
+
+14
+BRAD RODGERS, HARSHITH SAI VALLABHANENI
+we have
+lim
+N→∞ EACUE(N)
+�
+J
+�
+j=1
+det(1 − e−νj/Ng)
+det(1 − e−µj/Ng)
+�
+=
+1
+det
+�
+1
+νj−µi
+� det
+�
+1
+νj − µi
+e(µi, νj)
+�
+,
+for µj /∈ i
+2Z for all j, where
+e(µ, ν) := 1 − e−µ−ν
+1 − e−2µ .
+Hence on the assumption of the AGUE Hypothesis, one should instead expect for
+Re µj ̸= 0 for all j,
+lim
+T →∞
+1
+T
+� 2T
+T
+J
+�
+j=1
+ζ(1/2 + νj/ log T + it)
+ζ(1/2 + µj/ log T + it) dt
+= lim
+N→∞
+� 1
+0
+EACUE(N)
+�
+J
+�
+j=1
+det(1 − e−νj/Neiπr/Ng)
+det(1 − e−µj/Neiπr/Ng)
+�
+dr
+=
+� 1
+0
+1
+det
+�
+1
+νj−µi
+� det
+�
+1
+νj − µi
+e(µi − iπr, νj − iπr)
+�
+dr
+=
+1
+det
+�
+1
+νj−µi
+�
+�
+|z|=1
+det
+�
+1
+νj − µi
+1 − ze−µ−ν
+1 − ze−2µ
+� dz
+z .
+(20)
+(20) is of course a different expression than (19). Thus for averages of ratios of
+the Riemann zeta-function, an ACUE spacing would be distinguished from CUE
+spacing. In fact using the methods of [12, 24] it should be possible to demonstrate
+rigorously that (20) is equivalent to the AGUE Hypothesis, but we do not pursue
+this here.
+References
+[1] J. Baik, P. Deift, E. Strahov. Products and ratios of characteristic polynomials of random
+Hermitian matrices. Integrability, topological solitons and beyond. J. Math. Phys. 44 (2003),
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+[2] S.A.C. Baluyot. On the pair correlation conjecture and the alternative hypothesis. J. Number
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+[3] A. Borodin. Periodic Schur process and cylindric partitions. Duke Math. J. 140 (2007), no.
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+[4] A. Borodin, A. Okounkov, G. Olshanski. Asymptotics of Plancherel measures for symmetric
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+[5] A. Borodin, G. Olshanski, and E. Strahov. Giambelli compatible point processes. Adv. in
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+[6] A. Borodin, E. Strahov. Averages of characteristic polynomials in random matrix theory.
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+[8] D. Bump, and A. Gamburd. On the averages of characteristic polynomials from classical
+groups. Comm. Math. Phys. 265 (2006), no. 1, 227–274.
+[9] J. B. Conrey, D.W. Farmer, and M.R. Zirnbauer. Howe pairs, supersymmetry, and ratios of
+random characteristic polynomials for the unitary groups U(N). Preprint.
+[10] J.B. Conrey, P.J. Forrester, and N.C. Snaith. Averages of ratios of characteristic polynomials
+for the compact classical groups. Int. Math. Res. Not. IMRN (2005): 397-431.
+
+AUTOCORRELATIONS FOR THE ACUE
+15
+[11] J.B. Conrey, N.C. Snaith. Applications of the L-functions ratios conjectures. Proc. Lond.
+Math. Soc. (3) 94 (2007), no. 3, 594–646.
+[12] J.B. Conrey, N.C. Snaith. Correlations of eigenvalues and Riemann zeros. Commun. Number
+Theory Phys. 2 (2008), no. 3, 477–536.
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+zeta function: the microscopic landscape and a new approach to ratios. Invent. math. 207,
+23–113 (2017).
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+71–87.
+[15] D.W. Farmer. Mean values of ζ′/ζ and the GUE hypothesis. Int. Math. Res. Not. (1995):
+71 – 82.
+[16] K. Johansson. Non-intersecting paths, random tilings and random matrices. Probab. Theory
+Related Fields 123 (2002), no. 2, 225–280.
+[17] B. Jonnadula, J.P. Keating, and F. Mezzadri. On the moments of characteristic polynomials.
+Glasg. Math. J. (2022) 1-21.
+[18] J.P. Keating, N.C. Snaith. Random matrix theory and ζ(1/2 + it). Comm. Math. Phys. 214
+(2000), no. 1, 57–89.
+[19] M. Kieburg, and T. Guhr. Derivation of determinantal structures for random matrix ensem-
+bles in a new way. J. Phys. A: Math. Theor. 43 (2010): 31pp.
+[20] J. C. Lagarias, and B. Rodgers. Band-limited mimicry of point processes by point processes
+supported on a lattice. Ann. Appl. Probab. 31 (2021), no. 1, 351–376.
+[21] J. C. Lagarias, and B. Rodgers. Higher correlations and the alternative hypothesis. Q. J.
+Math. 71 (2020), no. 1, 257–280.
+[22] A. Medjedovic. Exact Formulas for Averages of Secular Coefficients. MSc Thesis. University
+of Waterloo. Available at http://hdl.handle.net/10012/17591.
+[23] G. P´olya, and G. Szeg˝o. Problems and theorems in analysis. II. Theory of functions, zeros,
+polynomials, determinants, number theory, geometry. Translated from the German by C.
+E. Billigheimer. Reprint of the 1976 English translation. Classics in Mathematics. Springer-
+Verlag, Berlin, 1998. xii+392 pp.
+[24] B. Rodgers. Tail bounds for counts of zeros and eigenvalues, and an application to ratios.
+Comment. Math. Helv. 92 (2017), no. 2, 311–347.
+[25] K. Soundararajan. The distribution of values of zeta and L-functions. arXiv preprint
+arXiv:2112.03389.
+[26] R.P. Stanley. Enumerative combinatorics. Vol. 2. Cambridge Studies in Advanced Mathe-
+matics, 62. Cambridge University Press, Cambridge, 1999.
+[27] E. Strahov, Y.V. Fyodorov. Universal results for correlations of characteristic polynomials:
+Riemann-Hilbert approach. Comm. Math. Phys. 241 (2003), no. 2-3, 343–382.
+[28] T.
+Tao,
+The
+alternative
+hypothesis
+for
+unitary
+matrices,
+weblog
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+Avail-
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+matrices/
+[29] H. Widom. Random Hermitian matrices and (nonrandom) Toeplitz matrices. Toeplitz op-
+erators and related topics (Santa Cruz, CA, 1992), 9–15, Oper. Theory Adv. Appl., 71,
+Birkh¨auser, Basel, 1994.
+Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario,
+K7L 3N6, Canada
+E-mail address: [email protected]
+Indian Institute of Technology Kharagpur, Kharagpur, West Bengal 721302, India
+E-mail address: [email protected]
+

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+arXiv:2301.03285v1  [math.LO]  9 Jan 2023
+REGAININGLY APPROXIMABLE NUMBERS
+AND SETS
+PETER HERTLING, RUPERT H¨OLZL AND PHILIP JANICKI
+Fakult¨at f¨ur Informatik, Universit¨at der Bundeswehr M¨unchen,
+85577 Neubiberg, Germany
+Abstract. We call a real number α regainingly approximable if
+there exists a computable nondecreasing sequence (an)n of rational
+numbers converging to α such that α−an < 2−n for infinitely many
+n ∈ N. We also call a c.e. set A ⊆ N regainingly approximable if the
+strongly left-computable number 2−A is regainingly approximable.
+We characterize this property directly in terms of enumerations of
+A and show that there exists a c.e. set A ⊆ N that is not regainingly
+approximable. Our main result is a splitting theorem: any c.e. set
+C ⊆ N can be split effectively into two disjoint c.e. sets A and B
+that are regainingly approximable. These results imply that the
+set of regainingly approximable numbers lies properly between the
+set of computable numbers and the set of left-computable numbers
+and that it is not closed under addition.
+Keywords:
+left-computable numbers; effective approximation;
+computably enumerable sets; splitting; Solovay reducibility.
+AMS classification: 03D78, 03D25, 03D30
+1. Introduction
+We call a sequence (an)n of real numbers increasing if, for all n ∈ N,
+an < an+1, and nondecreasing if, for all n ∈ N, an ≤ an+1. A real
+number is called left-computable if there exists a computable nonde-
+creasing sequence of rational numbers converging to it; in [1, 4] these
+real numbers are called left-c.e.. A real number α is called computable if
+there exists a computable sequence (an)n of rational numbers satisfying
+|α−an| < 2−n, for all n ∈ N. It is easy to see that any computable real
+number is left-computable. Computable and left-computable numbers
+E-mail address: [email protected], [email protected],
+[email protected].
+Date: November 18, 2022.
+1
+
+2
+are important both in computable analysis [6] and in the theory of al-
+gorithmic randomness [1, 4]. In this article, we study real numbers that
+are limits of computable, nondecreasing, converging sequences (an)n of
+rational numbers satisfying the condition |α−an| < 2−n not necessarily
+for all n ∈ N but for infinitely many n ∈ N.
+Definition 1. We call a real number α regainingly approximable if
+there exists a computable nondecreasing sequence of rational num-
+bers (an)n converging to α such that we have α − an < 2−n for infinitely
+many n ∈ N.
+Fact 2.
+(1) Every computable number is regainingly approximable.
+(2) Every regainingly approximable number is left-computable.
+Proof.
+(1) Let α be a computable number, and let (an)n be a computable
+sequence of rational numbers satisfying |α − an| < 2−n, for all
+n ∈ N. Then the sequence (bn)n of rational numbers defined by
+bn := an+3 − 2−(n+1) is computable and increasing, converges to
+α as well, and satisfies, for all n ∈ N, α − bn < 2−n. Hence, α
+is regainingly approximable.
+(2) This is clear from the definitions.
+□
+In Section 2 we begin by showing that Definition 1 is robust under
+several slight modifications, where the equivalences are effectively uni-
+form.
+In Section 3 we apply the idea of regaining approximability to c.e. sets
+of natural numbers. In fact, most of our results concerning regainingly
+approximable numbers involve strongly left-computable real numbers
+and can be expressed more naturally directly in terms of sets A ⊆ N
+of natural numbers. A real number x ∈ [0, 1] is called strongly left-
+computable if there exists a computably enumerable set A ⊆ N with
+x = 2−A :=
+�
+a∈A
+2−(a+1).
+We define different variations of regaining approximability for c.e. sets,
+and will again see that they coincide. However, in contrast to the sit-
+uation for regainingly approximable numbers, not all arguments are
+fully effectively uniform in this setting. Next we prove that there is
+a c.e. set that is not regainingly approximable and we prove a split-
+ting result, namely that there is an effectively uniform procedure that
+splits every c.e. set C ⊆ N into two disjoint regainingly approximable
+sets A, B ⊆ N. Note that this implies that there exists a regainingly
+
+3
+approximable set A ⊆ N that is not decidable, and that the union and
+intersection of two regainingly approximable sets need not be regain-
+ingly approximable.
+In Section 4 we again turn to regainingly approximable numbers. We
+observe that a set A ⊆ N is regainingly approximable if and only if
+the strongly left-computable number 2−A is regainingly approximable.
+Then we show that the set of regainingly approximable numbers is
+closed downwards under Solovay reduction and that regainingly appro-
+ximable numbers are not Martin-L¨of random.
+Finally, we observe
+that the results from the previous section imply that there exists a
+strongly left-computable number that is not regainingly approximable;
+that there exists a strongly left-computable and regainingly approxi-
+mable number that is not computable; and that every strongly left-
+computable number can be written as the sum of two strongly left-
+computable numbers that are regainingly approximable. We conclude
+that the set of regainingly approximable numbers is not closed under
+addition.
+2. Robustness
+In this section, we first show that slight changes to the definition of re-
+gainingly approximable numbers do not lead to a different notion. The
+following lemma will be useful; note that no computability assumptions
+are made.
+Lemma 3. Let (an)n be a nondecreasing sequence of real numbers con-
+verging to some real number α such that, for infinitely many n ∈ N,
+α − an < 2−n . Then, for every unbounded function f : N → N there
+exist infinitely many m with α − af(m+1) < 2−f(m).
+Proof. By assumption, the set
+A := {n ∈ N | α − an < 2−n and f(0) ≤ n}
+is infinite. We define a function g : A → N by
+g(n) := min{m ∈ N | n < f(m + 1)},
+for n ∈ A. The function g is well-defined because the function f is
+unbounded. For every n ∈ A we have f(g(n)) ≤ n < f(g(n) + 1).
+The set g(A) := {g(n) | n ∈ A} is infinite. Let us consider a number
+m ∈ g(A), and let n ∈ A be a number with m = g(n). Then
+α − af(m+1) = α − af(g(n)+1) ≤ α − an < 2−n ≤ 2−f(g(n)) = 2−f(m). □
+There are some obvious ways to modify Definition 1 that one could
+consider. First, instead of computable nondecreasing sequences (an)n
+
+4
+of rational numbers converging to the real number α one might consider
+only computable increasing sequences.
+Secondly, one might replace
+the condition α − an < 2−n by the condition α − an < 2−f(n) where
+f : N → N is an arbitrary computable, unbounded function of one’s
+choice; or, one might ask for this to hold only for some computable,
+nondecreasing, unbounded function f : N → N, a seemingly weaker
+requirement. However, it will turn out that none of these modifications
+make any difference.
+Proposition 4. For a real number α ∈ R the following statements are
+equivalent:
+(1) α is a regainingly approximable number.
+(2) There exists a computable, increasing sequence of rational num-
+bers (an)n converging to α such that, for infinitely many n ∈ N,
+α − an < 2−n.
+(3) For every computable, unbounded function f : N → N there ex-
+ists a computable increasing sequence of rational numbers (an)n
+converging to α such that, for infinitely many n ∈ N,
+α − an < 2−f(n).
+(4) There exist a computable, nondecreasing, and unbounded func-
+tion f : N → N and a computable nondecreasing sequence of ra-
+tional numbers (an)n converging to α such that, for infinitely
+many n ∈ N, α − an < 2−f(n).
+Note that this implies that it makes no difference whether we use “<” or
+“≤” in the definition of regaining approximability. We would also like
+to point out that all implications in the following proof are uniformly
+effective.
+Proof.
+(2) ⇒ (1): Trivial.
+(3) ⇒ (2): Trivial.
+(1) ⇒ (3): Let α be a regainingly approximable real number. Let (bn)n
+be a computable nondecreasing sequence of rational numbers converg-
+ing to α with α − bn < 2−n for infinitely many n ∈ N. Let f : N → N
+be a computable, unbounded function. Then the function g : N → N
+defined by
+g(n) := 1 + n + max{f(m) | m ≤ n}
+is computable, increasing, and satisfies g(n) ≥ f(n) + 1, for all n ∈ N.
+In particular, g is unbounded. The sequence (an)n of rational numbers
+defined by
+an := bg(n+1) − 2−g(n)
+
+5
+is computable and increasing and converges to α. By Lemma 3 there
+exist infinitely many n with α − bg(n+1) < 2−g(n).
+For all of these
+numbers n we obtain
+α − an = α − bg(n+1) + 2−g(n) < 2−g(n)+1 ≤ 2−f(n).
+(1) ⇒ (4): Trivial.
+(4) ⇒ (1): Let us assume that f : N → N is a computable, nondecreas-
+ing, and unbounded function and (bn)n is a computable nondecreasing
+sequence of rational numbers converging to α such that, for infinitely
+many n ∈ N, α − bn < 2−f(n). The function g : N → N defined by
+g(0) := max{m ∈ N | f(m) = f(0)}
+and
+g(n + 1) := max{m ∈ N | f(m) = f(g(n) + 1)},
+for n ∈ N, is computable and increasing and satisfies, for all n ∈ N,
+f(g(n)) ≥ n. Furthermore, for every k ∈ N there exists exactly one
+n ∈ N with f(k) = f(g(n)), and it satisfies k ≤ g(n). The sequence
+(an)n of rational numbers defined by
+an := bg(n),
+for all n ∈ N, is computable and nondecreasing and converges to α. By
+assumption, the set
+B := {k ∈ N | α − bk < 2−f(k)}
+is infinite. Hence, the set
+A := {n ∈ N | (∃k ∈ B) f(k) = f(g(n))}
+is infinite as well. Let us consider a number n ∈ A, and let k ∈ B be a
+number with f(k) = f(g(n)). Then k ≤ g(n) and
+α − an = α − bg(n) ≤ α − bk < 2−f(k) = 2−f(g(n)) ≤ 2−n.
+□
+As the final result in this section, we show that if a left-computable
+number α is regainingly approximable then this will be apparent no
+matter which of its effective approximations we look at.
+Proposition 5. Let α be a left-computable real number, and let (an)n be
+a computable, nondecreasing sequence of rational numbers converging
+to α. Then the following conditions are equivalent.
+(1) α is a regainingly approximable number.
+(2) There exists a computable, increasing function r: N → N such
+that, for infinitely many n, α − ar(n) < 2−n.
+Note that the proof is effectively uniform in both directions.
+
+6
+Proof. (2) ⇒ (1): Let us assume that there exists a computable, in-
+creasing function r: N → N such that we have α − ar(n) < 2−n for
+infinitely many n. Then the sequence (bn)n of rational numbers de-
+fined by bn := ar(n) is computable, nondecreasing, converges to α, and
+satisfies, for infinitely many n, α − bn < 2−n. Hence, α is regainingly
+approximable.
+(1) ⇒ (2): Let us assume that α is regainingly approximable.
+By
+Proposition 4 there exists a computable, increasing sequence (bn)n
+of rational numbers converging to α such that there exist infinitely
+many n with α − bn < 2−n. We define a computable, increasing func-
+tion r: N → N by r(0) := min{m ∈ N | am ≥ b0}, and
+r(n + 1) := min{m ∈ N | m > r(n) and am ≥ bn+1},
+for n ∈ N. For all n ∈ N we have ar(n) ≥ bn. For the infinitely many
+n ∈ N with α − bn < 2−n we obtain
+α − ar(n) ≤ α − bn < 2−n.
+□
+3. Regainingly Approximable Sets of Natural Numbers
+Let us call a total function f : N → N an enumeration of a set A ⊆ N
+if the following two conditions are satisfied:
+(1) A = {n ∈ N | (∃k ∈ N) f(k) = n + 1},
+(2) for every n ∈ A there exists exactly one k ∈ N with f(k) = n+1.
+If f(k) = n+1 then we say that at stage k the function f enumerates the
+number n into A. Note that here f(k) = 0 encodes that the function f
+does not enumerate anything into A at stage k.
+It is clear that a
+set A ⊆ N is computably enumerable if and only if there exists a
+computable enumeration of A. If f : N → N is an enumeration of a
+subset of N then, for t ∈ N, we write
+Enum(f)[t] := {n ∈ N | (∃k ∈ N)(k < t and f(k) = n + 1)}.
+Definition 6. Let r: N → N be a nondecreasing, unbounded function.
+(1) We call an enumeration f : N → N of a set A ⊆ N r-good if
+there exist infinitely many n such that
+{0, . . . , n − 1} ∩ A ⊆ Enum(f)[r(n)].
+(2) We call a set A ⊆ N regainingly r-approximable if there exists
+a computable enumeration f : N → N of A that is r-good.
+Example 7. Let A ⊆ N be a decidable set.
+Then the function
+f : N → N defined by f(n) := n + 1 if n ∈ A, f(n) := 0 if n ̸∈ A,
+is a computable and idN-good enumeration of A. Hence, A is regain-
+ingly idN-approximable.
+
+7
+Definition 8. We call a set A ⊆ N regainingly approximable if there
+exists a computable, nondecreasing, unbounded function r: N → N
+such that A is regainingly r-approximable.
+The following theorem says that in this definition one can replace the
+function r by the identity idN.
+Theorem 9. For a set A ⊆ N the following two conditions are equiv-
+alent.
+(1) There exists a computable, nondecreasing, unbounded function
+r: N → N such that A is regainingly r-approximable.
+(2) A is regainingly idN-approximable.
+The proof of this theorem is not fully effectively uniform, as it contains
+a noneffective case distinction. Therefore, we first formulate a partial
+result that does have a uniformly effective proof. It implies, for exam-
+ple, that from an r-good enumeration of a set A, where r: N → N is
+an arbitrary computable, nondecreasing, unbounded function, one can
+effectively switch to a 2n-good enumeration of the same set A.
+Lemma 10. Given two nondecreasing, unbounded functions r, s: N →
+N and an r-good enumeration f : N → N of a set A, one can compute an
+(idN +s)-good enumeration g : N → N of the same set A. In particular,
+if r, s: N → N are computable, nondecreasing, unbounded functions
+and a set A ⊆ N is regainingly r-approximable then it is regainingly
+(idN + s)-approximable as well.
+Proof. Let r, s: N → N be two nondecreasing, unbounded functions,
+and let f : N → N be an r-good enumeration of a set A ⊆ N. The
+function p: N → N defined by
+p(n) := min{m ∈ N | s(m) > n},
+for all n ∈ N, is nondecreasing and can be computed from s. It satisfies,
+for all n ∈ N,
+p(s(n)) = min{m ∈ N | s(m) > s(n)} > n.
+We define a function g : N → N recursively as follows. For t ∈ N let
+M[t] := Enum(f)[r(p(t))] \ Enum(g)[t]
+and
+g(t) :=
+�
+1 + min(M[t])
+if M[t] ̸= ∅,
+0
+if M[t] = ∅,
+
+8
+The function g is an enumeration of A.
+It is clear that it can be
+computed from r, s, and f. By assumption, there are infinitely many n
+such that
+{0, . . . , n − 1} ∩ A ⊆ Enum(f)[r(n)].
+Let us consider such a number n. We claim that
+{0, . . . , n − 1} ∩ A ⊆ Enum(g)[n + s(n)].
+To see this, first note that p(s(n)) > n implies
+{0, . . . , n − 1} ∩ A ⊆ Enum(f)[r(n)] ⊆ Enum(f)[r(p(s(n)))].
+These are at most n numbers, and those among them which have not
+yet been enumerated by g in stages strictly before stage s(n) are the
+smallest elements of M[s(n)]. Thus, because no further number smaller
+than n can enter M[t] for any t > s(n), they will be enumerated by g
+in one of the n stages s(n), . . . , s(n) + n − 1. Consequently, they are
+elements of Enum(g)[n + s(n)], as was to be shown.
+□
+Proof of Theorem 9. We prove the nontrivial direction “1 ⇒ 2”. Let
+us assume that r: N → N is a computable, nondecreasing, unbounded
+function such that A is regainingly r-approximable. Let f : N → N be
+a computable r-good enumeration of A. The function s: N → N de-
+fined by s(n) := ⌊n/2⌋, for n ∈ N, is computable, nondecreasing, and
+unbounded. By applying Lemma 10 we obtain a computable (idN + s)-
+good enumeration g : N → N of A. So, g is ⌊3n/2⌋-good. We distin-
+guish two cases for A.
+First case: For almost all n ∈ N, |{0, . . . , n − 1} ∩ A| ≤ ⌊n/2⌋. In this
+case, we proceed similarly as in the proof of Lemma 10. We define a
+function h: N → N recursively as follows. For t ∈ N let
+M[t] := Enum(g)[3t] \ Enum(h)[t]
+and
+h(t) :=
+�
+1 + min(M[t])
+if M[t] ̸= ∅,
+0
+if M[t] = ∅,
+The function h is a computable enumeration of A. Let N ∈ N be a
+number such that, for all n ≥ N, |{0, . . . , n − 1} ∩ A| ≤ ⌊n/2⌋. There
+are infinitely many n ≥ N with
+{0, . . . , n − 1} ∩ A ⊆ Enum(g)[⌊3n/2⌋].
+Let us consider such a number n. We claim that
+{0, . . . , n − 1} ∩ A ⊆ Enum(h)[n].
+
+9
+Indeed, there are at most ⌊n/2⌋ numbers in {0, . . . , n − 1} ∩ A, and all
+these numbers are elements of the set
+Enum(g)[⌊3n/2⌋] ⊆ Enum(g)[3 · ⌈n/2⌉].
+Furthermore, all those among these at most ⌊n/2⌋ numbers, that have
+not yet been enumerated by h in stages strictly before stage ⌈n/2⌉, are
+the smallest elements of M[⌈n/2⌉] and will be enumerated by h in one
+of the ⌊n/2⌋ stages ⌈n/2⌉, . . . , n−1 (because no further number smaller
+than n can enter M[t] for any t > ⌈n/2⌉). Thus, they are elements of
+Enum(h)[n]. That was to be shown.
+Second case: There exist infinitely many n ∈ N with
+|{0, . . . , n − 1} ∩ A| > ⌊n/2⌋.
+In this case we define two increasing and computable sequences (ni)i
+and (ti)i of natural numbers as follows. First we compute the smallest
+natural number t0 such that there exists a natural number n > 0 with
+⌊3n/2⌋ ≤ t0 and |{0, . . . , n − 1} ∩ Enum(g)[t0]| > ⌊n/2⌋.
+Then we let n0 be the smallest number n > 0 with this property. Let us
+consider i > 0. Once ni−1 and ti−1 have been determined, we compute
+the smallest natural number ti > ti−1 such that there exists a natural
+number n with
+2ni−1 ≤ n and ⌊3n/2⌋ ≤ ti and |{0, . . . , n − 1} ∩ Enum(g)[ti]| > ⌊n/2⌋.
+Then we let ni be the smallest number n with this property.
+Next we recursively define a function h: N → N which will be an
+enumeration of the infinite set A with h(t) ̸= 0 for all t ∈ N. For
+any i ∈ N, let mi be the number of elements of the following set
+Mi := {0, . . . , ni − 1} ∩ Enum(g)[ti] \ Enum(h)
+��
+j<i
+mj
+�
+.
+Then let k0, . . . , kmi−1 be the elements of this set in increasing order
+and, for t with 0 ≤ t ≤ mi − 1, define
+h
+�
+t +
+�
+j<i
+mj
+�
+:= 1 + kt.
+This function h is a computable enumeration of A. It is clear that
+⌊n0/2⌋ < m0 ≤ n0 and, for i > 0,
+0 ≤ ni−1 −
+�
+j<i
+mj ≤ ⌊ni/2⌋ −
+�
+j<i
+mj < mi ≤ ni −
+�
+j<i
+mj.
+
+10
+There are infinitely many n > n0 with
+{0, . . . , n − 1} ∩ A ⊆ Enum(g)[⌊3n/2⌋].
+Let us consider such a number n. We claim that
+{0, . . . , n − 1} ∩ A ⊆ Enum(h)[n].
+Let i := min{j ∈ N | n ≤ nj}. Then i > 0 and ni−1 < n ≤ ni. In the
+first �
+j<i mj stages exactly �
+j<i mj numbers smaller than ni−1 have
+been enumerated by h. During the next mi stages all numbers in the
+set Mi will be enumerated by h in increasing order. Note that
+{0, . . . , n − 1} ∩ A
+⊆
+{0, . . . , n − 1} ∩ Enum(g)[⌊3n/2⌋]
+⊆
+{0, . . . , ni − 1} ∩ Enum(g)[ti].
+Hence, the numbers in {0, . . . , n−1}∩A that have not been enumerated
+by h before stage �
+j<i mj are elements of Mi. In fact, they are the
+smallest elements of Mi. So, they will be enumerated by h in the next
+stages, starting with stage �
+j<i mj. As there are at most n−�
+j<i mj
+such numbers, the function h enumerates all of them before stage n.
+That was to be shown.
+□
+We also observe in analogy to Proposition 5 that if a set A is regainingly
+approximable then this will be apparent no matter which of its effective
+enumerations we look at.
+Lemma 11. Given two enumerations f : N → N and g : N → N of a
+set A ⊆ N and a nondecreasing, unbounded function r: N → N such
+that f is r-good, one can compute an increasing function s: N → N
+such that g is s-good.
+Proof. The function s: N → N defined by
+s(0) := 0
+s(n + 1) := min
+�
+m ∈ N
+����
+Enum(f)[r(n+1)] ⊆ Enum(g)[m]
+and m > s(n)
+�
+,
+for all n ∈ N, can be computed from f, g, r and has the desired prop-
+erties.
+□
+Corollary 12. Let A ⊆ N be a c.e. set, and let g : N → N be any com-
+putable enumeration of A. Then the following conditions are equivalent.
+(1) A is regainingly approximable.
+(2) There exists a computable, increasing function s: N → N such
+that g is s-good.
+Theorem 13. There exists a c.e. set A ⊆ N that is not regainingly
+approximable.
+
+11
+Proof. We use the Cantor pairing function ⟨·, ·⟩: N2 → N defined by
+⟨m, n⟩ := 1
+2 (m + n) (m + n + 1) + n,
+for all m, n ∈ N, and let π1: N → N and π2 : N → N denote the
+two components of its inverse function, that is, ⟨π1(n), π2(n)⟩ = n for
+all n ∈ N. Let ϕ0, ϕ1, ϕ2, . . . be a standard enumeration of all possibly
+partial computable functions with domain and range in N. As usual,
+we write ϕe(n)[t] ↓ to express that the e-th Turing machine (which
+computes ϕe) stops after at most t steps on input n.
+We shall construct a computable enumeration g : N → N of a set A ⊆ N
+such that the following requirements (Re) will be satisfied for all e ∈ N:
+(Re): if ϕe is total and increasing then
+(∃ne ∈ N)(∀n > ne)({0, . . . , n − 1} ∩ A ̸⊆ Enum(g)[ϕe(n)]).
+According to Corollary 12 this is sufficient.
+We construct g in stages; in stage t we proceed as follows: Define
+e := π1(π1(t)) and k := π2(π1(t)), hence, ⟨e, k⟩ = π1(t). Check whether
+the following conditions are satisfied:
+(∀n ≤ ⟨e, k + 1⟩) ϕe(n)[t] ↓
+and
+(∀n < ⟨e, k + 1⟩) ϕe(n) < ϕe(n + 1)
+and
+t ≥ ϕe(⟨e, k + 1⟩)
+and
+⟨e, k⟩ ̸∈ Enum(g)[t].
+If they are, set g(t) := 1 + ⟨e, k⟩, otherwise g(t) := 0.
+We come to the verification. It is clear that the function g is com-
+putable and an enumeration of some c.e. set A ⊆ N.
+We wish to
+show that the requirements Re are satisfied for all e ∈ N.
+Let us
+consider a number e such that ϕe is a total and increasing function
+as well as a number n > ⟨e, 0⟩.
+There exists a unique k ∈ N with
+⟨e, k⟩ < n ≤ ⟨e, k + 1⟩. The function g enumerates the number ⟨e, k⟩
+into A in some uniquely determined stage t, i.e., there exists exactly
+one number t with g(t) = 1 + ⟨e, k⟩. Then
+⟨e, k⟩ ∈ Enum(g)[t + 1] \ Enum(g)[t].
+Since n ≤ ⟨e, k + 1⟩, we have ϕe(n) ≤ ϕe(⟨e, k + 1⟩) ≤ t, and therefore
+⟨e, k⟩ ̸∈ Enum(g)[t] ⊇ Enum(g)[ϕe(n)].
+Thus ⟨e, k⟩ ∈ {0, . . . , n − 1} ∩ A witnesses that Re is satisfied with
+ne = ⟨e, 0⟩.
+□
+
+12
+The following theorem states that any c.e. set C ⊆ N can be split ef-
+fectively into two disjoint c.e. sets A and B that are regainingly appro-
+ximable.
+Theorem 14. Given an enumeration fC : N → N of a set C ⊆ N one
+can compute enumerations fA : N → N of a set A ⊆ N and fB : N → N
+of a set B ⊆ N such that
+(1) C is the disjoint union of A and B, and
+(2) there exist infinitely many t with
+A ∩ {0, . . . , t − 1} ⊆ Enum(fA)[t]
+and infinitely many t with
+B ∩ {0, . . . , t − 1} ⊆ Enum(fB)[t].
+In particular, for any c.e. set C ⊆ N there exist two disjoint, regainingly
+approximable sets A, B ⊆ N with C = A ∪ B.
+Proof. Let an enumeration fC of a set C ⊆ N be given. The algo-
+rithm that defines the desired enumerations fA and fB will work in
+stages −1, 0, 1, 2, . . . At the same time, we will also define a function
+s: N × (N ∪ {−1}) → N and write si[t] for s(i, t).
+At stage −1 we define si[−1] for all i ∈ N by si[−1] := i.
+At stage t with t ∈ N we proceed as follows:
+If fC(t) = 0 (recall that this means that fC does not enumerate any-
+thing into C at stage t) then we set fA(t) := 0 and fB(t) := 0. Fur-
+thermore, we set si[t] := si[t − 1] for all i ∈ N.
+If fC(t) > 0 then the number n := fC(t) − 1 enumerated by fC into C
+at stage t will be enumerated either into the set A or into the set B,
+as follows. If the number
+kt := min{j ∈ N | sj[t − 1] > n}
+is even then we set fA(t) := 0 and fB(t) := n + 1 (which means that
+n is enumerated into B); if kt is odd then we set fA(t) := n + 1 and
+fB(t) := 0 (which means that n is enumerated into A). Furthermore,
+we define si[t] for all i ∈ N by
+si[t] :=
+�
+si[t − 1]
+if i ≤ kt,
+si[t − 1] + t
+if kt < i.
+This ends the description of stage t and of the algorithm; we proceed
+with the verification.
+Claim 1:
+For every t ∈ N ∪ {−1}, the sequence (si[t])i is strictly
+increasing.
+
+13
+Proof: By induction over t. It is clear for t = −1. Let us consider some
+t ∈ N and assume that the sequence (si[t − 1])i is strictly increasing.
+If fC(t) = 0 then the sequence (si[t])i is identical to the sequence
+(si[t − 1])i, hence, it is strictly increasing well. Let us assume that
+fC(t) > 0; then kt is defined by induction hypothesis and we observe
+that the sequence (si[t])i is strictly increasing:
+• For any i < j ≤ kt we have
+si[t] = si[t − 1] < sj[t − 1] = sj[t].
+• For any i ≤ kt < j we have
+si[t] = si[t − 1] < sj[t − 1] ≤ sj[t − 1] + t = sj[t].
+• For any kt < i < j we have
+si[t] = si[t − 1] + t < sj[t − 1] + t = sj[t].
+This proves Claim 1.
+By Claim 1, for every t ∈ N with fC(t) > 0, the number kt is well
+defined. Now, it is clear that the functions fA and fB defined by the
+algorithm are enumerations of two disjoint sets A, B ⊆ N whose union
+is the set C. We still need to prove the second condition stated in
+Theorem 14.
+Claim 2: For every i, the sequence (si[t])t≥−1 is nondecreasing and
+eventually constant.
+Proof: It is clear that, for every i, the sequence (si[t])t is nondecreasing.
+We show by induction over i that the sequence (si[t])t is eventually
+constant. For all t ∈ N we have 0 ≤ kt, hence, s0[t] = s0[t − 1] =
+s0[−1] = 0. We consider any number i > 0. By induction hypothesis
+there exists a number t1 such that, for all j < i and for all t ≥ t1,
+sj[t] = sj[t1]. Let t2 be large enough so that t2 > t1 and
+C ∩ {0, . . . , si−1[t1] − 1} ⊆ Enum(fC)[t2]
+(meaning that fC does not enumerate any number smaller than si−1[t1]
+into the set C in any stage t ≥ t2).
+Then, for every t ≥ t2 with
+fC(t) > 0, we must have i ≤ kt and consequently si[t] = si[t − 1]. By
+induction we obtain si[t] = si[t2 − 1], for all t ≥ t2 − 1. Thus, (si[t])t is
+eventually constant, and Claim 2 is proven.
+Let the sequence (Si)i be defined by Si := limt→∞ si[t]. Due to Claim 1,
+(Si)i is strictly increasing.
+Claim 3: For every i ∈ N and every t ≥ Si, si[t] = Si.
+
+14
+Proof: If this were not true then there would be some t > Si with
+si[t] ̸= si[t − 1], hence, with Si ≥ si[t] = si[t − 1] + t ≥ t > Si, a
+contradiction.
+Claim 4: For every even i, A ∩ {0, . . . , Si − 1} ⊆ Enum(fA)[Si].
+Proof: Consider an even number i as well as some n ∈ A \ Enum(fA)[Si].
+It is sufficient to show that n ≥ Si. Let t be the unique number with n ∈
+Enum(fA)[t]\Enum(fA)[t−1]. Then t > Si and n+1 = fA(t) = fC(t).
+By construction, the number kt must be odd. Hence kt ̸= i. If kt were
+smaller than i then we would obtain si[t] = si[t−1]+t > si[t−1] = Si in
+contradiction to Claim 3. We conclude i < kt. This implies si[t−1] ≤ n
+by the definition of kt.
+As t > Si, using Claim 3 again we obtain
+Si = si[t − 1] ≤ n, which proves Claim 4.
+Claim 5: For every odd i, B ∩ {0, . . . , Si − 1} ⊆ Enum(fB)[Si].
+Proof: The proof is symmetric to that of Claim 4; it is enough to
+interchange the words “even” and “odd” and to replace “A” by “B”.
+□
+Corollary 15. There exists a regainingly approximable set A ⊆ N that
+is not decidable.
+Proof. Let C ⊆ N be a c.e. set that is not decidable.
+By Theo-
+rem 14 there exist two disjoint regainingly approximable sets A, B with
+C = A ∪ B. Not both of them can be decidable.
+□
+The set of all regainingly approximable sets is not closed under union,
+according to Theorems 13 and 14. The following limited closure prop-
+erties do hold, however, and will be useful in the proof of the next
+theorem.
+Lemma 16.
+(1) The union of a regainingly approximable set and a decidable set
+is regainingly approximable.
+(2) If A is a regainingly approximable set and f : N → N is a com-
+putable, nondecreasing function, then the set f(A) := {n ∈ N |
+(∃k ∈ A) n = f(k)} is regainingly approximable.
+Proof. Let A ⊆ N be a regainingly approximable set. By Lemma 10
+there exists a computable 2n-good enumeration g : N → N of A.
+For the first assertion, let B ⊆ N be a decidable set. Then the function
+h: N → N defined by h(2n) := g(n) and h(2n + 1) := n + 1 if n ∈ B,
+h(2n+1) := 0 if n ̸∈ B, is a computable and (4n−1)-good enumeration
+of A ∪ B.
+
+15
+For the second assertion, let f : N → N be a computable, nondecreasing
+function. Then the function h: N → N defined by
+h(n) :=
+�
+0
+if g(n) = 0,
+1 + f(g(n) − 1)
+if g(n) > 0,
+for n ∈ N, is a computable enumeration of f(A).
+If f is bounded
+then the set f(A) is finite, and the function h is trivially an idN-good
+enumeration of f(A). Let us assume that f is unbounded. Then the
+function r: N → N defined by
+r(n) := max{m ∈ N | f(m) ≤ n},
+for n ∈ N, is computable, nondecreasing, and unbounded. We claim
+that h is a (2r(n))-good enumeration of f(A). By assumption, the set
+B := {n ∈ N | {0, . . . , n − 1} ∩ A ⊆ Enum(g)[2n]}
+is infinite. So is the set C := f(B). Let us consider a number n ∈ C
+and a number m ∈ B with f(m) = n. Then m ≤ r(n). We obtain
+{0, . . . , n − 1} ∩ f(A)
+=
+{0, . . . , f(m) − 1} ∩ f(A)
+⊆
+{0, . . . , f(m − 1)} ∩ f(A)
+=
+f({0, . . . , m − 1} ∩ A)
+⊆
+f(Enum(g)[2m])
+=
+Enum(h)[2m]
+⊆
+Enum(h)[2r(n)].
+□
+Theorem 17. There exist two regainingly approximable sets A, B ⊆ N
+whose intersection A ∩ B is not regainingly approximable.
+Proof. For natural numbers a, b and a set D ⊆ N we write (a · D + b)
+for the set
+(a · D + b) := {n ∈ N | (∃d ∈ D) n = a · d + b}
+and (a · D) for the set (a · D + 0).
+By Theorem 13 there exists a
+c.e. set �C ⊆ N that is not regainingly approximable. By Theorem 14
+there exist two disjoint, regainingly approximable sets �A, �B ⊆ N with
+�A ∪ �B = �C. By Lemma 16 the sets
+A := (2 · �A) ∪ (2 · N + 1) and B := (2 · �B + 1) ∪ (2 · N)
+are regainingly approximable. We claim that their intersection A∩B =
+(2 · �A) ∪ (2 · �B + 1) is not regainingly approximable. Let the function
+g : N → N be defined by g(n) = ⌊n/2⌋ for all n ∈ N. We observe
+�C = g(A ∩ B). Thus, if A ∩ B were a regainingly approximable set,
+then so would be �C according to Lemma 16(2), a contradiction.
+□
+
+16
+To summarize our results, every decidable set is regainingly approxi-
+mable but the converse does not hold (by Example 7 and Corollary 15);
+every regainingly approximable set is computably enumerable but the
+converse does not hold (by Theorem 13); and the set of regainingly
+approximable sets is neither closed under union nor closed under inter-
+section (by Theorems 13, 14 and 17).
+4. Strongly Left-computable Numbers and Regainingly
+Approximable Numbers
+Lemma 18. For a c.e. set A ⊆ N the following two statements are
+equivalent.
+(1) The set A is regainingly approximable.
+(2) The real number 2−A is regainingly approximable,
+Proof.
+(2) ⇒ (1): Let A ⊆ N be a c.e. set such that the number 2−A is regain-
+ingly approximable. Let f : N → N be an arbitrary computable enu-
+meration of A. Then the sequence (an)n defined by an := 2−Enum(f)[n],
+for n ∈ N, is a computable nondecreasing sequence of rational numbers
+converging to 2−A. By Proposition 5 there exists a computable, increas-
+ing function r: N → N such that, for infinitely many n, 2−A − ar(n) <
+2−n. We obtain {0, . . . , n−1}∩A ⊆ Enum(f)[r(n)] for infinitely many
+n. Hence, A is regainingly r-approximable.
+(1) ⇒ (2): Let r: N → N be a computable, nondecreasing, unbounded
+function such that A is regainingly r-approximable. Let f : N → N be
+a computable r-good enumeration of A. Then by
+an := 2−Enum(f)[r(n+1)]
+a computable, nondecreasing sequence (an)n of rational numbers con-
+verging to 2−A is defined. For infinitely many n we have
+{0, . . . , n} ∩ A ⊆ Enum(f)[r(n + 1)],
+hence, 2−A − an ≤ 2−(n+1) < 2−n. This shows that 2−A is regainingly
+approximable.
+□
+Corollary 19.
+(1) There exists a strongly left-computable number that is not re-
+gainingly approximable.
+(2) There exists a strongly left-computable number that is regain-
+ingly approximable but not computable.
+Proof. The first assertion follows from Theorem 13 and Lemma 18.
+The second assertion follows from Corollary 15 and Lemma 18 and
+
+17
+from the well-known fact that, for any subset A ⊆ N, the number 2−A
+is computable if and only if the set A is decidable.
+□
+The regainingly approximable numbers are closed downwards with re-
+spect to Solovay reducibility. Let ≤S denote Solovay reducibility [5]
+between left-computable real numbers.
+Proposition 20. Let β be a regainingly approximable number, and let
+α be a left-computable number with α ≤S β. Then α is regainingly
+approximable as well.
+Proof. Let f : {q ∈ Q | q < β} → Q be a computable function and
+c ∈ N be a number such that, for all q ∈ {q ∈ Q | q < β}, f(q) < α and
+α − f(q) < 2c · (β − q).
+By Proposition 4 there exists a computable
+and increasing sequence (bn)n of rational numbers converging to β such
+that β − bn < 2−n−c for infinitely many n ∈ N. The sequence (an)n
+defined by
+an := max{f(bi) | 0 ≤ i ≤ n}
+is a nondecreasing, computable sequence of rational numbers converg-
+ing to α. For the infinitely many n with β − bn < 2−n−c we obtain
+α − an ≤ α − f(bn) < 2c · (β − bn) < 2−n.
+□
+Corollary 21. Regainingly approximable numbers are not Martin-L¨of
+random.
+Proof. We give two proofs. First, Kuˇcera and Slaman [3] showed that
+every left-computable number is below any Martin-L¨of random left-
+computable number with regards to Solovay reducibility. Thus, if a
+regainingly approximable number were Martin-L¨of random, then all
+left-computable numbers would be regainingly approximable according
+to Proposition 20. This contradicts Corollary 19(1).
+We also give an alternative direct proof: Let α be regainingly appro-
+ximable and let (an)n be a computable, nondecreasing sequence of ra-
+tional numbers converging to α such that α − an < 2−n for infinitely
+many n ∈ N. For every n ∈ N let Un be the interval (an, an + 2−n).
+Then (Un)n is a computable sequence of open intervals with rational
+endpoints such that �
+n∈N λ(Un) = 2 < ∞ where λ is the Lebesgue
+measure on R. Since α ∈ Un for infinitely many n, (Un)n is a Solovay
+test witnessing that α is not Martin-L¨of random.
+□
+Corollary 22.
+(1) If the sum of two left-computable real numbers is regainingly
+approximable, then both of them are regainingly approximable.
+
+18
+(2) The sum of a regainingly approximable number and a com-
+putable number is again a regainingly approximable number.
+Proof. The first assertion follows from Proposition 20 and from the fact
+that for any two left-computable numbers α, β one has α ≤S α+β and
+β ≤S α + β. Since adding a computable number to a left-computable
+number does not change its Solovay degree, the second assertion follows
+from Proposition 20 as well.
+□
+Corollary 23. Every strongly left-computable number can be written
+as the sum of two strongly left-computable numbers that are regainingly
+approximable.
+Proof. By Theorem 14 and Lemma 18.
+□
+Corollary 24. There exist two strongly left-computable and regainingly
+approximable numbers whose sum is not regainingly approximable.
+Proof. According to Corollary 19(1), there exists a strongly left-computable
+number γ that is not regainingly approximable. According to Corol-
+lary 23, there exist two strongly left-computable and regainingly appro-
+ximable numbers α, β with α + β = γ. They witness the truth of the
+assertion.
+□
+Corollary 23 raises the question whether every left-computable number
+can be written as the sum of two regainingly approximable numbers.
+The answer is no. This follows from Proposition 21, from the fact that
+there exist Martin-L¨of random left-computable numbers, and from the
+result of Downey, Hirschfeldt, Nies [2, Corollary 3.6] that the sum of
+two left-computable numbers that are not Martin-L¨of random is again
+not Martin-L¨of random.
+References
+[1] R. G. Downey and D. R. Hirschfeldt. Algorithmic randomness and complexity.
+Theory and Applications of Computability. Springer, New York, 2010.
+[2] R. G. Downey, D. R. Hirschfeldt, and A. Nies. Randomness, computability, and
+density. SIAM J. Comput., 31(4):1169–1183, 2002.
+[3] A. Kuˇcera and T. A. Slaman. Randomness and recursive enumerability. SIAM
+J. Comput., 31(1):199–211, 2001.
+[4] A. Nies. Computability and randomness, volume 51 of Oxford Logic Guides.
+Oxford University Press, Oxford, 2009.
+[5] R. M. Solovay. Draft of a paper (or series of papers) on Chaitin’s work. Unpub-
+lished notes, 1975.
+[6] K. Weihrauch. Computable analysis. Texts in Theoretical Computer Science.
+An EATCS Series. Springer-Verlag, Berlin, 2000. An introduction.
+

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+arXiv:2301.04629v1  [math.CA]  11 Jan 2023
+Prepared for submission to JHEP
+OU-HET-1167
+On some identities for confluent hypergeometric
+functions and Bessel functions
+Yoshitaka Okuyamaa,b
+aDepartment of Physics, Osaka University,
+Machikaneyama-Cho 1-1, Toyonaka 560-0043, Japan
+bDepartment of Physics, Faculty of Science, The University of Tokyo,
+Bunkyo-Ku, Tokyo 113-0033, Japan
+Abstract: We find a new integral representation of the Whittaker function of the first kind.
+We also show relevant summation formulas for Kummer’s confluent hypergeometric functions
+and Bessel functions.
+
+Contents
+1
+Introduction and summary
+1
+2
+Integral representation of the Whittaker function of the first kind
+2
+3
+Summation formula for confluent hypergeomertric functions
+3
+4
+Summation formula for Bessel functions
+5
+1
+Introduction and summary
+This paper concerns several identities related to confluent hypergeometric functions. We first
+show an integral representation of the Whittaker function:
+Mκ,µ(z) =
+√π Γ(2µ + 1)
+2µ Γ
+�
+µ+κ+1/2
+2
+�
+Γ
+�
+µ−κ+1/2
+2
+�
+× √z
+� 1
+0
+dξ ξ
+−κ+1/2
+2
+−1(1 − ξ)
+κ+1/2
+2
+−1 e(ξ−1/2)z Jµ
+��
+ξ(1 − ξ)z
+�
+,
+(1.1)
+for Re (µ ± κ + 1/2) > 0. Here, the Bessel function Jν(z) and the Whittaker function of the
+first kind Mκ,µ(z) are defined by [1]:
+Jν(z) =
+∞
+�
+n=0
+(−1)n
+Γ(ν + n + 1) n!
+�z
+2
+�ν+2n
+,
+(1.2)
+and
+Mκ,µ(z) = e−z/2 zµ+1/2
+∞
+�
+n=0
+(µ − κ + 1/2)n
+(2µ + 1)n n!
+zn .
+(1.3)
+It turns out that the identity (1.1) implies the following summation formula:
+M(2a; 2b; z) =
+∞
+�
+n=0
+(a)n(b)n(b − a)n
+(b)2n(2b)2n n!
+(−z2)n M(a + n; b + 2n; z) ,
+(1.4)
+where we use the standard definition of Kummer’s confluent hypergeometric function M(a; b; z):
+M(a; b; z) =
+∞
+�
+n=0
+(a)n
+(b)n n! zn .
+(1.5)
+– 1 –
+
+By specifying indices of (1.4) in a particular manner, one finds the following summation
+formula for Bessel-J:
+J2ν+1/2(z) =
+Γ(ν + 1)
+Γ(2ν + 3/2)
+∞
+�
+n=0
+(ν + 1/2)n
+(2ν + 3/2)n n!
+�z
+2
+�ν+1/2+n
+Jν+n(z) .
+(1.6)
+To the best of our effort, we could not find either of these three identities (1.4), (1.1) and
+(1.6) anywhere in literature.
+2
+Integral representation of the Whittaker function of the first kind
+We here show an integral representation of the Whittaker function of the first kind (1.1).
+Recall that the Whittaker function of the first kind is a solution to the Whittaker differential
+equation
+d2y
+dz2 +
+�
+−1
+4 + κ
+z + 1/4 − µ2
+z2
+�
+y = 0 ,
+(2.1)
+subject to the boundary condition;
+Mκ,µ(z) −−−→
+z→0 zµ+1/2 .
+(2.2)
+All we need to do is to check that the RHS of (1.1) satisfies the differential equation (2.1)
+and the boundary condition (2.2).
+Boundary condition.
+In taking z → 0 limit, the RHS of (1.1) becomes:
+(RHS of (1.1))
+−−−→
+z→0
+√π zµ+1/2 Γ(2µ + 1)
+22µ Γ
+�
+µ+κ+1/2
+2
+�
+Γ
+�
+µ−κ+1/2
+2
+�
+Γ(ν + 1)
+×
+� 1
+0
+dξ ξ
+−κ+µ+1/2
+2
+−1(1 − ξ)
+κ+µ+1/2
+2
+−1
+= zµ+1/2 .
+(2.3)
+We have used the series expansion of the Bessel function (1.2) in the second line. In going to
+the last line, we used the following two Gamma function identities;
+� 1
+0
+dt tx−1(1 − t)y−1 = Γ(x) Γ(y)
+Γ(x + y) ,
+(2.4)
+Γ(2z) = 22z−1 Γ(z) Γ(z + 1/2)
+√π
+.
+(2.5)
+– 2 –
+
+Differential equation.
+Let us check that the left of the RHS of (1.1) satisfies the Whittaker
+differential equation (2.1). After some manipulations, we see
+� d2
+dz2 +
+�
+−1
+4 + κ
+z + 1/4 − µ2
+z2
+��
+(RHS of (1.1))
+∝
+� 1
+0
+dξ ξ
+−κ+1/2
+2
+−1(1 − ξ)
+κ+1/2
+2
+−1 e(ξ−1/2)z
+×
+� d2
+dz2 + 1
+z
+d
+dz + ξ(1 − ξ) − µ2
+z2 + (2ξ − 1) d
+dz − 2ξ(1 − ξ) + κ + ξ − 1/2
+z
+�
+Jµ
+��
+ξ(1 − ξ)z
+�
+.
+(2.6)
+Notice that the first four terms in the last line add up to zero thanks to the Bessel differential
+equation:
+� d2
+dz2 + 1
+z
+d
+dz + 1 − ν2
+z2
+�
+Jν(z) = 0 .
+(2.7)
+We now focus on the fifth term in (2.6). Using the identity that follows from (1.2)
+(2ξ − 1) d
+dz Jµ
+��
+ξ(1 − ξ)z
+�
+= −2ξ(1 − ξ)
+z
+d
+dξ Jµ
+��
+ξ(1 − ξ)z
+�
+,
+(2.8)
+we perform integration by parts with respect to ξ:
+(The fifth term in (2.6))
+= −2
+z
+� 1
+0
+dξ ξ
+−κ+1/2
+2
+−1+1(1 − ξ)
+κ+1/2
+2
+−1+1 e(ξ−1/2)z
+� d
+dξ Jµ
+��
+ξ(1 − ξ)z
+��
+= 2
+z
+� 1
+0
+dξ Jµ
+��
+ξ(1 − ξ)z
+� d
+dξ
+�
+ξ
+−κ+1/2
+2
+(1 − ξ)
+κ+1/2
+2
+e(ξ−1/2)z�
+=
+� 1
+0
+dξ ξ
+−κ+1/2
+2
+−1(1 − ξ)
+κ+1/2
+2
+−1 e(ξ−1/2)z
+�
+2ξ(1 − ξ) − κ + ξ − 1/2
+z
+�
+Jµ
+��
+ξ(1 − ξ)z
+�
+.
+(2.9)
+Here, we assumed Re (µ±κ+1/2) > 0 to drop off the surface terms. Plugging this into (2.6),
+one can readily see that all terms cancel out and conclude that the RHS of (1.1) satisfies the
+Whittaker differential equation (2.1).
+3
+Summation formula for confluent hypergeomertric functions
+We then derive the summation formula for confluent hypergeometric functions (1.3).
+Assuming Re (µ ± κ + 1/2), Re (µ) > 0, we start by applying the Mellin-Barnes-type
+representation of Bessel-J (3.1) to (1.1) [2],
+Jν(z) =
+� i ∞
+−i ∞
+dt
+2πi
+Γ(−t)
+Γ(ν + t + 1)
+�z
+2
+�ν+2t
+Re (ν) > 0 .
+(3.1)
+– 3 –
+
+After the change of the order of integration, we find:
+(Second line of (1.1))
+= √z e−z/2
+� i ∞
+−i ∞
+dt
+2πi
+Γ(−t)
+Γ(µ + t + 1)
+�z
+2
+�µ+2t � 1
+0
+dξ ξ
+µ−κ+1/2
+2
++t−1(1 − ξ)
+µ+κ+1/2
+2
++t−1 ezξ .
+(3.2)
+Notice that the ξ-integral is nothing but the integral representation of the confluent hyper-
+geometric equation:
+M(a; b; z) =
+Γ(b)
+Γ(a)Γ(b − a)
+� 1
+0
+dt ezt ta−1(1 − t)b−a−1 .
+(3.3)
+Using (2.5), we have:
+(Second line of (1.1))
+= 2µ zµ+1/2
+√π
+e−z/2
+� i ∞
+−i ∞
+dt
+2πi
+Γ
+�
+µ−κ+1/2
+2
++ t
+�
+Γ
+�
+µ+κ+1/2
+2
++ t
+�
+Γ(µ + 1/2 + t) Γ(−t)
+Γ(2µ + 1 + 2t)Γ(µ + 1/2 + 2t)
+× z2t M
+�µ − κ + 1/2
+2
++ t, µ + 1
+2 + 2t, z
+�
+.
+(3.4)
+Deforming the integration contour to the right and picking up the residues coming from
+Γ(−t), we see:
+(Second line of (1.1)) =
+2µ Γ
+�
+µ−κ+1/2
+2
+�
+Γ
+�
+µ+κ+1/2
+2
+�
+√π Γ(2µ + 1)
+zµ+1/2 e−z/2
+×
+∞
+�
+n=0
+�
+µ−κ+1/2
+2
+�
+n
+�
+µ+κ+1/2
+2
+�
+n (µ + 1/2)n
+(µ + 1/2)2n(2µ + 1)2n n!
+(−z2)n M
+�µ − κ + 1/2
+2
++ n, µ + 1
+2 + 2n, z
+�
+.
+(3.5)
+Substituting this for (1.1), we arrive at (1.4) for Re (a), Re (b − a), Re (b + 1/2) > 0.
+We can verify that the identity (1.4) holds for any a, b ∈ C, by expanding in powers of z
+comparing both sides order by order using the following formula:
+(2a)k
+(2b)k
+=
+k!
+(b)k
+⌊k/2⌋
+�
+n=0
+(a)k−n (b)n (b − a)n
+(2b)2n n! (k − 2n)! (−1)n
+k ∈ Z≥0 .
+(3.6)
+where ⌊x⌋ is the floor function that returns the largest integer less than or equal to x. This
+formula (3.6) can be proven as follows. Firstly, short calculation leads:1
+(RHS of (3.6)) = (a)k
+(b)k
+3F2
+�
+b − a, − k−1
+2 , − k
+2
+b + 1
+2, 1 − a − k ; 1
+�
+,
+(3.7)
+1Use Euler reflection formula Γ(z)Γ(1 − z) = π/ sin πz and Legendre duplication formula (2.5).
+– 4 –
+
+with 3F2 being a generalized hypergeometric function defined by:
+3F2
+�a, b, c
+d, e ; z
+�
+=
+∞
+�
+n=0
+(a)n(b)n(c)n
+(d)n(e)n n! zn .
+(3.8)
+With the help of Saalsch¨utz’s theorem that asserts [3, equation (1), chapter II]:
+3F2
+�
+a, b, −n
+c, 1 + a + b − c − n; 1
+�
+= (c − a)n(c − b)n
+(c)n(c − a − b)n
+n ∈ Z≥0 ,
+(3.9)
+we can verify that (3.6) is an identity.
+4
+Summation formula for Bessel functions
+Lastly, we show the summation formula for Bessel functions (1.6). It follows from the relation
+between Kummer’s confluent hypergeometric functions and Bessel functions:
+Jν(z) =
+e∓iz
+Γ(ν + 1)
+�z
+2
+�ν
+M(ν + 1/2; 2ν + 1, ±2iz) ,
+(4.1)
+we have:
+(LHS of (1.6)) =
+e∓iz
+Γ(2ν + 3/2)
+�z
+2
+�2ν+1/2
+M(2ν + 1; 4ν + 2; ±2iz) .
+(4.2)
+Plugging the summation formula (1.4) into this and using (4.1) again, one finds:
+(LHS of (1.6))
+=
+e∓iz
+Γ(2ν + 3/2)
+�z
+2
+�2ν+1/2
+∞
+�
+n=0
+[(ν + 1/2)n]2(2ν + 1)n
+(2ν + 1)2n(4ν + 2)2n n!
+× (2z)2n M(ν + 1/2 + n; 2ν + 1 + 2n; ±2iz)
+=
+Γ(ν + 1)
+Γ(2ν + 3/2)
+∞
+�
+n=0
+24n [(ν + 1/2)n]2(ν + 1)n(2ν + 1)n
+(2ν + 1)2n(4ν + 2)2n n!
+�z
+2
+�ν+1/2+n
+Jν+n(z)
+= (RHS of (1.6)) .
+(4.3)
+We need to use (2.5) several times to reach the last line.
+When ν = 0, the LHS of (1.6) reduces to a triogeometric function divided by √z:
+J1/2(z) =
+�
+2
+πz sin z ,
+(4.4)
+whereas the RHS remains the summation of Bessel-J’s. This reproduces the known expansion
+of sin in terms of Bessel functions found in [4, equation (9.4.2.19)]:
+sin z
+z
+=
+∞
+�
+n=0
+1
+(2n + 1) n!
+�z
+2
+�n
+Jn(z) .
+(4.5)
+– 5 –
+
+Acknowledgments
+We are grateful to T. Nishioka for the valuable discussions. The work of Y. O. was supported
+by Forefront Physics and Mathematics Program to Drive Transformation (FoPM), a World-
+leading Innovative Graduate Study (WINGS) Program, the University of Tokyo. The work
+of Y. O. was also supported by JSPS fellowship for young students No. 21J20750, MEXT, and
+by JSR fellowship, the University of Tokyo.
+References
+[1] E. Whittaker and G. Watson, A Course of Modern Analysis, A Course of Modern Analysis: An
+Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an
+Account of the Principal Transcendental Functions. Cambridge University Press, 1996.
+[2] G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library.
+Cambridge University Press, 1995.
+[3] W. Bailey, Generalized Hypergeometric Series, Cambridge tracts in mathematics and
+mathematical physics. The University Press, 1935.
+[4] Y. L. Luke, The Special Functions and their Approximations. Vol. 2. Academic Press, New York,
+1969.
+– 6 –
+

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
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+page_content='CA]  11 Jan 2023  Prepared for submission to JHEP  OU-HET-1167  On some identities for confluent hypergeometric  functions and Bessel functions  Yoshitaka Okuyamaa,b  aDepartment of Physics, Osaka University,  Machikaneyama-Cho 1-1, Toyonaka 560-0043, Japan  bDepartment of Physics, Faculty of Science, The University of Tokyo,  Bunkyo-Ku, Tokyo 113-0033, Japan  Abstract: We find a new integral representation of the Whittaker function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  We also show relevant summation formulas for Kummer’s confluent hypergeometric functions  and Bessel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  Contents  1  Introduction and summary  1  2  Integral representation of the Whittaker function of the first kind  2  3  Summation formula for confluent hypergeomertric functions  3  4  Summation formula for Bessel functions  5  1  Introduction and summary  This paper concerns several identities related to confluent hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' We first  show an integral representation of the Whittaker function:  Mκ,µ(z) =  √π Γ(2µ + 1)  2µ Γ  �  µ+κ+1/2  2  �  Γ  �  µ−κ+1/2  2  �  × √z  � 1  0  dξ ξ  −κ+1/2  2  −1(1 − ξ)  κ+1/2  2  −1 e(ξ−1/2)z Jµ  ��  ξ(1 − ξ)z  �  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1)  for Re (µ ± κ + 1/2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Here, the Bessel function Jν(z) and the Whittaker function of the  first kind Mκ,µ(z) are defined by [1]:  Jν(z) =  ∞  �  n=0  (−1)n  Γ(ν + n + 1) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  �z  2  �ν+2n  ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='2)  and  Mκ,µ(z) = e−z/2 zµ+1/2  ∞  �  n=0  (µ − κ + 1/2)n  (2µ + 1)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='3)  It turns out that the identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1) implies the following summation formula:  M(2a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' 2b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' z) =  ∞  �  n=0  (a)n(b)n(b − a)n  (b)2n(2b)2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (−z2)n M(a + n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' b + 2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' z) ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='4)  where we use the standard definition of Kummer’s confluent hypergeometric function M(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' z):  M(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' z) =  ∞  �  n=0  (a)n  (b)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='5)  – 1 –  By specifying indices of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='4) in a particular manner, one finds the following summation  formula for Bessel-J:  J2ν+1/2(z) =  Γ(ν + 1)  Γ(2ν + 3/2)  ∞  �  n=0  (ν + 1/2)n  (2ν + 3/2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  �z  2  �ν+1/2+n  Jν+n(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6)  To the best of our effort, we could not find either of these three identities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='4), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1) and  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6) anywhere in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  2  Integral representation of the Whittaker function of the first kind  We here show an integral representation of the Whittaker function of the first kind (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  Recall that the Whittaker function of the first kind is a solution to the Whittaker differential  equation  d2y  dz2 +  �  −1  4 + κ  z + 1/4 − µ2  z2  �  y = 0 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1)  subject to the boundary condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  Mκ,µ(z) −−−→  z→0 zµ+1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='2)  All we need to do is to check that the RHS of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1) satisfies the differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1)  and the boundary condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  Boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  In taking z → 0 limit, the RHS of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1) becomes:  (RHS of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1))  −−−→  z→0  √π zµ+1/2 Γ(2µ + 1)  22µ Γ  �  µ+κ+1/2  2  �  Γ  �  µ−κ+1/2  2  �  Γ(ν + 1)  ×  � 1  0  dξ ξ  −κ+µ+1/2  2  −1(1 − ξ)  κ+µ+1/2  2  −1  = zµ+1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='3)  We have used the series expansion of the Bessel function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='2) in the second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' In going to  the last line, we used the following two Gamma function identities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  � 1  0  dt tx−1(1 − t)y−1 = Γ(x) Γ(y)  Γ(x + y) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='4)  Γ(2z) = 22z−1 Γ(z) Γ(z + 1/2)  √π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='5)  – 2 –  Differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  Let us check that the left of the RHS of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1) satisfies the Whittaker  differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' After some manipulations, we see  � d2  dz2 +  �  −1  4 + κ  z + 1/4 − µ2  z2  ��  (RHS of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1))  ∝  � 1  0  dξ ξ  −κ+1/2  2  −1(1 − ξ)  κ+1/2  2  −1 e(ξ−1/2)z  ×  � d2  dz2 + 1  z  d  dz + ξ(1 − ξ) − µ2  z2 + (2ξ − 1) d  dz − 2ξ(1 − ξ) + κ + ξ − 1/2  z  �  Jµ  ��  ξ(1 − ξ)z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6)  Notice that the first four terms in the last line add up to zero thanks to the Bessel differential  equation:  � d2  dz2 + 1  z  d  dz + 1 − ν2  z2  �  Jν(z) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='7)  We now focus on the fifth term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Using the identity that follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='2)  (2ξ − 1) d  dz Jµ  ��  ξ(1 − ξ)z  �  = −2ξ(1 − ξ)  z  d  dξ Jµ  ��  ξ(1 − ξ)z  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='8)  we perform integration by parts with respect to ξ:  (The fifth term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6))  = −2  z  � 1  0  dξ ξ  −κ+1/2  2  −1+1(1 − ξ)  κ+1/2  2  −1+1 e(ξ−1/2)z  � d  dξ Jµ  ��  ξ(1 − ξ)z  ��  = 2  z  � 1  0  dξ Jµ  ��  ξ(1 − ξ)z  � d  dξ  �  ξ  −κ+1/2  2  (1 − ξ)  κ+1/2  2  e(ξ−1/2)z�  =  � 1  0  dξ ξ  −κ+1/2  2  −1(1 − ξ)  κ+1/2  2  −1 e(ξ−1/2)z  �  2ξ(1 − ξ) − κ + ξ − 1/2  z  �  Jµ  ��  ξ(1 − ξ)z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='9)  Here, we assumed Re (µ±κ+1/2) > 0 to drop off the surface terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Plugging this into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6),  one can readily see that all terms cancel out and conclude that the RHS of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1) satisfies the  Whittaker differential equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  3  Summation formula for confluent hypergeomertric functions  We then derive the summation formula for confluent hypergeometric functions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  Assuming Re (µ ± κ + 1/2), Re (µ) > 0, we start by applying the Mellin-Barnes-type  representation of Bessel-J (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1) [2],  Jν(z) =  � i ∞  −i ∞  dt  2πi  Γ(−t)  Γ(ν + t + 1)  �z  2  �ν+2t  Re (ν) > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1)  – 3 –  After the change of the order of integration, we find:  (Second line of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1))  = √z e−z/2  � i ∞  −i ∞  dt  2πi  Γ(−t)  Γ(µ + t + 1)  �z  2  �µ+2t � 1  0  dξ ξ  µ−κ+1/2  2  +t−1(1 − ξ)  µ+κ+1/2  2  +t−1 ezξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='2)  Notice that the ξ-integral is nothing but the integral representation of the confluent hyper-  geometric equation:  M(a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' z) =  Γ(b)  Γ(a)Γ(b − a)  � 1  0  dt ezt ta−1(1 − t)b−a−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='3)  Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='5), we have:  (Second line of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1))  = 2µ zµ+1/2  √π  e−z/2  � i ∞  −i ∞  dt  2πi  Γ  �  µ−κ+1/2  2  + t  �  Γ  �  µ+κ+1/2  2  + t  �  Γ(µ + 1/2 + t) Γ(−t)  Γ(2µ + 1 + 2t)Γ(µ + 1/2 + 2t)  × z2t M  �µ − κ + 1/2  2  + t, µ + 1  2 + 2t, z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='4)  Deforming the integration contour to the right and picking up the residues coming from  Γ(−t), we see:  (Second line of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1)) =  2µ Γ  �  µ−κ+1/2  2  �  Γ  �  µ+κ+1/2  2  �  √π Γ(2µ + 1)  zµ+1/2 e−z/2  ×  ∞  �  n=0  �  µ−κ+1/2  2  �  n  �  µ+κ+1/2  2  �  n (µ + 1/2)n  (µ + 1/2)2n(2µ + 1)2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (−z2)n M  �µ − κ + 1/2  2  + n, µ + 1  2 + 2n, z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='5)  Substituting this for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1), we arrive at (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='4) for Re (a), Re (b − a), Re (b + 1/2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  We can verify that the identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='4) holds for any a, b ∈ C, by expanding in powers of z  comparing both sides order by order using the following formula:  (2a)k  (2b)k  =  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (b)k  ⌊k/2⌋  �  n=0  (a)k−n (b)n (b − a)n  (2b)2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' (k − 2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' (−1)n  k ∈ Z≥0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6)  where ⌊x⌋ is the floor function that returns the largest integer less than or equal to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' This  formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6) can be proven as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Firstly, short calculation leads:1  (RHS of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6)) = (a)k  (b)k  3F2  �  b − a, − k−1  2 , − k  2  b + 1  2, 1 − a − k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' 1  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='7)  1Use Euler reflection formula Γ(z)Γ(1 − z) = π/ sin πz and Legendre duplication formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  – 4 –  with 3F2 being a generalized hypergeometric function defined by:  3F2  �a, b, c  d, e ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' z  �  =  ∞  �  n=0  (a)n(b)n(c)n  (d)n(e)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' zn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='8)  With the help of Saalsch¨utz’s theorem that asserts [3, equation (1), chapter II]:  3F2  �  a, b, −n  c, 1 + a + b − c − n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' 1  �  = (c − a)n(c − b)n  (c)n(c − a − b)n  n ∈ Z≥0 ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='9)  we can verify that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6) is an identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  4  Summation formula for Bessel functions  Lastly, we show the summation formula for Bessel functions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' It follows from the relation  between Kummer’s confluent hypergeometric functions and Bessel functions:  Jν(z) =  e∓iz  Γ(ν + 1)  �z  2  �ν  M(ν + 1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' 2ν + 1, ±2iz) ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1)  we have:  (LHS of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6)) =  e∓iz  Γ(2ν + 3/2)  �z  2  �2ν+1/2  M(2ν + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' 4ν + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' ±2iz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='2)  Plugging the summation formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='4) into this and using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='1) again, one finds:  (LHS of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6))  =  e∓iz  Γ(2ν + 3/2)  �z  2  �2ν+1/2  ∞  �  n=0  [(ν + 1/2)n]2(2ν + 1)n  (2ν + 1)2n(4ν + 2)2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  × (2z)2n M(ν + 1/2 + n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' 2ν + 1 + 2n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' ±2iz)  =  Γ(ν + 1)  Γ(2ν + 3/2)  ∞  �  n=0  24n [(ν + 1/2)n]2(ν + 1)n(2ν + 1)n  (2ν + 1)2n(4ν + 2)2n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  �z  2  �ν+1/2+n  Jν+n(z)  = (RHS of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='3)  We need to use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='5) several times to reach the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  When ν = 0, the LHS of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='6) reduces to a triogeometric function divided by √z:  J1/2(z) =  �  2  πz sin z ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='4)  whereas the RHS remains the summation of Bessel-J’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' This reproduces the known expansion  of sin in terms of Bessel functions found in [4, equation (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='19)]:  sin z  z  =  ∞  �  n=0  1  (2n + 1) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  �z  2  �n  Jn(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='5)  – 5 –  Acknowledgments  We are grateful to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Nishioka for the valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' The work of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' was supported  by Forefront Physics and Mathematics Program to Drive Transformation (FoPM), a World-  leading Innovative Graduate Study (WINGS) Program, the University of Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' The work  of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' was also supported by JSPS fellowship for young students No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' 21J20750, MEXT, and  by JSR fellowship, the University of Tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  References  [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Whittaker and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Watson, A Course of Modern Analysis, A Course of Modern Analysis: An  Introduction to the General Theory of Infinite Processes and of Analytic Functions, with an  Account of the Principal Transcendental Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Cambridge University Press, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  Cambridge University Press, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  [3] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Bailey, Generalized Hypergeometric Series, Cambridge tracts in mathematics and  mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' The University Press, 1935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Luke, The Special Functions and their Approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content=' Academic Press, New York,  1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}
+page_content='  – 6 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE3T4oBgHgl3EQfoAol/content/2301.04629v1.pdf'}

39E3T4oBgHgl3EQfogqI/content/tmp_files/2301.04634v1.pdf.txt

@@ -0,0 +1,1004 @@
+Street-View Image Generation from a Bird’s-Eye View Layout
+Alexander Swerdlow
+Runsheng Xu
+Bolei Zhou
+University of California, Los Angeles
+{aswerdlow, rxx3386}@ucla.edu, [email protected]
+Abstract
+Bird’s-Eye View (BEV) Perception has received increas-
+ing attention in recent years as it provides a concise and
+unified spatial representation across views and benefits a
+diverse set of downstream driving applications. While the
+focus has been placed on discriminative tasks such as BEV
+segmentation, the dual generative task of creating street-
+view images from a BEV layout has rarely been explored.
+The ability to generate realistic street-view images that
+align with a given HD map and traffic layout is critical for
+visualizing complex traffic scenarios and developing robust
+perception models for autonomous driving. In this paper,
+we propose BEVGen, a conditional generative model that
+synthesizes a set of realistic and spatially consistent sur-
+rounding images that match the BEV layout of a traffic sce-
+nario. BEVGen incorporates a novel cross-view transfor-
+mation and spatial attention design which learn the rela-
+tionship between cameras and map views to ensure their
+consistency. Our model can accurately render road and
+lane lines, as well as generate traffic scenes under differ-
+ent weather conditions and times of day. The code will be
+made publicly available.
+1. Introduction
+BEV perception for autonomous driving is a fast-
+growing field, with the goal of learning a cross-view repre-
+sentation that transforms information between the perspec-
+tive and bird’s-eye view. Such representation can be used
+in downstream tasks such as path planning and trajectory
+forecasting [1, 55]. The recent successes in BEV percep-
+tion, whether for monocular images [9,13,14] or multi-view
+images [17,50,56], mostly focus on the discriminative side
+of BEV perception where the inputs are street-view images
+and the output is a semantic BEV layout. However, the gen-
+erative side of BEV perception, which aims at synthesizing
+realistic street-view images from a given BEV semantic lay-
+out, is rarely explored. A BEV layout concisely describes
+a traffic scenario at the semantic level, therefore generating
+its corresponding street-view images can help visualize the
+BEV Layout
+Generated Street-View Images
+Figure 1.
+The proposed BEVGen generates realistic and spa-
+tially consistent street-view images from BEV layout. There are
+six camera views surrounding the ego vehicle as indicated by the
+green rectangle in the BEV layout.
+scene in a more real-world setting.
+There are many potential applications for the BEV gen-
+eration task. For example, we can create synthetic train-
+ing data for BEV segmentation models. Whereas most cur-
+rent approaches to synthetic training data involve a complex
+simulator or 3D reconstructed meshes, it is simpler to adopt
+a controllable generative model for diverse image genera-
+tion. Another benefit provided by the BEV generation is
+the ease of visualizing and editing traffic scenes. In the case
+of self-driving vehicles, we often care about a small set of
+rare scenarios where an accident is most likely to happen.
+Human users can intuitively edit a BEV layout and then use
+a generative model to output the corresponding street-view
+images for training or testing a driving system.
+The fundamental question for BEV generation is: what
+could be a plausible set of street-view images that corre-
+spond to this BEV layout? One could think of numerous
+scenes with varying vehicle types, backgrounds, and more.
+For a set of views to be realistic, we need to consider several
+properties of the images. Similar to the problem of novel
+view synthesis, images must appear consistent, as if they
+were taken in the same physical location. For instance, cam-
+eras with an overlapping field-of-view (FoV) should have
+1
+arXiv:2301.04634v1  [cs.CV]  11 Jan 2023
+
+overlapping content, and objects partially visible in one
+frame should appear in a rotated frame. The visual styling
+of the scene also needs to be consistent such that all virtual
+views appear to be created in the same geographical area
+(e.g., urban vs. rural), time of day, with the same weather
+conditions, and so on. In addition to image consistency,
+the images must correspond to the HD map, faithfully re-
+producing the specified road layout, lane lines, and vehicle
+locations. Unlike image-to-image translation with a seman-
+tic mask, the BEV generation model must infer the image
+layout to account for occlusions between objects and the
+relative heights of objects in a scene. These two main chal-
+lenges, image consistency and correspondence, are critical
+to the task but can be difficult to reconcile. If we only desire
+image consistency, similar to the case of image outpainting,
+the model is free to generate any consistent image. How-
+ever, if we also wish to maintain correspondence between
+the virtual views and the HD map, portions of the virtual
+views are constrained to represent certain elements (e.g.,
+vehicles). On the other hand, if we only care about image
+correspondence, the model only needs the context of part
+of the HD map in its FoV and does not need to account for
+previously generated images or issues such as wraparound.
+In this work, we tackle the new task of generating street-
+view images from a BEV layout and propose a generative
+model called BEVGen to address the underlying challenges.
+We develop an autoregressive model called BEVGen that
+generates a set of n realistic and spatially consistent images.
+Fig. 1 shows generation examples. BEVGen has two tech-
+nical novelties: (i) it incorporates spatial embeddings using
+camera instrinsics and extrinsics to allow the model to at-
+tend to relevant portions of the images and HD map, and
+(ii) it contains a novel attention bias and decoding scheme
+that maintains both image consistency and correspondence.
+Thus the model can generate high-quality scenes with spa-
+tial awareness and scene consistency across camera views.
+Compared to baselines, the proposed model obtains sub-
+stantial improvement in terms of image synthesis quality
+and semantic consistency. The model can also render real-
+istic scene images from out-of-domain BEV maps, such as
+those provided by a driving simulator or edited by a user.
+We summarize our contributions as follows:
+• We tackle the new task of multi-view image generation
+from BEV layout. It is the first attempt to explore the
+generative side of BEV perception for driving scenes.
+• We develop a novel generative model BEVGen that
+can synthesize spatially consistent street-view images
+by incorporating spatial embeddings and a pairwise
+camera bias.
+• The model achieves high-quality synthesis results and
+shows promise for applications such as data augmen-
+tation and 3D simulation rendering.
+2. Related Work
+Cross-modal Image Generation.
+Cross-modal image
+generation has seen a lot of attention in recent years with
+work on text-to-image models [12, 32–34, 40], speech-to-
+image models [5, 20], and image-to-video models [42].
+Others have focused on using more direct representations
+to control generation, including generation from semantic
+masks [18,26,52], or worked to convert higher-level repre-
+sentations such as text [15,30], scene graphs [7,19,44,51],
+and bounding boxes [22] into such a semantic mask. There
+have also been several attempts at learning spatially disen-
+tangled scene representations by composing latent features
+that correspond to specific parts of a scene [10, 27]. Our
+task is conceptually similar to image generation from a se-
+mantic mask but distinct in that our semantic representation
+only provides minimal layout constraints, lacking height in-
+formation, direct occlusion representation, and background
+information.
+Image-to-image Generation.
+Direct image-to-image
+translation has also taken off in recent years with models
+such as pix2pix [18] and cycleGAN [57]. Several works
+have focused directly on the task of street-view synthe-
+sis from satellite views as a subset of the image-to-image
+translation problem [35, 41, 43, 54]. These works attempt
+to tackle viewpoint transformation, from a top-down to an
+ego-centric view, that is implicitly required for our task, but
+our task does not benefit from the rich RGB representation
+provided by a satellite view. Furthermore, large portions of
+our virtual camera views correspond to areas entirely un-
+labeled on our BEV map, requiring largely unconditional
+generation for these areas.
+Image Outpainting.
+Spatial consistency is important for
+tasks such as image outpainting, where the goal is to gener-
+ate or extend an image of the same scene. Early approaches
+for image outpainting used auto-regressive approaches on a
+pixel-wise level [6,25,45,46]. However, this approach can
+be computationally expensive and thus is limited to gen-
+erating low-resolution images. Subsequently, GANs were
+introduced to the task [16, 23, 39, 49] which do not suffer
+from the same computational limitations as pixel-wise au-
+toregressive approaches. More recent works have utilized a
+Vector Quantised-Variational Autoencoder (VQ-VAE) [47]
+to great success [2,4]. Similar to image outpainting, our task
+requires generated images to appear coherent in weather
+and location; however, we also seek to generate distinct,
+partially overlapping camera views and require that portions
+of these views are conditionally generated from a BEV lay-
+out.
+Novel View Synthesis.
+The same underlying VQ-VAE
+architecture has been used for the single-view novel view
+synthesis (NVS) task where the goal is to generate new vir-
+tual camera view given a source image. By conditioning
+2
+
+an autoregressive transformer with camera translation and
+rotation, [38] showed that a transformer-based model can
+learn the 3D relationship between images without explicit
+depth maps or warping as used in prior attempts for single-
+view NVS such as in [37, 48]. To improve the consistency
+between frames, [36] suggests a camera-aware bias for self-
+attention that encodes the similarity between consecutive
+image frames. Our task requires a similar 3D understand-
+ing between different viewpoints as in NVS, but lacks the
+conditioning information provided by a source view(s) and
+requires consistency not only between frames but also with
+an HD map. If we broaden our task to allow for a source
+view, as we demonstrate in Fig. 6, our task can be thought
+of as a conditional NVS task.
+3. Method
+In this section, we introduce the framework of the pro-
+posed BEVGen. We have a semantic layout in Birds-Eye
+View (BEV), B ∈ RHb×Hb×cb with the ego at the cen-
+ter and cb channels describing the locations of vehicles,
+roads, lane lines, and more (see Sec. 4.1). Given a set of
+n virtual camera views to generate, (Kk, Rk, tk)n
+k=1, where
+Kk, Rk, tk are the intrinsics, extrinsic rotation, and trans-
+lation of the kth camera, we generate n images, Ik ∈
+RHc×Wc×3.
+Fig. 2 illustrates the framework of the proposed BEV-
+Gen. BEVGen consists of two autoencoders modeled by
+VQ-VAE, one for images and one for the BEV representa-
+tion, that allow the causal transformer to model scenes at
+a high level. The key novelty lies in how the transformer
+can relate information between modalities and across differ-
+ent views. The cross-view transformation encodes a cross-
+modal inductive 3D bias, allowing the model to attend to
+relevant portions of the HD map and nearby image tokens.
+We explain each part in more detail below.
+3.1. Model Structure
+Image Encoder.
+To generate a globally coherent image,
+we model our distribution in a discrete latent space instead
+of pixel-space. We use the VQ-VAE model introduced by
+Oord et al. [47] as an alternative generative architecture to
+GANs1. Additionally, we incorporate a perceptual and a
+patch-wise adversarial loss as in [11]. The VQ-VAE archi-
+tecture consists of an encoder Ecam, a decoder Dcam, and a
+codebook Zc = {zm}Mc
+m=1 ⊂ Rnc where Mc is the number
+of code vectors and nc is the embedding dimension of each
+code. Given a source image, xk ∈ RHc×Wc×3 we encode
+ˆzk = E(xk) ∈ Rhc×wc×nc. To obtain a discrete, tokenized
+representation, we find the nearest codebook vector for each
+1Note that switching to the recently developed class of diffusion mod-
+els can potentially improve the image synthesis quality, but such models
+require an order of magnitude of additional data and computational re-
+sources for training and thus we leave it for future works.
+feature vector ˆzk,ij ∈ Rnc where i, j are the row, column in-
+dices in the discrete latent representation with size hc × wc:
+zk,ij = arg min
+m
+∥ˆzk,ij − zm∥ ∈ Rhc×wc×nc.
+(1)
+This creates a set of tokens zk ∈ Nhc×wc that we refer to
+as our image tokens. To generate an image from a set of
+tokens, we decode ˜zk ∈ Rhc×wc×nc with a convolutional
+decoder, Dcam(˜zk) ∈ RHc×Wc×3 using the same architec-
+ture as [11].
+BEV Encoder.
+To condition our model on a BEV lay-
+out, we use the same discrete representation as for cam-
+era images, except we replace the perceptual and adver-
+sarial losses with a binary cross entropy loss for binary
+channels and an L2 loss for continuous channels. We en-
+code our BEV map b as before with Ebev(b) ∈ Rhb×wb×nb
+and Zb = {zm}Mb
+m=1 ⊂ Rnb to obtain a set of tokens,
+zbev ∈ Nhb×wb. We discard the decoder stage, Dbev, after
+training the 1st stage as it is not needed for our transformer
+model or inference.
+Autoregressive Modeling.
+Given a BEV layout and k
+sets of camera parameters, we seek to generate k images by
+learning the prior distribution of a set of discrete tokens, z
+conditioned on zbev, K, R, t.
+p(z|zbev, K, R, t) =
+h×w×k
+�
+i=0
+p(zi|z<i, zbev, K, R, t).
+(2)
+We model p(.) by training a transformer τ that predicts the
+subsequent token based on the discretized BEV features,
+prior image tokens, and their respective camera parameters.
+This approach requires us to order all k × hc × wc cam-
+era tokens. Instead of using a camera-first, row-major order
+where the first wcam tokens are the first row of the 1st camera
+and the first hcamwcam tokens encompass the entire 1st cam-
+era, we choose to decode in a center-out order. We choose
+this order as we seek to maximize the influence of impor-
+tant scene semantics, which primarily lie directly ahead and
+behind of a vehicle while driving. We alternate between the
+front and back, as well as left to right, decoding starting
+at the top row and moving outward until we expand such
+that the front row meets the back row, and repeat this for all
+rows.
+3.2. Spatial Embeddings
+To help the model attend to relevant tokens both in the
+camera and BEV feature space, we introduce positional em-
+beddings. We take inspiration from work on BEV segmen-
+tation [56] on alignment between the BEV and first-person
+view (FPV) perspectives.
+Camera Embedding. In order to align image tokens with
+BEV tokens, we use the known intrinsics and extrinsics
+to reproject from image coordinates to world coordinates.
+3
+
+Learned Pos Emb
+Source Multi-view images
+Token Direction Vectors
+Encoder
+4
+3
+6
+1
+Flatten (Row-
+Major)
+World Coords
+Encoder
+Flatten (Center-
+Outward)
+Autoregressive Transformer
+Learned Pos Emb
+Weighted  
+CE loss
+Decoder
+BEV Layout
+Generated Multi-view images
+3
+1
+4
+3
+6
+1
+4
+6
+3
+Pairwise Similarity
+Camera Bias
+Figure 2. BEVGen framework. A BEV layout and source multi-view images are encoded to a discrete representation and are flattened
+before passed to the autoregressive transformer. Spatial embeddings are added to both camera and BEV tokens inside each transformed
+bloc, the learned pairwise camera bias are added to the attention weights. Weighted CE loss is applied during training, and we pass the
+tokens to the decoder to obtain generated images during inference.
+Given a token in image space, zk,ij, we convert to homoge-
+neous coordinates and obtain a direction vector in the ego
+frame as follows:
+dk,ij = R−1
+k K−1
+k zi,jk + tk.
+(3)
+We use a 1D convolution, θc(d) ∈ Rn×hc×wc×nemb, to en-
+code our direction vector in the latent space of the trans-
+former. We encode our image tokens using a shared learn-
+able embedding λc(zk,ij) ∈ Rnemb, and add a per-token
+learnable parameter, Λc
+k,ij ∈ Rnemb, across image tokens:
+lk,ij = λ(zk,ij) + θ(dk,ij) + Λk,ij.
+(4)
+BEV Embedding.
+To align our BEV tokens with our
+image tokens, we perform a similar operation as in Eq. (4)
+and use the known BEV layout dimensions to obtain co-
+ordinates in the ego frame, txy, for each token and en-
+code these into our transformer latent space, with θb(t) ∈
+Rhb×wb×nemb. We similarly use a shared learnable embed-
+ding for our discrete tokens, λ(zxy) ∈ Rnemd, and a per-
+token learnable parameter, Λxy:
+lxy = λ(zxy) + θb(txy) + Λxy.
+(5)
+3.3. Camera Bias
+In addition to providing the model with aligned embed-
+dings, we add a bias to our self-attention layers that provides
+both an intramodal (image to image) and intermodel (im-
+age to BEV) similarity constraint. This draws inspiration
+Learned Pos Emb
+Self Attn
+Output
+Pairwise  
+Camera
+Bias
+Figure 3. Camera Bias Overview. We construct a pairwise matrix
+that encodes relationship between a given image token and another
+BEV/image token.
+from [36], but instead of providing a blockwise similarity
+matrix that is composed of encoded poses between frames,
+we provide a per-token similarity based on their relative di-
+rection vectors. Our approach also encodes the relationship
+between image and BEV tokens. For self-attention in any
+layer between some query qr and key/value kc/vc we have:
+Attention(qr, kc, vc) = vc softmax
+�arc
+√
+d
+�
+,
+(6)
+arc = qrkc + βrc.
+(7)
+4
+
+940940940940米The transformer sequence is composed of hbwb condi-
+tional BEV tokens followed by hcwc camera tokens.
+If
+r, c > hbwb, both positions correspond to image tokens and
+thus we have two direction vectors, dr, dc, computed as in
+Eq. (3). As discussed in Sec. 3.1, we have a mapping be-
+tween the sequence index and image token (i, j) in camera
+k. If r > hbwb > c, we have a query for some image token
+and a key/value pair corresponding to BEV token. Thus, we
+again construct two direction vectors. In this case our BEV
+direction vector consists of the 2D World coordinates (in
+the ego-center frame) and our image direction vector is the
+same as in Eq. (3) except with the row value as the center
+of the image. Given these two direction vectors, dr, dc, we
+add the cosine similarity and a learnable parameter, θrc, as
+shown in Fig. 3:
+βrc =
+dr · dc
+∥dr∥∥dc∥ + θrc.
+(8)
+3.4. Random Masking
+A key problem that arises when generating multiple im-
+ages in parallel is the quadratic complexity of the self-
+attention mechanism. One solution to this issue would be to
+limit the sequence length of our transformer by using per-
+forming image extrapolation as in [2]. However, this lim-
+its the scene context and can cause later images to appear
+far different from the first image, despite having local im-
+age consistency. Instead, we implement a version of sparse
+attention as in [8, 53]. As opposed to a uniform random
+attention mask, we instead unmask regions of the image
+near the token we attend. Using the same formulation as
+in Eq. (8), we create a pairwise similarity matrix for image
+tokens only. As sparse attention groups the input sequence
+into discrete blocks, we perform an average pooling on this
+matrix down to the resolution of the sparse attention opera-
+tion and use these values as weights for sampling. Addition-
+ally, we have a sliding window in which we always attend
+to the last r tokens, and we attend to all BEV tokens.
+4. Experiments
+4.1. Dataset
+We evaluate the proposed method using the NuScenes
+dataset [3], one of the popular driving datasets used for BEV
+segmentation and detection. We chose NuScenes as it is
+among the only large driving dataset to provide full 360 deg
+camera coverage with a consistent camera resolution, but
+our method can be easily adapted to other datasets with
+different camera arrangements as demonstrated in Sec. 4.4.
+NuScenes consists of 1000, 20-second scenes, captured in
+Boston and Singapore. There are a total of 40k annotated in-
+stances that are labeled every 2Hz, split into 34k, 6k, and 6k
+instances for the train, validation, and test sets respectively.
+Each instance contains ground-truth 3D bounding boxes,
+6 camera images covering a 360 deg FoV, calibrated cam-
+era intrinsics and extrinsics, as well as LiDAR and Radar
+scans. We project these 3D bounding boxes onto a BEV
+layout following standard practice used in BEV segmenta-
+tion [17,29,56].
+Preprocessing.
+The BEV layout representation used
+in training and testing is a 256 × 256 mask representing
+80m × 80m around the ego center and containing 21 chan-
+nels. 14 channels are binary masks representing map in-
+formation (lane lines, dividers, etc.) and actor annotations
+(cars, trucks, pedestrians, etc.). The remaining 7 channels
+provide instance information including the visibility of an
+annotation within the camera view, the height, width, and
+orientation of the annotation, and the pixel offset from the
+center point of the annotation. We resize our cropped cam-
+era images to 224 × 400 and appropriately modify the in-
+trinsics passed to our model. To enable our weighted cross-
+entropy loss, we project the provided 3D annotations onto
+the camera frame and weight the corresponding tokens in
+our discrete camera frame representation, zk ∈ Nhc×wc.
+4.2. Training Details
+VQ-VAE. We train the 1st stage camera VQ-VAE with ag-
+gressive augmentation consisting of flips, rotations, color
+shifts, and crops. Similarly, we train our 1st stage BEV
+VQ-VAE with flips and rotations. For the 2nd stage, we
+add minimal rotations and scaling but perform cropping and
+modify the corresponding intrinsics that are passed to the
+model.
+Transformer. We crop all images to H × W = 224 × 400
+and our 4 encoder/decoder stages create a discrete latent
+representation of hc × wc = 14 × 25. Our BEV layout has
+a discrete latent representation of hb × hb = 16, 16. Both
+the BEV and image codebooks have |Zc| = |Zb| = 1024
+codes with an embedding dimension, nc = nb = 256. Our
+transformer is GPT-like [31] with 16-heads and 24-layers.
+We use DeepSpeed to facilitate sparse self-attention and
+16-bit training. We clip gradients at 50 to prevent insta-
+bility during training and use the AdamW optimizer [24]
+with β1, β2 = 0.9, 0.95 and a learning rate of λ = 5e-7.
+For our sparse models, we have an attention mask density
+of 35% with a sliding window length of r = 96. Except
+as described in Sec. 4.4, our sparse model is derived from
+fine-tuning our full-attention model for 10 epochs.
+4.3. Results
+We show generation results for all six camera views
+trained from the NuScenes dataset. For all visualizations,
+we flip the back left and right cameras along the vertical
+axis to highlight the image consistency of our model. Thus,
+the side, front, and back cameras meet at their outer edges in
+all figures. Since our work is, to our knowledge, the first at-
+5
+
+Method
+FID↓
+Road mIoU↑
+Vehicle mIoU↑
+Baseline
+43.18
+45.80
+4.44
+BEVGen
+25.54
+50.20
+5.89
+Sparse BEVGen
+28.67
+50.92
+6.69
+Table 1. Baseline Comparison over all 6 views on NuScenes Vali-
+dation.
+tempt at conditional street-view synthesis from a BEV lay-
+out, we find no existing method to directly compare with.
+Instead, we compare with a baseline model consisting of
+the same underlying GPT architecture and using the same
+1st stage encoders/decoders as our BEVGen model. We use
+a row-major decoding order and employ only a learnable
+position embedding, but do not add the spatial embeddings
+(Sec. 3.2) or camera bias (Sec. 3.3) and use full-attention.
+Qualitative result. Fig. 5 exhibits the generation examples
+from BEVGen. Our model is able to generate a diverse set
+of scenes including intersections, parking lots, and boule-
+vards. We observe that each camera view not only correctly
+displays the surrounding of the same location, but also pre-
+serves the spatial perspective. BEVGen synthesizes images
+under various weather conditions, with the same weather
+apparent in all images, including physical artifacts such as
+rain. We also demonstrate that our model is capable of gen-
+erating diverse scenes corresponding to the same BEV lay-
+out. We see at the bottom of Fig. 5 the same location ren-
+dered in the day and at night by the model.
+We compare generation quality of BEVGen to our base-
+line using the same BEV layout in Fig. 5. We see that BEV-
+Gen can not only render a more accurate scene with nearby
+vehicles present in the correct camera views, but our spatial
+consistency is significantly improved. Our model is able to
+correctly synthesize a vehicle partially present in multiple
+camera views. We also see that the background of the scene
+is consistent between cameras, unlike the baseline model.
+Additionally, we apply a BEV segmentation model on
+the synthesized images and analyze the semantic content.
+As seen in Fig. 5, our images allow the model to correctly
+infer the road layout whereas our baseline images do not.
+Quantitative result.
+We use the Fr´echet Inception Dis-
+tance (FID) to evaluate our synthesized quality compared to
+the source images. Unless otherwise noted, all metrics are
+calculated on a subset of the NuScenes validation set. We
+sample 4 images from each scene, with 600 instances over-
+all, and synthesize a set of images with no post-generation
+filtering. For calculating FID scores, we use clean-fid [28].
+To differentiate between the performance of our 1st and
+2nd stage, we compare our results to the results obtained by
+feeding the encoded tokens of the source images directly to
+the decoder, as is done when training the 1st stage. This
+represents the theoretical upper bound of our model’s per-
+formance and it allows us to largely remove the effect of the
+first stage which is not the focus of this paper. However, it
+should be noted that the design of the 1st stage and the prop-
+erties of the learned codebook can have a significant impact
+on the 2nd stage [11,12].
+As seen in Tab. 1, our BEVGen model achieved an FID
+score of 25.54 compared to the baseline score of 43.18. This
+is in comparison to our reference upper-bound FID score of
+9.37. Our model utilizing our sparse masking design from
+Sec. 3.4 achieved an FID score of 28.67. This sparse vari-
+ant is approximately 48% faster during inference and 40%
+faster for training.
+While FID is a common metric to measure image syn-
+thesis quality, it fails to entirely capture the design goals of
+our task and cannot reflect the synthesis quality of different
+semantic categories. Since we seek to generate multi-view
+images consistent with a BEV layout, we wish to measure
+our performance on this consistency. To do this, we lever-
+age a BEV segmentation network CVT from [56], trained
+entirely on source data for a fair comparison. We use the
+same set of generated images conditioned on a ground-truth
+BEV layout as before and for each set we apply the CVT to
+the generated images and then compare the predicted lay-
+out with the ground-truth BEV layout. We report both the
+road and vehicle class mean intersection-over-union (mIoU)
+scores. As shown in Tab. 1, we beat our baseline by 4.4 and
+1.45 for road and vehicle classes respectively. Note that the
+performance of the BEV segmentation model on the val-
+idation set is 66.31 and 27.51 for road and vehicle mIoU
+respectively. This reveals that though the model can gen-
+erate road regions in the image in a reasonable manner, it
+still has a limited capability of generating high-quality indi-
+vidual vehicles that can be recognized correctly by the seg-
+mentation network. This is a common problem for scene
+generation where it remains challenging to synthesize the
+small objects entirely. Our work is a starting point and we
+plan to improve small object synthesis in the future work.
+View-conditioned generation.
+We test the ability of our
+model to synthesize other views when provided a view from
+a single camera as seen in Fig. 6. Due to the chosen center-
+out decoding order, not all image tokens are able to attend to
+the source image and, instead, we simply skip inference for
+provided camera views. Despite this, we observe that our
+model is able to generate consistent imagery both in scene
+content and time of day.
+4.4. Ablation Study
+To verify the effectiveness of our design choices, we run
+an ablation study on key features of our model. We run
+these experiments on the same subset of the NuScenes vali-
+dation set as in Sec. 4.3, but only consider the 3 front-facing
+views to reduce training time. The 3 front-facing views have
+a larger FoV overlap than the rear view and capture more
+6
+
+Figure 4. Synthesized multi-view images from BEVGen. Image contents are diverse and realistic. The two instances in the bottom row
+use the same BEV layout for synthesizing the same location in day and night.
+Figure 5. Qualitative comparison to baseline. Left is the instance from the baseline and right is from BEVGen. We also show the predicted
+layout (only for the road class) from the generated multi-view images.
+relevant scene features such as cars and lane-lines when
+compared to the side-facing rear views that capture a sig-
+nificant amount of background. This is more relevant to our
+task as it allows us to better verify the design objectives of
+our model.
+We test four variants of our model, one with only center-
+out decoding, one with our camera bias, one with the cam-
+era bias and spatial embeddings, and a final model that we
+train from scratch using our sparse masking, instead of fine-
+tuning. Tab. 2 shows a steady improvement in FID scores
+as we add the camera bias, and spatial embeddings.
+5. Applications
+Generating realistic images from BEV layout has many
+applications. In this section we explore the applications of
+data augmentation for BEV segmentation and image gener-
+ation from simulated BEV.
+Method
+FID↓
+Center-out decoding
+42.32
++ Camera Bias
+41.20
++ Camera Bias, Spatial Embedding
+40.48
++ Camera Bias, Spatial Embedding, Sparse Mask
+48.31
+Table 2. Ablation of the key model components.
+Data augmentation for BEV segmentation.
+An impor-
+tant application of our BEV conditional generative model
+is generating synthetic data to improve prediction models.
+Thus, we seek to verify the effectiveness of our model by
+incorporating our generated images as augmented samples
+during training of a BEV segmentation model.
+We use
+CVT [56] as our model, which is also used in Sec. 4.3, and
+compare our results to training without any synthetic sam-
+7
+
+Figure 6. View-conditioned generation. Green box indicates the provided source tokens.
+Figure 7. Generating images based on the BEV layouts provided by the MetaDrive driving simulator
+Road mIoU
+Vehicle mIoU
+CVT (w/o augmentation)
+71.3
+36.0
+CVT (w/ augmentation)
+71.9
+36.6
+Table 3. Application of data augmentation. We report the segmen-
+tation results on the validation set of NuScenes trained from the
+original training set and the one augmented with synthetic data.
+ples. We generate 6,000 unique instances using the BEV
+layout from the train set on NuScenes. These synthetic in-
+stances are associated with the ground truth BEV layout for
+training, with no relation to results from Sec. 4.3. To reduce
+the effect of randomness during training, we set the random
+seed and disable non-deterministic operations for all train-
+ing. As seen in Tab. 3, our data improves validation mIoU
+by 0.6 for both the road category and the vehicle category.
+Image generation from simulated BEV. Since one mo-
+tivation for our task definition lies in the simplicity of the
+BEV layout, we wish to determine whether this enables our
+model to generate new scenes from out-of-domain (OOD)
+HD maps. We use MetaDrive simulator [21] to generate
+random traffic scenarios and their associated BEV layouts
+in simulation, and then input the BEV layouts in our BEV-
+Gen. Generated images are shown in Fig. 7. We can see that
+our model can turn the simulated scenes into realistic street
+images using the BEV layout as a bridge. It has potential to
+address the sim2real gap.
+6. Discussion and Conclusion
+Limitations and Future work.
+Despite the performance
+achieved with sparse attention, future work may benefit
+from use of a bidirectional transformer to allow for paral-
+lel decoding as demonstrated in [4]. We will also explore
+replacing the encoder with a diffusion model to improve the
+image synthesis quality. The proposed model still struggles
+on generating small objects like pedestrians and some vehi-
+cles. We plan to decouple the generation of foreground and
+background to address this issue in the future work.
+In this work we tackle the BEV generation task by de-
+veloping a generative model called BEVGen. After training
+on real-world driving dataset, the proposed model can gen-
+erate spatially consistent multi-view images from a given
+BEV layout. We further show its application on data aug-
+mentation and simulated BEV generation.
+8
+
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+LINKS IN ORTHOPLICIAL APOLLONIAN PACKINGS
+JORGE L. RAM´IREZ ALFONS´IN† AND IV´AN RASSKIN‡
+Abstract. In this paper, we introduce a connection between Apollonian packings and links. We present
+new representations of links embedded in the tangency graph of orthoplicial Apollonian packings and
+show that any algebraic link can be projected onto the tangency graph of a cubic Apollonian packing.
+We use these representations to improve the upper bound on the ball number of an infinite family of
+alternating algebraic links, to reinterpret the correspondence of rational tangles and rational numbers,
+and to find primitive solutions of the Diophantine equation x4 + y4 + z4 = 2t2.
+1. Introduction
+Apollonian packings and their generalizations appear in many different fields of science: in the mod-
+elling of granular systems [AM95], fluid emulsions [Kwo+20], in number theory [Gra+03], etc. In this
+paper, we push further the applications of Apollonian packings into the novel direction of knot theory
+by introducing new representations of links based on a generalization of the classic Apollonian packing.
+1.1. Main results. We begin by proving that any link can be realized in the tangency graph of any
+orthoplicial Apollonian packing (Theorem 3.1). We then focus our attention to algebraic links and show
+that any algebraic link can be regularly projected onto the tangency graph of a cubic Apollonian packing
+(Theorem 4.1). The diagrams arising from the latter construction, called orthocubic representations,
+have the following interesting applications.
+1.1.1. Ball number. A necklace representation of a link L is a sphere packing containing a collection
+of disjoint cycles in its tangency graph realizing L. Necklace representations have been used for the
+study of the volume of hyperbolic 3-manifolds [Gab+21]. The ball number of L, denoted by ball(L),
+is defined as the minimum number of spheres needed to construct a necklace representation of L. It
+is known that ball(22
+1) = 8 and 9 ≤ ball(31) ≤ 12 [Mae07], where 22
+1 denotes the Hopf link and 31 the
+trefoil knot. Nowadays, the Hopf link remains the only link such that its ball number is known. In
+[RR21b], the authors gave a constructive proof showing that for every non-trivial and non-splittable link
+L, ball(L) ≤ 5cr(L) and put forward the following.
+Conjecture 1. For any nontrivial and nonsplittable link L, ball(L) ≤ 4cr(L). Moreover, the equality
+holds if L is alternating.
+Orthocubic representations allow us to show the validity of the inequality in the Conjecture 1 for
+an infinite family of alternating algebraic links (Theorem 4.2), containing the family of rational links,
+alternating Pretzel links or, more generally, alternating Montesinos links (see Figure 1).
+2010 Mathematics Subject Classification. 52C26, 57K10, 11D72.
+Key words and phrases. Apollonian sphere packings, Ball number, Knots, Links, Diophantine equations.
+† Partially supported by grant IEA-CNRS
+‡ Supported by the Austrian Science Fund (FWF), projects F-5503 and P-34763.
+1
+arXiv:2301.03089v1  [math.GT]  8 Jan 2023
+
+Figure 1. (Left) A necklace representation of the “Figure-Eight” knot 41 obtained by the
+method of [RR21b] with 20 spheres, (right) an orthocubic representation of the same knot
+with 16 spheres.
+1.1.2. A new visualization of the slope of rational tangles. It is well-known that rational tangles are in
+correspondence to Q ∪ {∞}. Orthocubic representations allow us to reinterpret this correspondence.
+Indeed, we show that the slope of rational tangle, i.e.
+the corresponding rational number, can be
+obtained from the coordinates of the intersection of an orthocubic representation of the rational tangle
+with a certain circle in a cubic circle packing (Theorem 5.1).
+1.1.3. Primitive solutions of a Diophantine equation. By combining the coordinates of the intersection
+point described above with the Lorentz model of the space of spheres, we shall find infinitely many
+primitive solutions of the Diophantine equation x4 + y4 + z4 = 2t2 (Corollary 5.1).
+1.2. Organization. The paper is organized as follows.
+In section 2, we present the background on
+polytopal Apollonian packings and rational tangles needed throughout the paper. In section 3, we show
+that every link can be embedded in the 1-skeleton of several polytopal Apollonian packings, and discuss
+about the optimality of the orthoplicial case regarding the number of spheres used for the constructions.
+In section 4, we introduce and study the orthocubic representations of rational links and show the
+existence of orthocubic representations of algebraic links. Finally, in Section 5, we discuss a geometric
+visualization of rational tangles as well as its connection with the solutions of Diophantine equations.
+1.3. Acknowledgements. We would like to thank Alex Kontorovich for enlightening conversations on
+several aspects of Apollonian packings.
+2. General background
+In this section, we shall review notions and definitions needed in the rest of the paper. We refer the
+reader to [RR21b; RR21a] for more details.
+2.1. Lorentz model of the space of spheres. An oriented hypersphere (in short, sphere) of �
+Rd is
+the image of spherical cap of Sd under the stereographic projection. Let Ld+1,1 be the Lorentz space of
+dimension d+2, i.e. the real vector space endowed with an inner product of signature (d+1, 1). It is well-
+known that there is a bijection between spheres and points of �
+Rd and vectors of Ld+1,1 with Lorentzian
+norm 1 and 0, respectively. M¨obius transformations of �
+Rd corresponds to linear maps of Ld+1,1 preserving
+the Lorentz product and the time-direction. The inversive coordinates of a sphere (resp. point) are the
+Cartesian coordinates of the corresponding vector in Ld+1,1. There are several equivalent ways (up to
+basis exchange) to compute the inversive coordinates. We use the Wilker’s convention ([Wil81]). For a
+sphere (resp. half-space) b with curvature κ ̸= 0 and center c (resp. κ = 0, normal vector �n and signed
+2
+
+distance to the origin δ) its inversive coordinates are
+i(b) =
+�
+�
+�
+κ
+2 (2c, ∥c∥2 − 1 − κ−2, ∥c∥2 + 1 − κ−2)T
+if κ ̸= 0,
+(�n, δ, δ)T
+if κ = 0.
+(1)
+For points η ∈ �
+Rd, the inversive coordinates are
+i(η)
+=
+�
+�
+�
+(2η, ∥η∥2 − 1, ∥η∥2 + 1)T
+if η ̸= ∞
+(0d, 1, 1)T
+if η = ∞.
+(2)
+Reciprocally, if i(η) = (x1, . . . , xd+2), then η =
+1
+xd+2−xd+1 (x1, . . . , xd).
+We recall that the inversive
+coordinates of points are homogeneous, in the sense that for every λ ̸= 0, λi(η) are valid inversive
+coordinates of the same point of �
+Rd [Wil81]. Under the Wilker’s convention, the matrix of the Lorentz
+product is the diagonal matrix Qd+2 with diagonal entries (1, . . . , 1, −1), and M¨obius transformations
+are represented by the group of matrices
+O↑
+d+1,1(R) = {M ∈ GLd+2(R) | MT Qd+2M = Qd+2
+and
+Md+2,d+2 > 0}.
+(3)
+The matrix of O↑
+d+1,1(R) representing the inversion sb through the boundary of a sphere b is given by
+Sb := Id+2 − 2i(b)T i(b)Qd+2
+(4)
+where Id+2 is the identity matrix of size d + 2.
+2.2. Polytopal Apollonian packings. A polytopal sphere packing BP in dimension d ≥ 1, is the image,
+up to M¨obius transformations, of the ball-arrangement projection β of an edge-scribed (d + 1)-polytope
+P on �
+Rd. The mapping β sends vertices of P to spheres of BP and the tangency relations are encoded
+by the edges of P. For every 1 ≤ n ≤ d, there is a natural realization of the n-skeleton of P as a CW-
+complex contained in BP, which we call the n-skeleton of BP, and is made by realizing the vertices of P
+as the centers of BP, and then, for every face f of P, taking the convex hull of the centers corresponding
+to the vertices of f. The 1-skeleton of BP corresponds to the natural realization of the tangency graph
+of BP usually called the carrier of the packing [Ste05]. Every polytopal sphere packing admits a dual
+arrangement B∗
+P induced by the ball arrangement projection of the polar of P. The Apollonian group
+A(BP) is the Klenian group generated by the inversions through the dual spheres of BP, i.e. the spheres of
+B∗
+P. If we add the symmetries of BP to the set of generators, then we obtain the symmetrized Apollonian
+group of BP, denoted by SA(BP). When the interiors of every pair of spheres in P(BP) := A(BP) · BP
+are disjoint, then we obtain an infinite sphere packing that we call polytopal Apollonian packing. This
+class of infinite sphere packings can be seen as a particular case of the crystallographic sphere packings
+introduced in [KN19], where they are called polyhedral packings.
+Remark 1. For d ≥ 2, every polytopal sphere packing and its endowed structures are unique up to M¨obius
+transformations. This can be seen as a consequence of the Mostow Rigidity Theorem [KN19]. In other
+words, any two edge-scribed realizations of a d-polytope are connected by a M¨obius transformation.
+2.3. The hyperoctahedral group. We denoted by T d, Od and Cd, the analogue of the regular tetra-
+hedron, octahedron and cube in dimension d ≥ 2, respectively (we refer to [RR21a; Ras21] for results
+on polytopal sphere packings arising from these polytopes). We recall that, for every d ≥ 2, Od and Cd
+are dual from each other, while T d is self-dual. Among these families of polytopes, two of them are of
+special relevance for this paper: the cube C3 and the hyperoctahedron O4, also called orthoplex. The cor-
+responding polytopal packings induced from these two polytopes are called cubic packings BC3 [Sta15] and
+orthoplicial packings BO4 [Nak14]. We shall index the elements by an antipodal labelling, where sphere
+bi and bi correspond to antipodal vertices in the polytope, and we shall use the bar notation ¯i := −i.
+The vertices of Od will be labelled by {1, . . . , d, 1, . . . , d}, where the facets are the (d − 1)-simplices with
+vertices {±1, . . . , ±d}. Since facets of Od corresponds to vertices of Cd, we shall label each vertex of
+Cd by the concatenation of the labelling of the vertices in Od incident to the corresponding facet. The
+symmetry group of Od (or equivalently Cd) is called the hyperoctahedral group, which corresponds to the
+Coxeter group Bd. Under the antipodal labelling, the hyperoctahedral group is generated by the signed
+permutations rij := (ij)(ij).
+3
+
+2.4. Apollonian sections. An Apollonian section of P(BP) is a subset S (BP) = Γ · X ⊂ P(BP)
+where Γ < SA(BP) and X ⊂ BP. Two Apollonian sections S (BP) = Γ · X and S (BQ) = Γ′ · X′ of two
+different Apollonian packings are said to be algebraically equivalent if Γ and Γ′ are isomorphic and there
+is an equivariant bijection φ : S (BP) → S (BQ) with respect to the actions. With this notion in the
+hand, the second author proved in [Ras21] that any orthoplicial Apollonian packing P(BO4) contains
+a tetrahedral ST 3(BO4), octahedral SO3(BO4) and cubic section SC3(BO4), i.e. an Apollonian section
+which is algebraically equivalent to a tetrahedral P(BT 3), octahedral P(BO3) and cubic Apollonian
+packing P(BC3), respectively. We shall use a cubic section SC3(BO4) as a geometric framework for the
+constructions introduced in section 4.
+2.5. Algebraic links. A 2-tangle (in short tangle) is a pair (U, t) where U is a compact set of R3
+homeomorphic to a 3-ball and t is a collection {γ1, γ2, . . . , γm} of m ≥ 2 disjoint arcs contained in U
+satisfying that γ1 and γ2 are open arcs whose endpoints lie on the boundary of U, and the rest of the
+arcs are closed. Two tangles (U, t) and (U′, t′) are said to be equivalent if there is an isotopy of R3
+carrying U to U′, t to t′ and the endpoints of (U, t) to the endpoints of (U′, t′). We shall denote this
+equivalence relation t ≃ t′. Up to equivalence, we may consider that the endpoints of t lie on a same
+plane H. A tangle diagram of (U, t) is a regular projection of t on H, together with U ∩ H and the
+crossing information. If it is not required, we shall refer to a tangle (U, t) by t. We shall name the
+endpoints in a tangle diagram by the cardinal points NE, NW, SE and SW. The elementary tangles t0,
+t1 and t∞ are the tangles illustrated in Figure 2.
+NE
+NW
+SW
+SE
+t0
+NE
+NW
+SW
+SE
+t1
+NE
+NW
+SW
+SE
+t∞
+Figure 2. The elementary tangles.
+For any two tangles t and t′, we have the following operations:
+(i) the sum t + t′, obtained by connecting the East endpoints of t to the West endpoints of t′,
+t′
+t
+t + t′
+Figure 3. Sum of tangles.
+(ii) the mirror −t: the image of t under the reflection on the plane containing the equator,
+(iii) the flip F(t): the image of t under the reflection on the plane perpendicular to the equator and
+passing through the endpoints SW and NE,
+(iv) the positive half-twist H+ : t �→ t1 + t,
+(v) the negative half-twist H− : t �→ −t1 + t.
+t
+−t
+F(t)
+H+(t)
+H−(t)
+Figure 4. Mirror, flip and half-twist operations of tangles.
+4
+
+Rational tangles were introduced by Conway in his work on enumerating and classifying knots and
+links [Con70]. For a given sequence of integers a1, . . . , an all non-zero except maybe a1, we denote by
+t(a1, · · · , an) the rational tangle given by the following Conway’s algorithm [Cro04] (see Figure 5).
+t(a1, · · · , an) := Ha1F · · · HanF(t∞).
+(5)
+H−3F
+t∞
+H−2F
+t(−3)
+H2F
+t(−2, −3)
+t(2, −2, −3)
+Figure 5. The rational tangle t(2, −2, −3) obtained by the Conway’s algorithm.
+The slope of a rational tangle t(a1, . . . , an) is the rational number p/q obtained by the continued
+fraction expansion
+[a1, . . . , an] := a1 +
+1
+... +
+1
+an
+= p
+q .
+(6)
+The origin of the name of rational tangle came from the connection established by the Conway’s theorem
+[Con70], between the family of tangles produced by the Conways’s algorithm and rational numbers,
+which states that two rational tangles are equivalent if and only if they have the same slope. We shall
+denote by tp/q the class of rational tangles with slope p/q up to isotopy. The closure of a tangle (U, t) is
+the link formed by joining the endpoints by two disjoint and unlinked paths at the exterior of U. Up to
+equivalence, there are two possible closures, the numerator N(t), obtained by joining the northern and
+the southern endpoints separately, and the denominator D(t), obtained by joining the western and the
+eastern endpoints (see Figure 6).
+D
+N
+t
+t
+t
+Figure 6. The tangle closures.
+A rational link is the closure of a rational tangle. Algebraic tangles are those obtained by sums and
+flips of rational tangles [Ada94]. Equivalently, links which are obtained by the closure of algebraic tangles
+are said to be algebraic or arborescent [GT86]. Pretzel links P(q1, . . . , qn) := N(t1/q1 + · · · + t1/qn) are
+a particular case of algebraic links, see Figure 7.
+Figure 7. The Pretzel knot P(3, −2, 3) which corresponds to the knot 819 in the Alexander-
+Briggs notation.
+5
+
+3. Necklace representations in polytopal Apollonian packings
+In this section, we investigate the following question: given a link L and a polytopal Apollonian sphere
+packing P(BP4), can we find a necklace representation of L contained in P(BP4)? We answer positively
+this question for some 4-polytopes. We first have the following
+Theorem 3.1. Let L be a link and let P(BO4) be an orthoplicial Apollonian sphere packing. There is
+a necklace representation of L contained in P(BO4).
+Let us first introduce a previous notion. Let P(BP) be a polytopal Apollonian sphere packing, where
+P is an edge-scribed (d + 1)-polytope. For every edge {i, j} ∈ P, we define the edge-figure section of
+P(BP) as the Apollonian section Sij(BP) := Γij · BP where Γij is the stabilizer subgroup of the Apollo-
+nian group of P for {bi, bj}. The subgroup Γij corresponds to a Euclidean reflection group. Indeed, we
+may apply an inversion to BP through a sphere centered at the tangency point of bi and bj mapping these
+two spheres into two parallel half-spaces tangent at the infinity. We then observe that every generator
+in Γij must be a reflection on a hyperplane orthogonal to bi and bj (see Figure 9, left).
+Proof of Theorem 3.1. Let B12
+O4 the orthoplicial packing depicted in Figure 8. The edge-figure section
+S12(B12
+O4) := Γ12 ·B12
+O4 is generated by the action of the parabolic subgroup of the orthoplicial Apollonian
+group Γ12 := ⟨s1234, s1234, s1234, s1234⟩.
+B12
+O4
+κ (δ if κ = 0)
+c (�n if κ = 0)
+i(b)T
+b1
+0 (1)
+0
+0
+1
+0
+0
+1
+1
+1
+b2
+0 (1)
+0
+0
+−1
+0
+0
+−1
+1
+1
+b3
+1
+1
+1
+0
+1
+1
+0
+0
+1
+b4
+1
+−1
+1
+0
+−1
+1
+0
+0
+1
+b1
+2
+0
+0
+−1/2
+0
+0
+−1
+−1
+1
+b2
+2
+0
+0
+1/2
+0
+0
+1
+−1
+1
+b3
+1
+−1
+−1
+0
+−1
+−1
+0
+0
+1
+b4
+1
+1
+−1
+0
+1
+−1
+0
+0
+1
+3
+4
+1
+2
+2
+1
+4
+3
+Figure 8. The orthoplicial packing B12
+O4.
+We notice that the 1-skeleton of S12(B12
+O4) contains an infinite square-grid, with two vertices lying in
+the orthogonal line to each square and connected to every corner (see Figure 9).
+3
+4
+4
+3
+2
+s1234
+s1234
+s1234
+s1234
+1
+2
+Figure 9. (Left) B12
+O4 with the mirrors of the generators of Γ12, view from above; (right)
+S12(B12
+O4) with its 1-skeleton.
+6
+
+The well-known Alexander’s Theorem [Ale23] implies that there is a braid γ such that its closure is
+isotopically equivalent to L. We can always draw a diagram of γ in a regular square-grid, where the
+crossings are drawn at the intersections of the diagonals of the squares, and the rest of arcs use the edges
+of the grid, as in Figure 10 (center). This square-grid diagram induces a polygonal closed path in the
+1-skeleton of S12(B12
+O4), as in Figure 10 (right), which gives us a necklace representation NL ⊂ P(B12
+O4).
+Since M¨obius transformations preserving the orientation are ambient isotopies of �
+R3 then, by Remark
+1, we have that there is a M¨obius transformation µ carrying P(B12
+O4) to P(BO4) and NL to a necklace
+representation of L contained in P(BO4).
+□
+Figure 10. (Left) A diagram of the trefoil obtained as a closed braid; (center) a square-
+grid diagram of the same closed braid; (right) a necklace representation of the trefoil in
+S12(B12
+O4).
+We wonder if Theorem 3.1 can be proved without invoking Alexander’s Theorem. The construction
+used in the proof of above can be used to show the inequality in the Conjecture 1 for 2-braid links.
+Corollary 3.1. For any 2-braid link L, we have that ball(L) ≤ 4cr(L).
+Proof. The necklace representation induced by the square-grid diagram of an alternating 2-braid with n
+crossings has 4n + 2 spheres (see Figure 11 (left)). For the closure, we can exchange the last 4 spheres
+with the two half-spaces of S12(B12
+O4) (Figure 11 (right)).
+□
+Figure 11. (Left) A necklace representation of the 2-braid of 4 crossings in the square-grid
+section, with 18 spheres; (right) a necklace representation of the closure of the 2-braid of 4
+crossings, in the square-grid section, with 16 spheres.
+The upper bound of Corollary 3.1 cannot be extended to n-braid links when n ≥ 3. The main reason
+is that the half-spaces of the square-grid section cannot be used to close all the strands of the braid.
+The latter might increase the number of spheres to more than 4 times the number of crossings.
+A
+similar strategy as used in the proof of Theorem 3.1 can be employed to prove that every link admits
+a necklace representation in other polytopal Apollonian sphere packings P(BP4). For instance, if P4
+has a regular triangle as edge-figure, then the 1-skeleton of the edge-figure section contains a subgraph
+topologically equivalent to a triangular grid. In this case, two tangent triangles in the triangular grid
+made up a rhombus which can play the same role as the square in the square-grid. Indeed, if there is a
+7
+
+chain of spheres connecting the opposite vertices in the great diagonal of the rhombus, then we can use
+them to construct a crossing. It turns out that this is the case for the 4-simplex, hypercube, 24-cell or
+the 120-cell (see Figure 12). Although these triangular constructions produce necklace representations
+with more spheres that the orthoplicial one, these could be interesting for other issues like constructing
+4-polytopes containing a given link in its graph [Epp14].
+Figure 12. A necklace representation of the trefoil knot in P(BT 4) (left) and P(BC4)
+(right).
+4. Orthocubic representations of algebraic links
+Let BC3 and BO4 be the cubic and the orthoplicial packing given in Figures 13 and 14, respectively.
+We point out that the labelling of BO4 has been given in such a way that for every bi ∈ BO4, the label i
+is positive if and only if the third coordinate of the center of bi is positive.
+BC3
+κ
+c
+i(b)T
+b123
+1 +
+√
+2 (−1 +
+√
+2) (
+1 −1) ( 1 −1 −1
+√
+2)
+b123
+1 +
+√
+2 (−1 +
+√
+2) (−1
+1) (−1
+1 −1
+√
+2)
+b123 −1 +
+√
+2 (
+1 +
+√
+2) (−1 −1) (−1 −1
+1
+√
+2)
+b123 −1 +
+√
+2 (
+1 +
+√
+2) (
+1
+1) ( 1
+1
+1
+√
+2)
+b123 −1 +
+√
+2 (
+1 +
+√
+2) (−1
+1) (−1
+1
+1
+√
+2)
+b123 −1 +
+√
+2 (
+1 +
+√
+2) (
+1 −1) ( 1 −1
+1
+√
+2)
+b123
+1 +
+√
+2 (−1 +
+√
+2) (
+1
+1) ( 1
+1 −1
+√
+2)
+b123
+1 +
+√
+2 (−1 +
+√
+2) (−1 −1) (−1 −1 −1
+√
+2)
+123
+123
+123
+123
+123
+123
+123
+123
+Figure 13. The cubic packing BC3.
+BO4
+κ
+c
+i(b)T
+b1
+1 + 1/
+√
+2 (−1 +
+√
+2) (
+1 −1
+1) 1/
+√
+2 (
+1 −1
+1 −1
+√
+2)
+b2
+1 + 1/
+√
+2 (−1 +
+√
+2) (−1
+1
+1) 1/
+√
+2 (−1
+1
+1 −1
+√
+2)
+b3
+1 − 1/
+√
+2 (
+1 +
+√
+2) (−1 −1
+1) 1/
+√
+2 (−1 −1
+1
+1
+√
+2)
+b4
+1 − 1/
+√
+2 (
+1 +
+√
+2) (
+1
+1
+1) 1/
+√
+2 (
+1
+1
+1
+1
+√
+2)
+b1
+1 − 1/
+√
+2 (
+1 +
+√
+2) (−1
+1 −1) 1/
+√
+2 (−1
+1 −1
+1
+√
+2)
+b2
+1 − 1/
+√
+2 (
+1 +
+√
+2) (
+1 −1 −1) 1/
+√
+2 (
+1 −1 −1
+1
+√
+2)
+b3
+1 + 1/
+√
+2 (−1 +
+√
+2) (
+1
+1 −1) 1/
+√
+2 (
+1
+1 −1 −1
+√
+2)
+b4
+1 + 1/
+√
+2 (−1 +
+√
+2) (−1 −1 −1) 1/
+√
+2 (−1 −1 −1 −1
+√
+2)
+4
+1
+3
+2
+4 1
+3
+2
+Figure 14. The orthoplicial packing BO4.
+Let SC3(BO4) := ΓC3 · BO4 be a cubic Apollonian section of P(BO4), where
+ΓC3 := ⟨s1234, s1234, s1234, s1234, s1234, s1234⟩.
+8
+
+The equivariant bijection φ : P(BC3) → SC3(BO4) is induced by the following isomorphisms (see Figure
+15).
+A(BC3)
+−→
+ΓC3
+BC3
+−→
+BO4
+s±1
+�→
+s±(1234)
+b±(123)
+�→
+b±1
+s±2
+�→
+s±(1234)
+b±(123)
+�→
+b±2
+s±3
+�→
+s±(1234)
+b±(123)
+�→
+b±3
+b±(123)
+�→
+b±4
+4
+1
+3
+2
+4
+1
+3
+2
+1234
+1234
+1234
+1234
+1234
+1234
+123
+123
+1
+2
+3
+2
+1
+123
+123
+123
+123
+123
+123
+3
+Figure 15. (Right) the cubic packing BC3 with its dual, (left) BO4 with the mirrors of the
+generators of the cubic section.
+An alternative geometric way to obtain the bijection between the cubic section and cubic Apollonian
+packing results by taking the intersection of BO4 and its dual with the XY -plane. The relative position
+of the centers of the spheres in the cubic section, with respect to the XY -plane, induces a 2-coloring
+of the cubic Apollonian packing where two disks of same color never intersect (see Figure 16). We call
+this coloring the z-coloring. By extending the z-coloring to the vertices of the 1-skeleton of P(BC3), we
+obtain a proper 2-coloring of the tangency graph.
+Figure 16. (Left) P(BC3) with the z-coloring, (right) SC3(BO4) with the XY -plane.
+9
+
+On the same direction as Theorem 3.1, we present the following result allowing us to prove the
+inequality of the Conjecture 1 for an infinite family of alternating algebraic links (containing, in particular,
+2-braid links).
+Theorem 4.1. For any algebraic link L, there is a necklace representation of L contained in SC3(BO4).
+4.1. Orthocubic shifts. Let BO3 be the octahedral packing which is the dual arrangement of the cubic
+packing BC3. The former can be also obtained by intersecting the dual arrangement of BO4 with the
+XY -plane. Let us consider the symmetries r12, r13, r23, r13, r23, r33, of BO3. By duality, these are also
+symmetries of BC3. We recall that rij denotes the signed permutation (ij)(ij). In the octahedral packing
+BO3, we have that r12 corresponds to the reflection on the line {x = y}, r±13 is the inversion through
+the circle centered at (±1, 0) and radius
+√
+2, and r33 is the inversion through the unit circle centered at
+the origin (see Figure 17).
+123
+123
+r13
+r23
+r23
+r13
+r33
+r12
+123
+123
+123
+123
+123
+123
+3
+1
+2
+3
+2
+1
+Figure 17. BC3 with the mirrors of the generators of the cubic shifts.
+We define the cubic shifts as the following six elements belonging to the symmetrized Apollonian
+group of BC3
+µi := siri3
+for every i ∈ {±1, ±2, −3}
+and
+µ3 := s3r33.
+(7)
+In Figure 18, we show the action of the cubical shifts on the 1-skeleton of BC3 with the z-coloring. We
+notice that µ±1 and µ±2 (resp. µ±3) preserves (resp. reverses) the z-coloring. The bijection φ : BC3 →
+BO4 induces the following morphisms:
+φ :
+Sym(BC3)
+−→
+Sym(BO4)
+φ:
+SA(BC3)
+−→
+SA(BO4)
+r12
+�−→
+r12
+µ1
+�−→
+s1234 r13
+r13
+�−→
+r13
+µ−1
+�−→
+s1234 r24
+r23
+�−→
+r23
+µ2
+�−→
+s1234 r23
+r13
+�−→
+r24
+µ−2
+�−→
+s1234 r14
+r23
+�−→
+r14
+µ3
+�−→
+s1234 r12r34
+r33
+�−→
+r12r34
+µ−3
+�−→
+s1234 r12r34
+For every i = ±1, ±2, ±3, we call the elements φ(µi) ∈ SA(BO4) the orthocubic shifts.
+10
+
+µ1
+µ2
+µ−1
+µ−2
+µ3
+µ−3
+Figure 18. The action of the cubic shifts on the 1-skeleton of BC3 with the z-coloring.
+4.2. Orthocubic coordinates. The cubic Apollonian packing P(BC3) can be seen as a Coxeter system
+(W, S) where W = A(BC3) and system of generators S = {s±1, s±2, s±3}. Its Coxeter graph is the graph
+of the cube with ∞ label at each edge. Therefore, the reduced words of (W, S) are the words without
+consecutive repeated letters. We have that for each b ∈ SC3(BO4), there is a reduce word of w = sj1 · · · sjn
+and an element bi ∈ BC3 such that b = w · bi. The depth of b is the minimal length of w in terms of the
+generators. By combining the reduced words of (W, S) with the bijection φ : P(BC3) → SC3(BO4) we
+can give a coordinate system to the spheres is the cubic section. We define the orthocubic coordinates of
+every b ∈ SC3(BO4) as the label
+ij1···jn := φ(sj1) · · · φ(sjn) · bi = b
+(8)
+where i ∈ {±1, ±2, ±3, ±4} and jl ∈ {±1, ±2, ±3}. In Figure 19, we show the orthocubic coordinates of
+the elements of SC3(BO4) with depth ≤ 1.
+11
+
+33
+23
+43
+13
+4
+1
+3
+2
+3
+2
+4
+1
+21
+11
+31
+41
+12
+22
+32
+42
+31
+41
+21
+11
+32
+42
+12
+22
+43
+13
+33
+23
+Figure 19. The orthocubic coordinates of the elements of SC3(BO4) of depth≤ 1.
+4.3. Orthocubic representations. We define an orthocubic path γ as a polygonal curve in the 1-
+skeleton of SC3(BO4). A cubic diagram of γ will be its orthogonal projection on the XY -plane. The
+orthogonal projection of the 1-skeleton of SC3(BO4) on the XY -plane is the 1-skeleton of P(BC3) plus
+the diagonal edges of each square-face, which join two vertices of same color under the z-coloring. The
+crossings of any cubic diagram are obtained by the intersection of the two diagonal edges of a same square-
+face. With the information given by the z-coloring, the over/under crossing information can be deduced
+from the color of the vertices of the diagonal edges (black=over/white=under). We define an orthocubic
+representation of a link L as a collection of disjoint closed orthocubic paths isotopically equivalent to L.
+Every orthocubic representation induces a necklace representation in SC3(BO4). In Figure 20, we show
+an orthocubic representation of the trefoil knot, and its corresponding cubic diagram.
+Figure 20. (Left) An orthocubic representation of the trefoil knot and its corresponding
+cubic diagram (right).
+12
+
+An orthocubic path will be encoded by a sequence of the orthocubic coordinates �iw1, · · · , iwn� of
+the elements given in the linear order induced by γ. Since we shall consider unoriented paths, and the
+concatenation of two paths gives another path, vectors encoding orthocubic paths must be quotient by
+the following relations:
+(i) (Symmetry) �iw1, · · · , iwn� = �iwn, . . . , iw1�.
+(ii) (Concatenation) {�iw1, · · · , iwn�, �iwn, · · · , iwm�} = {�iw1, · · · , iwn, · · · , iwm�}.
+4.4. Orthocubic tangles. Let T be the tetrahedron in the 3-skeleton of BO4 with vertices {1, 2, 3, 4}.
+We define an orthocubic tangle as a tangle (T , t ) where t is a collection {γ1, γ2, . . . , γm} of m ≥ 2
+disjoint orthocubic paths contained in T satisfying that the endpoints of γ1 and γ2 lie in the corners of
+T , and the rest of the orthocubic paths are closed. In what follows, we construct the respective analog
+of the elementary tangles, sum, mirror, flip and half-twists for orthocubic tangles by using elements of
+the symmetrized Apollonian group of BO4.
+(i) The orthocubic elementary tangles:
+t0
+:= {�1, 4�, �3, 2�},
+t1
+:= {�1, 2�, �3, 4�} and t∞
+:=
+{�1, 3�, �2, 4�}.
+4
+1
+3
+2
+t0
+4
+1
+3
+2
+t1
+4
+1
+3
+2
+t∞
+Figure 21. The elementary orthocubic tangles.
+(ii) The orthocubic flip FO t := r12 t , where r12 ∈ Sym(BO4) acting as the reflection on the plane
+{y = x} in R3.
+(iii) The orthocubic mirror − t := φ(µ−3) t ∪ {�1, 2�, �1, 2�, �3, 4�, �3, 4�}.
+4
+1
+2
+3
+t
+t
+4
+1
+2
+3
+t
+FO t
+4
+1
+2
+3
+3
+2
+1
+4
+t
+− t
+Figure 22. The orthocubic flip and mirror.
+(iv) The orthocubic sum t′ + t := φ(µ−1) t′ ∪ {�1, 4�, �2, 3�} ∪ φ(µ1) t .
+(v) The orthocubic half-twists H+
+O t := t1 + t and H− t := − t1 + t .
+13
+
+t′
+t
+4
+1
+2
+3
+3
+2
+1
+4
+t′ + t
+4
+1
+2
+3
+3
+2
+1
+4
+t
+H+
+O t
+4
+1
+2
+3
+3
+2
+1
+4
+t
+H−
+O t
+Figure 23. The orthocubic sum and half-twists.
+We define the orthocubic tangle closures by:
+(vi) The orthocubic numerator NO t := t ∪ {�1, 23, 33, 4�, �2, 13, 43, 3�}
+(vi) The orthocubic denominator DO t := t ∪ {�1, 23, 43, 3�, �2, 13, 33, 4�}
+DO
+NO
+t
+4
+1
+2
+3
+t
+33
+23
+13
+43
+4
+1
+2
+3
+t
+33
+23
+13
+43
+4
+1
+2
+3
+Figure 24. The orthocubic tangle closures.
+Since the orthocubic elementary tangles, operations and closures are isotopically equivalent to their
+homonym in the classic framework of tangles, we can mimic the Conway’s method to define an orthocubic
+rational tangle tO[a1, · · · , an] ≃ t[a1, · · · , an], by
+tO[a1, · · · , an] := Ha1
+O FO · · · Han
+O FO t∞
+(9)
+We have now all the elements to prove the Theorem 4.1.
+Proof of Theorem 4.1. Every rational tangle admits a necklace representation in S (BO4), via the or-
+thocubic version of Conway’s construction. By combining the latter with the orthocubic tangle operations
+we obtain that any algebraic link admits an orthocubic representation.
+□
+4.5. Improvement of the upper bound of the ball number. The orthocubic Conway’s algorithm
+can be slightly adapted in order to improve the get the upper bound of Theorem 4.2. For every a1 ≥ 0,
+a2, . . . , an > 0, we define the reduced orthocubic Conway’s algorithm �tO[a1, · · · , an] by
+�tO[a1, · · · , an] := Ha1
+O FO · · · Han−1
+O
+t1
+(10)
+Clearly, for every a1 ≥ 0, a2, . . . , an > 0, we have tO[a1, · · · , an] ≃ �tO[a1, · · · , an].
+Theorem 4.2. Let L be an algebraic link obtained by the closure of the algebraic tangle
+tp1/q1 + · · · + tpm/qm
+where all the pi/qi have same sign. Then, ball(L) ≤ 4cr(L).
+Proof. Let L be an algebraic link made by the closure N(t) where t is the algebraic tangle
+tp1/q1 + · · · + tpm/qm.
+The condition that all pi/qi have the same sign implies that we have alternating diagram of L induced
+by the closure of t, and thus, by the Tait conjecture on the crossing number of alternating diagrams
+[Kau87; Thi87; Mur87], the crossing number of L is equal to the sum of the crossing numbers of each
+14
+
+tpi/qi. Without loss of generality, we can consider that all pi/qi are positive. For every pi/qi with positive
+continued fraction [a1, · · · , an], let tpi/qi := �tO[a1, · · · , an]. Since the FO does not change the necklace
+length, and H+
+O increases the necklace length by 4, we have that
+| tpi/qi | = 4(a1 + . . . + an − 1) + | t1 |
+= 4(a1 + . . . + an) = 4cr(tpi/qi)
+Let t be the orthocubic tangle made by the orthocubic sums tp1/q1 +· · ·+ tpm/qm . By the equivalence
+between the orthocubic and tangle operations we have that t ≃ t. Since the necklace length is additive
+for the sum,
+| t | = | tp1/q1 | + · · · + | tpm/qm |
+= 4cr(tp1/q1) + · · · + 4cr(tpm/qm)
+= 4cr(L).
+Finally, we notice that the exterior orthocubic paths �1, 4� and �2, 3� are not included in any orthocubic
+tangle obtained after applying an orthocubic sum. Therefore, we can use the exterior paths to close t ,
+and in this way obtain a necklace representation of L with 4cr(L) spheres.
+□
+4.6. No tightness for non-alternating links. The family of algebraic links considered in Theorem
+4.2 contains all the rational links and other well-known families as the Montesinos links with positive
+coefficients. These are the links obtained by the closure of
+tp1/q1 + · · · + tpn/qn + tr
+with pi/qi > 0 and r ≥ 0. If r = 0 and every pi = 1, then we obtain the Pretzel link P(q1, . . . , qn).
+In the non-alternating case, it is possible to construct orthocubic algebraic tangles with necklace length
+strictly less than 4 times the crossing number. The first non-trivial example that we have found satisfying
+this property, is the Pretzel knot P(3, −2, 3), which corresponds to the knot 819 in the Alexander-Briggs-
+Rolfsen notation. This knot is not alternating [Cro04] and it admits an orthocubic necklace representation
+with 28 spheres (= 3
+2cr(819), see Figure 25). However, it becomes more tricky to establish a relation
+with the crossing number in the non-alternating case since, in general, the crossing number does not
+correspond to the sum of the crossings of its rational factors.
+Figure 25. An orthocubic representation of the knot 819 with 28 spheres (left) and its
+cubic diagram (right).
+15
+
+5. A new visualization of the slope of rational tangles
+The slope p/q of a rational tangle tp/q can be identified with the slope of the meridian of a solid torus
+that is the branched double covering of a rational tangle [Cro04]. We shall present a new geometric in-
+terpretation of the correspondance between rational tangles and rational numbers. We do so by relating
+the slope of a tangle with the slope of the line passing through the origin and the last tangency point
+in the orthocubic Conway’s construction. Astonishingly, this approach turns out to be helpful to find
+infinitely many primitive solutions of the Diophantine equation x4 + y4 + z4 = 2t2.
+Let p/q be a positive fraction with positive continued fraction expansion [a1, · · · , an]. We define the
+orthocubic point ηp/q of the rational tangle tp/q as the tangency point of the two disks in the cubic
+diagram of tO(a1, · · · , an) corresponding to the last edge of the orthocubic tangle. By last edge, we mean
+the edge connecting the disk in the upper-right corner (see Figure 26). We point out that the disk in
+the upper-right corner corresponds to the sphere b123 ∈ BC3 which remains fixed under the orthocubic
+Conway’s algorithm. We can naturally extend the notion of orthocubic point to tangles with negative
+fractions, by applying a reflection through the plane {x = 0} to the whole setting.
+Theorem 5.1. For every p/q ∈ Q± ∪ {∞}, ηp/q is the first intersection of the line passing through the
+origin and having slope ±(p/q)−2, with the boundary of the disk b±123 ∈ BC3.
+Proof. It is enough to prove the positive case. Let p ≥ 0 and q ≥ 1 be two coprime integers. We claim
+that
+i(ηp/q) =
+�
+�
+�
+�
+p2
+q2
+(p − q)2
+√
+2(p2 − pq + q2)
+�
+�
+�
+� .
+(11)
+This would imply that the Cartesian coordinates of ηp/q are
+1
+√
+2pq − (1 −
+√
+2)(p − q)2 (p2, q2),
+which is exactly the first point of intersection of the line {p2y = q2x} and the circle centred at
+(1 +
+√
+2, 1 +
+√
+2) and radius (1 +
+√
+2), which is the boundary of b123 ∈ BC3.
+Let us prove the equality (11).
+The positiveness of p and q implies that we can find a positive
+continued fraction expansion [a1, · · · , an] = p/q with a1 ≥ 0 and ai ≥ 1 for every 1 < i ≤ n. Let
+tp/q
+the orthocubic tangle tO[a1, . . . , an]. Let ηp/q and η∞ be the orthocubic points of tp/q and t∞,
+respectively. Now, by the definitions of the orthocubic operations HO and FO, the isomorphism φ :
+SA(BC3) −→ SA(BO4) and the definition of orthocubic rational tangles given in (9), we have that
+tp/q = Ha1
+O FO · · · Han
+O FO t∞ ⇒ ηp/q = µa1
+1 r12 · · · µan
+x r12(η∞)
+= (s1r13)a1r12 · · · (s1r13)anr12(η∞)
+where s1, r13 and r12 are the elements of SA(BC3) described in subsection 4.1. The inversive coordinates
+of η∞ and the matrices representing s1, r13 and r12 can be computed by using the equations (2) (with
+λ = 1+
+√
+2
+2
+) and (4), giving
+i(η∞) =
+�
+�
+�
+1
+0
+1
+√
+2
+�
+�
+�,
+s1 �→ S1 =
+�
+�
+�
+−3
+0
+0
+2
+√
+2
+0
+1
+0
+0
+0
+0
+1
+0
+−2
+√
+2
+0
+0
+3
+�
+�
+�,
+r13 �→ R13 =
+�
+�
+�
+0
+0
+1
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+0
+0
+1
+�
+�
+�,
+r12 �→ R12 =
+�
+�
+�
+0
+1
+0
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+1
+�
+�
+�.
+16
+
+Let M(k) := (S1R13)kR12. By induction on k, it can be found that
+M(k) =
+�
+�
+�
+�
+0
+1 − k2
+−k(k + 2)
+√
+2k(k + 1)
+1
+0
+0
+0
+0
+−k(k − 2)
+1 − k2
+√
+2k(k − 1)
+0
+−
+√
+2k(k − 1)
+−
+√
+2k(k + 1)
+2k2 + 1
+�
+�
+�
+�
+We will finally prove the equality (11) by induction on the number of coefficients n in the fraction
+expansion of p/q. For n = 1 (that is p = a1 and q = 1) we have
+i(ηa1) = M(a1)
+�
+�
+�
+1
+0
+1
+√
+2
+�
+�
+� =
+�
+�
+�
+a2
+1
+1
+(a1 − 1)2
+√
+2(a2
+1 − a1 + 1)
+�
+�
+�.
+We suppose equality (11) to be true for n − 1 ≥ 1. Let r/s = a2 +
+1
+···+ 1
+an . Then,
+i(ηp/q) = M(a1)M(a2) · · · M(an)
+�
+�
+�
+�
+1
+0
+1
+√
+2
+�
+�
+�
+� = M(a1)
+�
+�
+�
+�
+r2
+s2
+(r − s)2
+√
+2(r2 − rs + s)
+�
+�
+�
+�
+=
+�
+�
+�
+�
+(ra1 + s)2
+r2
+(ra1 + s − r)2
+√
+2((ra1 + s)2 − r(ra1 + s) + r2)
+�
+�
+�
+�
+We finally notice that
+ra1 + s
+r
+= a1 + s
+r = a1 +
+1
+r/s = a1 +
+1
+a2 +
+1
+···+ 1
+an
+= p
+q
+and therefore, equality (11) holds.
+□
+Corollary 5.1. The Diophantine equation
+x4 + y4 + z4 = 2t2
+(12)
+has an infinite number of primitive solutions.
+Proof. Since points of �
+R2 correspond to light-like vectors of L3,1, we can use the inversive coordinates
+of the orthocubic point of every rational tangle given in equation (11) to produce primitive solutions of
+the Diophantine equation by taking
+x = p,
+y = q,
+z = p − q,
+t = p2 − pq + q2.
+(13)
+□
+We hope and expect the above approach to be helpful to investigate solutions of other type of Dio-
+phantine equations.
+17
+
+32y = 22x
+η3/2
+Figure 26. The orthocubic point (red) of the rational tangle t3/2 corresponding to the
+primitive solution 34 + 24 + 14 = 2 × 72.
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+19
+

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@@ -0,0 +1,945 @@
+Draft version January 9, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Simulations of high-redshift [OIII] emitters: Chemical evolution and multi-line diagnostics
+Yurina Nakazato,1 Naoki Yoshida,1, 2, 3 and Daniel Ceverino4, 5
+1Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
+2Kavli Institute for the Physics and Mathematics of the Universe (WPI), UT Institute for Advanced Study, The University of Tokyo,
+Kashiwa, Chiba 277-8583, Japan
+3Research Center for the Early Universe, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
+4Universidad Autonoma de Madrid, Ciudad Universitaria de Cantoblanco, E-28049 Madrid, Spain
+5CIAFF, Facultad de Ciencias, Universidad Autonoma de Madrid, E-28049 Madrid, Spain
+ABSTRACT
+Recent observations by James Webb Space Telescope discovered a number of high-redshift galaxies
+with strong emission lines from doubly ionized oxygen. Combined with ALMA observations of far-
+infrared lines, multi-line diagnostics can be applied to the high-redshift galaxies in order to probe the
+physical conditions of the inter-stellar medium. We study the formation and evolution of galaxies
+using the FirstLight simulation suite, which provides outputs of 62 high-resolution, zoom-in galaxy
+simulations.
+We devise a physical model of Hii regions and calculate spatially resolved [Oiii] line
+emission. We show that massive galaxies with stellar masses of M∗ > 109M⊙ chemically evolve rapidly
+to z = 9. Young stellar populations in the star-forming galaxies boost the [Oiii] line emission, rendering
+the ratio of line luminosity to star formation rate larger than that for low-redshift galaxies, which is
+consistent with recent observations. Measuring the flux ratios of rest-frame optical and far-infrared
+lines allows us to estimate the physical conditions such as density and metallicity of the star-forming
+gas in high-redshift [Oiii] emitters.
+1. INTRODUCTION
+Understanding the formation and evolution of the first
+galaxies is one of the key scientific goals of new genera-
+tion telescopes including James Webb Space Telescope
+(JWST) and Atacama Large Millimetre/Submillimetre
+Array (ALMA). High-redshift galaxies can be detected
+and identified using strong emission lines, among which
+[Oiii] 88µm line is thought to be promising (Inoue et al.
+2014). A number of galaxies have been found at z > 7 by
+ALMA observations targeting the [Oiii] 88µm line (e.g.
+Inoue et al. 2016; Hashimoto et al. 2018), including the
+most distant galaxy candidate at z = 13.27 with a 4σ
+[Oiii] 88µm detection (Harikane et al. 2022). Since the
+[Oiii] line emission originates from Hii regions around
+young massive stars, it can be used to trace the star for-
+mation activities and also the physical properties of the
+inter-stellar medium (ISM) in the early galaxies.
+JWST is opening a new window into the early universe
+through its superb observational capability in near-
+infrared.
+For example, JWST Early Research Obser-
+Corresponding author: Yurina Nakazato
+[email protected]
+vation (ERO) in the lensing field SMACS 0723 already
+reported three galaxies confirmed spectroscopically by
+NIRSpec (Schaerer et al. 2022; Curti et al. 2022; Heintz
+et al. 2022). NIRSpec instrument is capable of detect-
+ing and identifying various rest-frame optical lines such
+as [Oii] 3727˚A, [Oiii] 4959˚A and [Oiii] 5007˚A. The rela-
+tively weak [Oiii] 4363˚A line has been detected for all the
+three galaxies, enabling us to estimate the ISM metal-
+licity in a direct manner.
+Detailed numerical simulations are indispensable to
+study the physical conditions of the ISM. There have
+been several studies focusing on [Oiii] emission lines
+from high-z galaxies (Hirschmann et al. 2017; Olsen
+et al. 2017; Moriwaki et al. 2018; Katz et al. 2019;
+Arata et al. 2020; Ceverino et al. 2021; Pallottini et al.
+2022). Moriwaki et al. (2018) use a cosmological sim-
+ulation with a large boxsize of 50 Mpc (Shimizu et al.
+2016) to calculate the [Oiii] 88µm line intensities for a
+few hundred galaxies with stellar masses of ∼ 108 M⊙.
+High-resolution, zoom-in simulations have also been per-
+formed to study the internal structure of early galaxies
+(Katz et al. 2019; Arata et al. 2020). For the upcoming
+observations conducted by JWST, it is urgently needed
+to study the population of high-redshift galaxies with
+arXiv:2301.02416v1  [astro-ph.GA]  6 Jan 2023
+
+2
+Nakazato et al.
+high resolution in a fully cosmological context. In this
+Letter, we use the outputs of FirstLight simulation (Cev-
+erino et al. 2017).
+The simulation suite is motivated
+to produce a statistically significant number of galaxies
+with very high resolution at the epoch of reionization.
+Thanks to the mass and volume complete sample of
+more than 60 massive galaxies and to the high-resolution
+of ∼ 20 pc, we can investigate the internal structure as
+well as statistics of the high-redshift galaxies.
+Throughout this Letter, we assume Z⊙ = 0.02 as the
+solar metallicity (Anders & Grevesse 1989).
+2. METHOD
+2.1. Cosmological Simulation
+We use mass-limited galaxy samples selected from the
+FirstLight simulation suite (Ceverino et al. 2017). The
+simulations are performed with ART code (Kravtsov
+et al. 1997; Kravtsov 2003; Ceverino & Klypin 2009;
+Ceverino et al. 2014), which follows gravitational N-
+body dynamics and Eulerian hydrodynamics using an
+adaptive mesh refinement method.
+Besides the two
+processes, the code incorporates astrophysical processes
+relevant for galaxy formation.
+The so-called subgrid
+physics includes atomic and molecular cooling of hydro-
+gen and helium, photoionization heating by a cosmolog-
+ical UV background with partial self-shielding, and star
+formation and the associated stellar feedback. Details
+are described in Ceverino et al. (2017). The simulations
+track metals released from SNe-Ia and from SNe-II, us-
+ing supernovae yields from Woosley & Weaver (1995).
+Our simulated galaxies are hosted by dark matter
+haloes with maximum circular velocity (Vmax) higher
+than 178 km/s at z = 5 in a cosmological volume of 40
+h−1Mpc on a side. The host haloes are selected in a low-
+resolution N-body only simulation, for which refined ini-
+tial conditions are generated using a standard zoom-in
+technique (Klypin et al. 2011). The refinement achieves
+the dark matter particle mass of mDM = 8 × 104 M⊙,
+the minimum star particle mass of 103 M⊙, and the
+maximum spatial resolution is a few tens proper parsec
+depending on the refinement level.
+We calculate the stellar mass distribution for the se-
+lected 62 massive galaxies at z = 9, 8, 7, 6. The max-
+imum stellar mass is 9.5, 9.7, 10.1, 10.7×109 M⊙, re-
+spectively.
+The sample allows us to study the evolu-
+tion of more massive galaxies than in previous simula-
+tions, e.g., Moriwaki et al. (2018), SERRA simulation
+(Pallottini et al. 2022) and S´IGAME simulation (Olsen
+et al. 2017), and thus is well-suited to compare with
+observed massive galaxies by HST, ALMA, and JWST
+(e.g. Tacchella et al. 2022; Graziani et al. 2020; Topping
+et al. 2022; Trussler et al. 2022; Barrufet et al. 2022;
+Leethochawalit et al. 2022).
+2.2. Line emissivity calculation
+We generate emission-line maps for our galaxy sam-
+ples by choosing a region enclosed by 0.3 times the virial
+radius of the host halo as same as Mandelker et al.
+(2014, 2017). We configure a uniform 3D grid with a
+side length of 100 pc. We locate the star particles and
+gas elements within each grid, and store the mass of
+stars younger than 10 Myr, the average density of the
+gas with nH > 0.1 cm−3, and the average metallicity of
+the cold/warm gas with T < 5 × 104 K. These physi-
+cal quantities assigned to the individual grids are then
+used to compute the line emissivities in a similar man-
+ner to those in Hirschmann et al. (2017); Moriwaki et al.
+(2018); Ceverino et al. (2021). We generate a library of
+emission lines using CLOUDY (Ferland et al. 2013).The
+library covers a wide range of gas metallicity Z and ion-
+ization parameter U as given in Table 1.
+The library lists the individual line luminosity, Lline,
+normalized by the Hβ line luminosity calculated with
+the case-B approximation (Dopita & Sutherland 2003),
+LcaseB
+Hβ
+, as
+Lline = (1 − fesc) Cline(Zgas, U, nHii) LcaseB
+Hβ
+,
+(1)
+LcaseB
+Hβ
+= 4πjHβV = hνHβ
+�
+αeff
+Hβ
+αB
+�
+Q,
+(2)
+where fesc is the Lyman continuum escape fraction and
+Cline is the line luminosity ratio. The Hβ emission rate
+per unit volume per unit time per unit solid angle is
+denoted as jHβ, and αeff
+Hβ is an effective recombination
+coefficient, Q is the production rate of ionizing photons
+from each star particle, and αB is the case-B hydrogen
+recombination coefficient given by
+αB = 2.6 × 10−13
+�
+Te
+104 K
+�−0.85
+cm3s−1
+(3)
+with a constant electron temperature Te = 104 K.
+We set fesc = 0.1, which is consistent with previous
+radiative transfer simulations for massive galaxies with
+Mhalo > 1010−11M⊙ (Yajima et al. 2011; Kimm & Cen
+2014; Wise et al. 2014; Paardekooper et al. 2015; Xu
+et al. 2016). It is also consistent with recent observa-
+tional estimates at z ∼ 6 − 8 (Castellano et al. 2017;
+Robertson et al. 2013).
+We note that some galaxies
+have been reported to have an even higher escape frac-
+tion of over 20 percent (e.g. Marques-Chaves et al. 2022;
+Vanzella et al. 2016; Fletcher et al. 2019; Bian & Fan
+2020; Flury et al. 2022) at z < 4.
+
+[Oiii] Luminosity calculation in First Light
+3
+Since individual Hii regions are not resolved in our
+simulations, we resort to a physical model of the ISM
+structure to calculate the line emissivities of Hii regions.
+We characterize the ISM by the local gas density n and
+metallicity Z, and also by a volume-averaged ionization
+parameter
+⟨U⟩ = 3α2/3
+B
+4c
+�3Qϵ2nHii
+4π
+�1/3
+.
+(4)
+Our fiducial model assumes a constant gas density nHii
+in a spherical Hii region surrounding a star particle (see,
+e.g. Panuzzo et al. 2003; Gutkin et al. 2016). We set the
+Hii region density nHii = 100 cm−3 (e.g. Osterbrock &
+Ferland 2006; Hirschmann et al. 2017, 2022). We define
+the volume-filling factor of the gas as
+ϵ = ngas,grid
+nHii
+,
+(5)
+where ngas,grid is the gas number density in each grid.
+In rare cases where the volume-averaged gas density ex-
+ceeds nHii (ϵ > 1), we set the filling factor to unity. Note
+that a larger ngas,grid for a fixed nHii yields a larger fill-
+ing factor ϵ. Hence the resulting line emissivity depends
+only weakly on the assumed nHii in our model. We have
+tested with a few variations with nHii = 50, 300 cm−3,
+and explicitly checked that our main findings in the fol-
+lowing sections are not sensitively affected by this choice.
+log10 (Zgas/Z⊙)
+-1.30, -0.70, -0.40, 0., 0.30
+log10 U
+-4.0, -3.9, ..., -1.1, -1.0
+log10 (nHii/cm−3)
+2.0 (fixed)
+Table 1. The parameters used to calculate the line lumi-
+nosities with CLOUDY.
+We compute the production rate of ionizing photons
+Q of the simulated galaxies using publicly available ta-
+bles from the Binary Population and Spectral Synthesis
+(BPASS) model (Byrne et al. 2022). Our simulations
+adopt a stellar initial mass function represented by bro-
+ken power laws as
+N(M < Mmax) ∝
+� M1
+0.1
+� M
+M⊙
+�α1
+dM + M α1
+1
+� Mmax
+M1
+� M
+M⊙
+�α2
+dM
+(6)
+with α1 = −1.3, α2 = −2.35, M1 = 0.5, Mmax =
+300 M⊙ as in Ceverino et al. (2019). We use a grid of
+13 values of metallicity, from Z = 10−5 to 0.04, and
+50 logarithmic bins in stellar population ages between 1
+Myr and 100 Gyr.
+We re-assign ”fine” ages to star particles in order to
+mitigate the discreteness effect caused by our simula-
+tion set up. Our simulations produce new star particles
+with a fixed time step of ∆tSF = 5 Myr, and the simula-
+tion output timings are not synchronized with ∆tSF. In
+a snapshot, young stars typically have discretized ages
+such like tage = 2 Myr, 7 Myr, etc. The apparently mi-
+nor gap in stellar ages causes a large impact when we
+compute the line emissivities because the ionization pho-
+ton production rate quickly decreases with age. For in-
+stance, in the BPASS SED of a single stellar population
+that we use, the number of ionizing photons decreases
+over a factor of 100 from age 1 Myr to 10 Myr (Xiao
+et al. 2018). We thus re-assign the stellar age as follows.
+We consider star particles younger than 15 Myr, with
+stamped ages at T1, T2, T3 (T1 < T2 < T3) Myr. We do
+random sampling within each age interval. For instance,
+to a star with T1, we randomly draw a new age within
+[1, T1] Myr and re-assign to it. Finally, we select star
+particles younger than 10 Myr for our emission line cal-
+culation. We calculate the ionizing photon production
+rate Q for each stellar particle using the BPASS table.
+We consider stellar atmosphere models with different
+elemental compositions, i.e., different values of [α/Fe].
+In the BPASS v2.3 (Byrne et al. 2022), there are five
+models with the mass fractions in α-elements relative to
+iron of ∆(log(α/Fe)) = −0.2, +0.0, +0.2, +0.4 and +0.6.
+For the calculation of [α/Fe], the α-element abundance
+is approximated by the oxygen abundance (log NO) as-
+suming that a half of the mass in metals produced by
+SNII are in the form of oxygen atoms;
+log NO = log(fOzSNII/AO),
+(7)
+where fO, zSNII are the fraction of oxygen released by
+Type-II SNe, and the mass fraction of metals released
+from Type-II SNe, respectively. Here, the atomic weight
+of oxygen is AO = 16 and we assume fO = 0.5 (Woosley
+& Weaver 1995). We calculate the iron abundance ratio
+considering both contributions from Type-Ia and II SNe
+as
+NFe = (fFe,Ia zSNIa + fFe,II zSNIa)
+AFe
+,
+(8)
+where zSNIa is the mass fraction of metals released from
+Type-Ia SNe and AFe = 56. We set the fractions fFe,Ia =
+0.5 (Thielemann et al. 1986) and fFe,II = (0.026, 0.033)
+for metal mass ratio between zero and solar metallicity
+(Nomoto et al. 2006; Ceverino et al. 2019), respectively.
+Finally, [α/Fe] is obtained from
+[α/Fe] = log NO − log NFe − log(NO/NFe)⊙,
+(9)
+where (NO/NFe)⊙ = 1.17 is the solar value of O/Fe
+abundance ratio.
+
+4
+Nakazato et al.
+Figure 1. The [Oiii] 88 µm luminosity versus SFR for our 62 simulated galaxies at z = 9 (top left), z = 8 (top right), z = 7
+(bottom left), and z = 6 (bottom right). The solid circles are colored with the gas metallicity (see the colorbar on the right).
+For comparison, we show the [Oiii] -SFR relation derived from observations of local galaxies by De Looze et al. (2014). Gray
+points are the observational results of high-z (z > 6) galaxies from Hashimoto et al. (2018); Laporte et al. (2017); Tamura et al.
+(2019), Inoue et al. (2016)(I16), Hashimoto et al. (2019)(H19), Carniani et al. (2017)(C17), Wong et al. (2022)(WG22), Witstok
+et al. (2022)(WT22) and Harikane et al. (2020).
+3. RESULTS
+We focus on rest-frame sub-millimeter and optical
+[Oiii] lines from high-redshift galaxies, which are de-
+tected by ALMA and JWST.
+3.1. L[Oiii] vs SFR
+Figure 1 shows the [Oiii] 88µm luminosity against star
+formation rate (SFR) for our galaxy samples. The color-
+bar indicates the nebular metallicity Zneb, which is the
+line luminosity-weighted gas metallicity.
+We compare
+with the observed local galaxies (De Looze et al. 2014)
+and with the observed [Oiii] 88µm luminosities of high-
+redshift galaxies (see the caption). At z = 9 to z = 7,
+most of our simulated galaxies are located above the
+local galaxy relation (solid line), similar to the results
+of Moriwaki et al. (2018); Arata et al. (2020); Pallottini
+et al. (2022).
+At z = 7 − 9, our galaxy samples are distributed
+around the observed galaxies. It is interesting that lu-
+minous galaxies are already chemically enriched with
+log(Z/Z⊙) ∼ −0.5 at the early epochs.
+Our simula-
+tions predict a slightly steeper relation at z = 7−9 than
+the local relation:
+L[Oiii] ,88 ∝
+�
+SFR
+M⊙ yr−1
+�0.9−1.2
+.
+(10)
+We find three galaxies with L[Oiii] > 109 L⊙ at z = 7,
+which are as bright as several observed galaxies.
+We
+study the structure of one of them (sample FL964) in de-
+tail. It has Mgas = 6.41×109 M⊙, M⋆ = 9.96×109 M⊙,
+and a specific SFR of 11 Gyr at z = 7. Figure 2 shows
+the projected maps of number density of gas, ionization
+parameter, and [Oiii] 88µm. Clearly, regions with high
+ionization parameters of log U ∼ −2 cause high emissiv-
+
+6=2
+1010
+0
+109
+L(88μm)[Lo ]
+-0.5
+log10(ZIZo)
+108
+-1.0
+107
+-1.5
+106
+MACS1149-JD1
+-2
+10-1
+100
+101
+102
+103
+SFR [M。 yr-1]z=8
+1010
+0
+A2744-YD4
+MACS0416-Y1
+109
+L(88μm)[L ]
+-0.5
+log10(ZIZo)
+108
+-1.0
+107
+-1.5
+106
+-2
+10-1
+100
+101
+102
+103
+SFR [M。 yr-1]Z=7
+1010
+0
+109
+L(88μm)[Lo]
+-0.5
+log10(ZIZo)
+108
+-1.0
+107
+}
+116
+WG22
+-1.5
+H19
+WT22
+106
+C17
+-2
+10-1
+100
+101
+102
+103
+SFR [Mo yr-1]9=2
+1010
+0
+J0235-0532
+X
+109
+L(88μm)[Lo ]
+-0.5
+108
+-1.0
+107
+-1.5
+106
+-2
+10-1
+100
+101
+102
+103
+SFR [Mo yr-1][Oiii] Luminosity calculation in First Light
+5
+Figure 2. Projected gas density (left), averaged ionization parameter (middle), and [Oiii] 88µm distribution (right) for a galaxy
+sample FL964 at z = 7. Each panel shows a region with a side length and depth of 0.3Rvir(= 7.4 kpc).
+ities, consistent with the observation by Harikane et al.
+(2020) and also with recent simulations by Kohandel
+et al. (2022). The total luminosity of [Oiii] 5007˚A of
+FL964 is 7.60 × 109L⊙, which is about 5 times larger
+than L[Oiii] ,88.
+3.2. The mass-metallicity relation
+It is important to examine the metallicity evolution
+of our simulated galaxies. We study the so-called mass-
+metallicity relation (MZR) by calculating the gas-phase
+metallicity for individual galaxies. Figure 3 shows the
+stellar mass-gas phase oxygen abundance relation. We
+calculate the gas phase oxygen abundance by adopting
+the conversion equation of Mandelker et al. (2014);
+O
+H =
+fO
+zSNII
+XAO.
+(11)
+We set the hydrogen mass fraction X = 0.755 and
+the other values of fO and AO are the same as those
+of eq.(7), which adopts the solar oxygen abundance
+12 + log(O/H) = 8.9. We then calculate the averaged
+zSNII, weighted by the [Oiii] luminosity of each grid.
+This weighting is compatible with observational meth-
+ods such as direct method or strong line method, which
+use oxygen emission lines (e.g. Bian et al. 2018; Izotov
+et al. 2019).
+We calculate the mass of stars within the region of 0.3
+Rvir. In Figure 3, we also plot the MZR for local galaxies
+from Curti et al. (2020) (dashed line) and recent JWST
+observation results of high-redshift galaxies (Sun et al.
+2022; Curti et al. 2022; Langeroodi et al. 2022; Williams
+et al. 2022). Curti et al. (2022) estimated metallicities
+of SMACS field galaxies by direct method, Sun et al.
+(2022) adopt strong line calibration by Bian et al. (2018)
+using O32, and Langeroodi et al. (2022) and Williams
+et al. (2022) adopt strong line method by Izotov et al.
+(2019).
+Our simulated galaxies have similar metallicities (oxy-
+gen abundance) and stellar masses to the observed ones.
+Note that Figure 3 shows the evolution for a fixed sam-
+ple of simulated galaxies, rather than for all the galaxies
+at respective epochs. Namely, we select the galaxies at
+z = 5 by mass and plot their progenitors at z = 6 − 9.
+Hence we likely miss low-mass, low-metallicity galax-
+ies at z = 9 (see Langan et al. (2020) for the mass-
+metallicity of low-mass galaxies in FirstLight).
+Some
+galaxies with M⋆ > 109 M⊙ have gas-phase metallici-
+ties of 12 + log (O/H) ∼ 8.5 even at z = 9, suggesting
+that metal-enrichment can proceed rapidly in the early
+galaxies.
+3.3. Far-IR/optical line ratios
+It is interesting and timely to explore line-ratio di-
+agnostics using three [Oiii] lines; 88 µm, 52 µm and
+5007˚A. The former two fine-structure lines are observed
+by ALMA whereas the latter is to be observed by JWST.
+Hereafter we denote the line luminosity ratios using the
+wavelength such as R5007/88 = L5007˚
+A/L88µm. Figure 4
+shows R5007/88 against R52/88 for our simulated galax-
+ies. We also show the model line ratios obtained by our
+set of CLOUDY calculations (Table 1).
+The ratio R5007/88 is commonly thought to be a sensi-
+tive temperature indicator (e.g. Fujimoto et al. 2022).
+Interestingly, Figure 4 shows that R5007/88 may also
+trace the mean gas metallicity of a galaxy. We argue
+that it is a model-dependent, indirect indicator because
+
+1 kpc
+0
+2
+3
+-5
+-4
+6
+8
+9
+10
+log <U)
+log [ Density / cm-3
+log [ Z[o]/L/kpc? 6
+Nakazato et al.
+Figure 3. Gas-phase metallicity versus stellar mass for our galaxy samples from z = 9 to z = 6. The solid lines show the
+median and the colored bands indicate the sample dispersion in the range of 5-95%. The dashed line is the local mass-metallicity
+relation from Curti et al. (2020). Red, blue, and purple symbols show the mass and metallicity of the z > 7 galaxies observed
+in SMACS J0723 field (Curti et al. 2022), z ∼ 6 galaxies observed by JWST/ NIRCam WFSS mode (Sun et al. 2022), and
+z = 8.1 − 9.5 galaxies observed in the cluster RX J2129.4+0009 field (two galaxies at z ∼ 8.15 from Langeroodi et al. (2022)
+and one at z = 9.51 from Williams et al. (2022)), respectively.
+of the complex dependence of the line emissivities on the
+relevant physical quantities. Typically, the oxygen line
+emissivity increases with increasing oxygen abundance
+(metallicity), but there is a critical abundance beyond
+which the emissivity decreases because of the tempera-
+ture decrease of Hii regions owing to metal line cooling.
+The critical ”peak” abundance is different for different
+lines and thus line ratios vary non-trivially as metallicity
+increases.
+In Figure 4, we plot local metal-rich galaxies ob-
+served with both FIR (Brauher et al. 2008) and op-
+tical emission lines (Moustakas et al. 2006).
+Most of
+the plotted local galaxies have high metallicities with
+Z > 1Z⊙ and are located in the lower portion (low
+R52/88) in the figure.
+Only NGC 1569, the left most
+symbol with R5007/88 = 4.9, has a sub-solar metallic-
+ity of log(Z/Z⊙) = −0.6 (Israel 1988), which is located
+near the same metallicity line as our high-redshift galaxy
+samples. The local planetary nebulae data from Din-
+erstein et al. (1985) are also plotted as red stars.
+It
+can be easily estimated that the planetary nebulae have
+electron densities of ne([Oiii] ) ≳ 103cm−3, which are
+consistent with those derived from [Oii] line ratios.
+The line emissivities and hence the ratios have im-
+plicit dependence on ionization parameter through other
+quantities such as electron temperature, but the de-
+pendence is weak at log U ∼ (−3, −2). Our simulated
+galaxies have generally high ionization parameter with
+log U ≃ −2 (Figure 2), and thus we may use R5007/88 as
+a metallicity indicator as well.
+In our emission line model (Section 2.2), the Hii re-
+gions have a fixed density of nHii = 100 cm−3. Hence
+our galaxy samples are populated in the left-upper por-
+tion with R52/88 ≲ 1. Since R52/88 varies weakly with
+Z and U (Yang & Lidz 2020), galaxies with high Z and
+high U are distributed toward bottom/right in Figure 4.
+4. DISCUSSION
+In this Letter, we have studied the chemical evolu-
+tion of early star-forming galaxies from z = 9 to z = 6
+by using zoom-in hydrodynamics simulations. We find
+that oxygen line emission galaxies with stellar masses
+of M⋆ = 109−9.5 M⊙ have large ionization parameter of
+log U = −2 and metallicity of log(Z/Z⊙) ∼ (−1, −0.5).
+In these galaxies, metal-enrichment occurs early and
+quickly over a few hundred million years.
+We have examined line diagnostics using [Oiii] 5007˚A,
+88 µm, and 52 µm for future observation synergies of
+JWST and ALMA. There have already been a few inter-
+esting observations of high-redshift galaxies. Killi et al.
+(2022) use ALMA and detect [Oiii] 52 µm line from a
+galaxy at z = 7 for the first time. The derived value
+of R52/88 ∼ 0.7 is close to our galaxy samples (Fig-
+ure 4), and indicates a relatively low electron density
+
+9.0
+Curti+20 (z = 0)
+z=9
+z=8
+z= 8.50, S04590
+z=7
+z = 7.67, S06355
+8.5
+z=6
+z= 7.66, S10612
+- log(O/H)
+P330E-z6.15
+P330E-z6.28
+8.0
+P330E-z6.35
++
+z= 8.16, ID11002
+2
+7.5
+z= 8.15, ID11022
+z=9.51, ID11027
+7.0
+7
+8
+9
+10
+stellar mass 「 M*/Mo[Oiii] Luminosity calculation in First Light
+7
+Figure 4. Line luminosity ratio R5007/52 against R52/88. Our simulated galaxies at z = 7 are represented by solid circles
+colored with gas metallicity. Gray star symbols show the local galaxies from Brauher et al. (2008); Moustakas et al. (2006) and
+red ones show the local planetary nebulae from Dinerstein et al. (1985). The results of CLOUDY calculations are represented
+by lines colored with metallicity (log(Z/Z⊙) = −1.30, −0.70, −0.40, 0.0). Solid, dashed, and dotted lines are the case of log U =
+−1.5, − 2, − 3 respectively. The number densities of Hii region log nHII[cm−3] = 1, 2, 3 are also marked by ticks from left to
+right on each CLOUDY line.
+of ne ∼ 50 − 260 cm−3. Observations of SMACS0723-
+4590 at z = 8.5 by Fujimoto et al. (2022) show a
+large line ratio R5007/88 = 15.8, which is slightly larger
+than our galaxy samples, suggesting a low-metallicity
+of Z ∼ 0.04 Z⊙. Combining R52/88 from future obser-
+vation will constrain the values of metallicity and ion-
+ization parameter at the same time according to Fig-
+ure 4. Planned observations using JWST NIRSpec are
+targetted to several [Oiii] 88 µm emitters (e.g., GO-
+1740, PI: Harikane, and GO-1840, PI:´Alvarez-M´arquez
+& Hashimoto). Multi-line diagnostics such as those pre-
+sented in this Letter holds promise to reveal the physical
+conditions of the ISM in the high-redshift galaxies.
+Our simulations show rapid chemical evolution at
+high redshift. The resulting MZR relation is consistent
+with up-to-date JWST observations (Figure 3).
+Lan-
+gan et al. (2020) use 300 less massive galaxy samples
+with M⋆ ≤ 108.5 M⊙ at z = 8 and derive MZR from
+z = 8 to z = 5 (see also a similar study by Noel et al.
+(2022)). Our galaxy samples with larger stellar masses
+with M⋆ = 108.5−10.0 M⊙ show a steeper MZR, which
+indicates rapid chemical evolution at the early epoch. It
+would be highly interesting to study relatively massive
+galaxies with JWST observations such as B14–65666
+(Roberts-Borsani et al. 2020), A2744–YD4 (Morishita
+et al. 2022), and MACS1149–JD1 (Hashimoto et al.
+2018).
+There are a few caveats in our emission line model.
+Most notably we do not account for dust extinction. Re-
+cent ALMA surveys report the existence of a substantial
+amount of dust in star-forming galaxies at z ∼ 6−8 (Fu-
+damoto et al. 2020; Burgarella et al. 2022; Bakx et al.
+2021; Schouws et al. 2022; Tamura et al. 2019; Inami
+et al. 2022). Given the importance of emission line ra-
+tios including [Oiii] 5007˚A, accurate modeling of dust
+extinction may be needed for future studies.
+We have studied the statistics of early emission-line
+galaxies and compared with recent observations. It will
+be possible and important to study the internal struc-
+ture of galaxies using both JWST observations and nu-
+merical simulations. We have shown in Figure 2 that
+there are large variations/fluctuations of line emissiv-
+ities, metal and density distributions within a galaxy.
+Cameron et al. (2022) argue that unresolved variations
+of the electron temperature within a galaxy results in a
+biased estimate when the so-called Te-method is applied.
+
+0
+102
+planetary
+nebulae
+-0.5
+101
+L5007/L88
+-1.0
+100
+要
+★ 
++
+10-1
+-1.5
+ logU= -1.5
+-- logU= - 2.0
+local galaxies
+logU= - 3.0
+10-2
+-2
+0.5
+1
+2
+3
+４
+5
+L52/L888
+Nakazato et al.
+JWST’s NIRSpec IFU can resolve with a pixel scale of
+0.1 [arcsec/pixel]1. For our configuration shown in Fig-
+ure 2, the 7.4 kpc region at z = 7 can be resolved with
+13 × 13 pixels. Gravitational lensing magnification will
+greatly help resolving further the structure of individ-
+ual galaxies. In our future work, we will generate mock
+two-dimensional maps for our simulated galaxies with
+the same resolution of NIRSpec IFU, and will address
+how well the physical quantities such as gas density and
+temperature distribution can be reconstructed.
+5. ACKNOWLEDGEMENTS
+We thank Kana Moriwaki and Yuichi Harikane for
+fruitful discussions.
+This work made use of v2.3 of
+the Binary Population and Spectral Synthesis (BPASS)
+models as described in Byrne et al. (2022) and Stanway
+& Eldridge (2018).
+The authors thankfully acknowl-
+edges the computer resources at MareNostrum and the
+technical support provided by the Barcelona Supercom-
+puting Center (RES-AECT-2020-3-0019).
+Numerical
+analyses were carried out on the analysis servers at
+Center for Computational Astrophysics, National As-
+tronomical Observatory of Japan.
+YN has been sup-
+ported by International Graduate Program for Excel-
+lence in Earth-Space Science (IGPEES) of the Univer-
+sity of Tokyo.
+DC is a Ramon-Cajal Researcher and
+is supported by the Ministerio de Ciencia, Innovaci´on
+y Universidades (MICIU/FEDER) under research grant
+PGC2018-094975-C21.
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+Enriching the scholarly metadata commons with citation metadata and
+spatio-temporal metadata to support responsible research assessment
+and research discovery
+Daniel Nüst1*, Gazi Yücel2, Anette Cordts2, Christian Hauschke2
+1Institute for Geoinformatics, University of Münster, Germany
+2TIB – Leibniz Information Centre for Science and Technology, Hannover, Germany
+* Correspondence: Daniel Nüst, [email protected]
+Keywords: open research information, citation metadata, spatio-temporal metadata, scholarly
+publishing, research assessment, discovery
+Abstract
+In this article, we focus on the importance of open research information as the foundation for
+transparent and responsible research assessment and discovery of research outputs. We introduce
+work in which we support the open research information commons by enabling, in particular,
+independent and small Open Access journals to provide metadata to several open data hubs (Open
+Citations, Wikidata, Open Research Knowledge Graph). In this context, we present The OPTIMETA
+Way, a means to integrate metadata collection, enrichment, and distribution in an effective and
+quality-ensured way that enables uptake even amongst small scholar-led publication venues. We have
+designed an implementation strategy for this approach in the form of two plugins for the most widely
+used journal publishing software, Open Journal Systems (OJS). These plugins collect, enrich, and
+automatically deliver citation metadata and spatio-temporal metadata for articles. Our contribution to
+research assessment and discovery with linked open bibliographic data is threefold. First, we enlarge
+the open research information data pool by advocating for the collection of enriched, user-validated
+metadata at the time of publication through open APIs. Second, we integrate data platforms and
+journals currently not included in the standard scientometric practices because of their language or
+lack of support from big publishing houses. Third, we allow new use cases based on location and
+temporal metadata that go beyond commonly used discovery features, specifically, the assessment of
+research activities using spatial coverage and new transdisciplinary connections between research
+outputs.
+This document is published under
+CC BY (4.0): Creative Commons Attribution
+
+BYEnriching the scholarly metadata commons
+1
+Introduction
+Open research information (ORI) provides a foundation for transparent and responsible research
+assessment and effective discovery of research outputs. Therefore, the quality, openness, extent, and
+scope of ORI should be as high as possible. However, especially for small and independent Open
+Access journals, it is difficult to collect publication metadata and deposit it in open research
+information data hubs. This is especially the case for more complex metadata that goes beyond the
+simple properties of individual records, such as title or publication date, by making connections
+between scientific publications. In this work, we introduce a concept for eliciting the rapid
+publication of verified open publication metadata, at the most effective and efficient moment in the
+publication process. We call this concept The OPTIMETA Way and implement it with two plugins
+for the publishing software Open Journal Systems (OJS) developed by the Public Knowledge Project
+(PKP; https://pkp.sfu.ca/). These plugins cover two distinct types of metadata that have very high
+potential to support more responsible research assessment and more powerful research discovery, but
+that are not yet widely available: verified and complete open citation metadata and spatio-temporal
+metadata.
+The main contribution of this work is a concept for effective publication metadata collection and
+publication that (a) balances data quality with the need for manual user interaction, (b) targets
+specific points in the publication process when the motivation and expertise of the stakeholders are
+provided, and (c) enables independent journals to create innovative metadata that goes beyond the
+common standard of even large commercial publishers. This concept is here demonstrated through
+prototypical implementations, the OPTIMETA services. The OPTIMETA services target two areas of
+publication metadata that unleash the integrative power of spatial and temporal relations between
+research outputs and facilitate the much-needed availability of open citation information to tackle the
+overwhelming amount of scientific literature.
+This paper is structured as follows. First, we briefly introduce the foundational background work
+considering, in particular, the potential diversity of the audience. Then, we present a concept and
+implementation strategy for innovative publication metadata. To conclude, we relate our ideas and
+products to the scholarly metadata commons, research assessment, and research discovery concepts
+before discussing the benefits, limitations, and future directions of the work.
+2
+Related Work
+2.1
+Open bibliographic metadata as a basis for responsible research assessment
+Research is resource-intensive, and the actors organising and funding it have an interest in screening
+and evaluating the research results generated from these efforts. Thus, research assessment is being
+conducted for various reasons, among them, because of a "lack of trust" in academia and a desire to
+facilitate improved resource allocation. Such assessments are often presented as a metadata-driven
+quantification of the research process based on elaborate frameworks and assessment rules. They are
+usually based on output and reception-based indicators, which in turn can only be produced if
+suitable metadata is available for this purpose.
+2
+
+Enriching the scholarly metadata commons
+In recent years, the traditional methods of assessing research have been fundamentally challenged
+from various sides. The role of bibliographic metadata is one important issue that has been raised, for
+example, by two widely discussed initiatives, as such material has to be made available through open
+licences in order to enable fair and responsible research evaluation. Both initiatives comment on the
+data basis required for responsible assessments. In 2012, several influential players in scholarly
+publishing met at the Annual Meeting of The American Society for Cell Biology in San Francisco,
+CA, USA (The American Society for Cell Biology 2012) to discuss emerging issues related to
+research evaluation. The result of this meeting was the San Francisco Declaration of Research
+Assessment (DORA, see Cagan 2013). By July 2022, almost 22,000 individuals and organisations in
+158 countries had signed the declaration. DORA gives clear instructions for publishers and suppliers
+of metrics to be "open and transparent". One key action recommended by DORA is that publishers
+should "remove all reuse limitations on reference lists in research articles and make them available
+under the Creative Commons Public Domain Dedication licence" (recommendation 9), while metrics
+suppliers should make the data and methods underlying their metrics available under an open licence
+(recommendation 11 and 12). In the second initiative, launched in 2015, the Leiden Manifesto (Hicks
+et al. 2015) took this further with more detailed instructions on how and, in particular, why the
+metadata used for assessment should be shared. Two of the resulting principles, 4 ("Keep data
+collection and analytical processes open, transparent and simple") and 5 ("Allow those evaluated to
+verify data and analysis"), are particularly relevant in the context of the present article.
+In order to comply with the principles and recommendations for responsible research assessment,
+data sources must meet various criteria. The research information - that is, metadata about actors,
+events, processes, and output related to research activities - must be findable and accessible if the
+publications are to be evaluated, and the records must contain the metadata in a form that allows legal
+and technically easy reuse.  These criteria (findability, accessibility, interoperability, and reusability)
+are described by Wilkinson et al. (2016) as the so-called "FAIR Guiding Principles for scientific data
+management and stewardship". Taken to its logical conclusion, this means that any research
+assessment should be based exclusively on metadata that is derived from publicly available data
+sources, is openly licenced, and is published using open standards.
+2.2
+Open bibliographic metadata and spatio-temporal metadata as part of open research
+information
+Open Research Information (ORI) is an emerging term used to describe metadata that complies with
+the above criteria for data sources intended for use in responsible research assessment. Recently,
+various actions have been taken and initiatives launched with the aim of more precisely defining and
+promoting ORI. For example, in 2018, the 14th International Conference on Current Research
+Information Systems (euroCRIS 2018) focused on the “FAIRness of Research Information”. In 2020,
+a German-Ukrainian project discussing the topic of "FAIR Research Information in Open
+Infrastructures" with international experts (Kaliuzhna and Altemeier 2021) led to the development of
+high-level criteria that applied the FAIR principles to research information (Hauschke et al. 2021a).
+Bijsterbosch et al. (2022) described the "Seven Guiding Principles for Open Research Information"
+and provided a more detailed analysis of "Trusted and transparent provenance", "Openness of
+3
+
+Enriching the scholarly metadata commons
+Metadata", "Openness of Algorithms", "Enduring access and availability", "Open Standards &
+Interoperability", "Open collaboration with Third parties", and "Academic sovereignty through
+governance".
+Open bibliographic metadata is an important part of ORI, especially in relation to output-oriented
+research assessment. From a broader perspective, this relates to any metadata that describes published
+works, e.g., journal articles, conference proceedings, monographs, or edited books. Being the
+foundation and fuel of the publishing and library worlds, bibliographic metadata is produced by
+authors, editors, librarians, and many others involved in the dissemination of scholarly output. The
+conventional bibliographic metadata types are, e.g., reference lists, abstracts, author affiliations,
+author identifiers, and licences. Besides research assessment, bibliographic metadata is used in
+several other ways such as the creation of scholarly knowledge graphs (e.g., Jaradeh et al. 2019;
+Manghi et al. 2021; Priem et al. 2022; Turki et al. 2021) and in library catalogues and bibliographic
+discovery services (Gonzales 2014).
+Recently, further types of bibliographic metadata, spatial and temporal metadata, have gained
+attention (Niers and Nüst 2020). Spatial and temporal metadata can deliver precise information about
+the location and time period that is covered in a publication. These metadata enable connections to be
+drawn between different research outputs, however, the availability and use of spatio-temporal
+metadata in ORI are currently sparse. The integrative potential of time and space is underutilised for
+publications (Niers and Nüst 2020) as well as for data (Garzón and Nüst 2021a) and current research
+information systems (CRIS) platforms.
+Several initiatives and projects are working on enriching the scholarly metadata commons. While an
+exhaustive review of all activities can not be given here, some key examples include Rasberry et al.
+(2019), who show how Wikidata can be used as a source for scholarly metadata through its frontend
+Scholia. They even discuss location data, though primarily in relation to the author and their
+institutional address as coordinates. Nielsen et al. (2018) expand on the use cases for discovery based
+on location information and also describe the opportunities for querying using locations mentioned as
+topics in articles in Scholia using point and polygon features from Wikidata. Lauscher et al. (2018)
+argue that libraries should play an important role in the production and curation of scholarly metadata
+and prove the feasibility and effectiveness of the strategy for the case of citation metadata from
+printed books in the social sciences. The Ukrainian Open Citation Index is an example of the
+nationwide collection of citation metadata from academic publishers (Nazarovets 2019). On an
+international level two initiatives have gained a lot of traction: The Initiative for Open Citations
+(https://i4oc.org/) and the Initiative for Open Abstracts (https://i4oa.org/). Nevertheless, for all the
+merits of these projects and initiatives, there is still much to be done, especially for small,
+independent and scholar-led journals. For these journals, the citation metadata and spatio-temporal
+metadata have great potential because given these metadata are available in a structured and
+machine-readable format and are accessible in open bibliographic databases, they enable connections
+to be drawn between different publications and, thus, improve the visibility of the contributions made
+by the large number of scholar-led Open Access journals.
+4
+
+Enriching the scholarly metadata commons
+2.3
+Citation metadata
+Citation metadata is metadata that expresses how one document refers to another. The idea of
+recording cross-references between documents was raised as early as 1952 by Eugene Garfield in a
+talk to the Maryland Section of the American Chemical Society. He stated “If authors would provide
+the CA abstract number with each bibliographical citation, I can assure you that CA abstracts in the
+future would be much more informative by providing cross-references to related abstracts'' (Garfield
+1952). Later, he went on to create the Science Citation Index, which evolved into the citation
+database Web of Science and inspired many similar products in the decades that followed.
+The description of citation metadata is a simple relation between two objects and several standards
+and definitions for different applications have been developed over time. To illustrate the diversity of
+the various efforts, we discuss three examples involving different types of citation metadata. Starr
+and Gastl (2011) present the initial way in which DataCite depicts relations between publications and
+research datasets. The basic assumption is that every entity, every research output in the DataCite
+metadata schema is described using a minimum number of mandatory fields: Creator, Title,
+Publisher, Identifier and Publication Year. Relationships between individual entities can then be
+constructed in various ways. IsCitedBy (and its inverse property Cites) tracks relationships between
+works that cite each other. This approach is content-agnostic. Peroni and Shotton (2012) developed
+the Citation Typing Ontology (CiTO), in which the citation still connects two works, but the citation
+itself is considered an entity that can be described by various properties independently. This allows
+for a much more detailed description of various types of relations and the representation of
+characteristics such as in-text citation frequency.
+Over time, the standards for describing citation relationships for specific types of works have also
+emerged. A recent example is the citation of research software, for which Smith et al. (2016) have
+developed principles that seek to capture the specifics of this type of work. For example, it must be
+possible to address the versioning common in software development, to describe specifically whether
+a particular version of a software is cited, any of its versions, or its latest version.
+2.4
+Spatio-temporal metadata
+Geospatial metadata is metadata for geographic data and information (Wikipedia 2022). There is a
+wide variety of standards, formats, and tools, ranging from public and industry standards for
+encoding geospatial metadata, such as the complex ISO 191** suite of standards, to catalogues
+collecting and serving geospatial metadata online
+(Federal Geographic Data Committee). While
+these types of data are often relevant, research output and publishing datasets are becoming more
+common and more widely acknowledged. Our study focuses on the geospatial properties of more
+classical research outputs: research papers. Therefore, we use the term spatio-temporal metadata
+when referring to the metadata of a spatial or temporal nature that describes textual and graphical
+research outputs. This type of metadata can refer, for example, to the spatial extent or the so-called
+area of interest that a scientific article investigates, or to the time period for which data were
+analysed. This approach has been demonstrated previously by JournalMap (Karl et al. 2013), albeit
+5
+
+Enriching the scholarly metadata commons
+with considerable limitations (Hauschke et al. 2021b). Furthermore, the most commonly used
+research data repositories (e.g., Zenodo, OSF, and Figshare) do not explicitly support spatial
+metadata. Only a few research data repositories are tailored to handle georeferenced data, such as
+Pangaea
+(https://www.pangaea.de/about/)
+and
+CKAN
+with
+its
+spatial
+extension
+(https://ckan.org/features/geospatial/). Dataverse only supports vector data in the outdated format
+Shapefile
+(https://guides.dataverse.org/en/latest/developers/geospatial.html?highlight=geospatial).
+This lack of support is possibly due to the fact that the handling of geospatial data was not a common
+feature of database software (except with the additional software extensions) or the focus of expert
+specialisation.
+Metadata of such a kind is relevant for a broad variety of scientific fields. In the natural and life
+sciences, observation data, model data, habitats or the sites of finds represent the most obvious points
+of connection (concerns, e.g., Earth science, geology, oceanography, meteorology, ecology, zoology,
+and botany). For the social and applied sciences, the connections to humans and their physical areas
+of activity are relevant in, for example, the medical and health sciences, agricultural science,
+economics, engineering, sociology, political science, and more. The theoretical work in the formal
+sciences (e.g., mathematics, logic, computer science) or the small-scale and theoretical physical
+sciences (physics, chemistry) are, as expected, less interested in geospatial metadata, although
+interdisciplinary work or research that features some element of application often has a real-world
+connection, i.e., a location in space and time.
+Turning our attention back to platforms for handling research publications, neither big commercial
+solutions for CRIS, such as Pure (Elsevier) or Converis (Clarivate Analytics), nor widely used open
+projects, such as DSpace (Smith et al. 2003), support spatio-temporal metadata for any record type.
+VIVO (Conlon et al. 2019) has a property for (vivo:geographicFocus) describing the spatial
+component of a given research activity or output, but it is not widely used. Developer documentation
+and code repositories show interest in the topic, for example, DSpace lists several occurrences where
+a spatial search was suggested or prototyped (cf. https://github.com/DSpace/DSpace/pull/511). The
+same lack of support occurs in relation to discovery platforms (e.g., ScienceDirect, Google Scholar)
+and publishers' websites. The only explicit modelling of spatio-temporal information in the
+publishing domain is the spatio-temporal fields in the Dublin Core specification DCMI Metadata
+Terms, coverage being the most important among them (https://www.dublincore.org/specifications/
+dublin-core/dcmi-terms/terms/coverage/).
+However,
+this
+field
+is
+not
+particularly
+useful
+for
+machine-readable information exchange, as it may hold any type of spatial or temporal metadata, be
+it prose, coordinate pairs, or textual encoding of complex geometries or time periods. DCMI
+Metadata Terms use some alternative terms, e.g., jurisdiction or location, but these do not seem to be
+implemented in the platforms mentioned here. Thus, all these platforms include inexplicit or not
+directly usable spatial information, e.g., in the form of addresses for researchers, location names in
+paper abstracts, excavation site coordinates in the full text, or remote sensing imagery as figures in
+the supplementary material. Temporal metadata is more common, not least because it is simpler in
+nature and readily supported by any database management system. However, the temporal fields are
+often focused on publication metadata (when was the resource created or published?) rather than on
+the content. Similarly, this kind of temporal information is hidden in titles, abstracts, or full texts.
+6
+
+Enriching the scholarly metadata commons
+This completely neglects the potential for spatial and temporal information to act as an integrator,
+e.g., as Kuhn (2012) argued in relation to transdisciplinary research. The potential has been pointed
+out in the past, both in relation to creating new connections between publications (Niers and Nüst
+2020) and in relation to data (Garzón and Nüst 2021a; Garzón and Nüst 2021b). In general,
+spatio-temporal metadata currently does not play an important role in ORI.
+3
+The OPTIMETA Way
+3.1
+Approach
+The introduction presented the challenges and benefits involved in enriching the scholarly metadata
+commons and how this connects with the responsible assessment and effective discovery of research.
+In order to tackle some of the challenges of capturing and disseminating high-quality and useful
+metadata for research outputs, we developed an approach to strengthen the Open Access publishing
+system through open citations and spatio-temporal metadata - The OPTIMETA Way. This concept
+recognises the conflict involved in metadata creation and usage during the publication phases of the
+research cycle.
+Firstly, the benefits of creating high-quality metadata are intangible for authors, although they are the
+most knowledgeable source for most of the relevant information. However, this may change if the
+reasons for providing the metadata are communicated clearly, such as enabling a more responsible
+assessment of their work and better connection with other disciplines for evaluation and discovery
+purposes. Nevertheless, the way the metadata are captured should be intuitive and engaging, keeping
+in mind James Frew's laws: "Frew’s first law: scientists don’t write metadata. Frew’s second law: any
+scientist can be forced to write bad metadata." (Hey 2015). Secondly, the large metadata owners are
+currently the big scholarly publishers. They have a strong interest in building their business and
+making a profit based on such data (Pooley 2022; Brembs 2021; Franceschi-Bicchierai 2022) and
+little interest in contributing everything they can to the creation of knowledge or addressing the need
+for technical innovation. Instead, innovation must be pursued by those who have an interest in having
+an open and free scholarly metadata commons, such as university publishers or independent journals,
+even though they have limited resources.
+In acknowledgement of these conflicts and challenges, The OPTIMETA Way is output-oriented and
+focuses on enabling the essential, if currently relatively powerless, stakeholders in academic
+publishing to capture and distribute scholarly publication metadata themselves. The approach is
+"OPTImal" in the sense that it begins with small improvements with the potential to generate the
+biggest benefits: the potential impact of high-quality citation metadata on research assessment and
+quality-ensured metascience is huge. Furthermore, the novelty of spatio-temporal metadata and its
+integrative potential make it attractive even at a rudimentary level, for example, when simple
+geometries representing articles are shown on the same map for visual inspection and discovery.
+7
+
+Enriching the scholarly metadata commons
+Fig. 1. Stages of the publication process, the generic research process, and The OPTIMETA Way
+with their connections.
+This approach is output-oriented as it will assist metadata creators through automation and intuitive
+user interfaces and, thereby, avoid requiring additional strenuous, time-consuming, or unwelcome
+tasks. By tapping into open data sources, the software enriches the user input which is then provided
+for inspection. The approach further targets the quickest gains compared to the required effort and
+thus does not have to be comprehensive. The metadata will be created at a point in the publication
+process when authors are most willing to fulfil all administrative requirements, that is, while
+submitting an article for review and then, eventually, publication. This is also the last point in time at
+which engaged professionals (reviewers, editors, publishing staff) examine the metadata critically
+and at which the publication of output, including metadata, is already a core part of the process.
+Furthermore, the output is always checked by humans during this process and, as the data are not
+being produced by an algorithm, higher-quality metadata can be returned to the data sources used for
+enrichment.
+Finally, the facilitation happens through The OPTIMETA Way. This approach enables independent
+journals and small publishers, who often work with open-source software platforms, to actively
+engage in structured metadata collection and distribution without expert knowledge. A crucial step in
+enabling this is the deposition of metadata in open data hubs, which will allow journals with the
+shared mission of disseminating knowledge to contribute to the bigger goal of an open scholarly
+publication metadata commons. Thus, the impact of The OPTIMETA Way will be bigger than the
+sum of its parts.
+Fig. 1 is a schematic representation of The OPTIMETA Way. One side shows the simplified research
+cycle including the research activities and their translation into written output. The other side shows a
+journal’s publication process from submission to editing to publication and the associated or
+subsequent dissemination of metadata. The OPTIMETA Way means that we support the generation,
+8
+
+Research process
+Publishing process
+Submission
+Idea
+Editorial process
+Read
+Publish
+Publishing
+Analyse,
+Develop,
+Metadata dissemination
+Write
+OPTIMETA
+Metadata
+Metadata
+Services
+enrichment
+collection
+ScholarlyMetadataCommons
+TheOPTIMETAprocessEnriching the scholarly metadata commons
+enrichment, and dissemination of metadata during the stages of the journal publication process in
+which the metadata is already a focus. In this way, this approach avoids any retrospective editing and
+post-publication tasks for researchers.
+3.2
+Implementation examples
+3.2.1 Overview
+Open Journal Systems (OJS) is the most widely used journal publishing software in the world. It was
+developed to make scientific publishing easier and more effective (Willinsky 2005). It organises the
+complete publishing workflow from submission to review, from proofreading to production. The
+software is open source and is continuously being improved by a global community. It is currently
+available in version 3.3. OJS provides the core functionalities discussed above, including metadata
+creation, editing and export. Additional functionality can be added to OJS through plugins that may
+be
+installed
+manually
+or
+using
+the
+so-called
+Plugin
+Gallery
+(https://docs.pkp.sfu.ca/plugin-inventory/en/). These plugins can extend or alter any step in the OJS
+submission and publication workflows and add new features to the editorial backend or the system's
+front end.
+We have realised The OPTIMETA Way through two plugins: the OPTIMETA citation plugin and the
+OPTIMETA geoplugin (“citation plugin” or “geoplugin” for short). Together, these two plugins
+implement the OPTIMETA Services described in Fig. 1. Both plugins collect metadata during the
+submission workflow and enhance it with data from open scholarly data sources. They then present
+the person submitting the data with the enhanced information before, ultimately, exposing the
+information on both the OJS website and the external data sinks. The connection to external data
+sinks can be made in near real-time, in the sense that information is deposited actively in connection
+with events in the publishing workflow. Alternatively, other platforms can be used as a harvesting
+mechanism to regularly retrieve the published metadata from OJS. Fig. 2 shows the data sources and
+sinks that are currently supported by the plugins, as well as those that may be supported in the future.
+The data sources on the left are marked in red. Both plugins rely on user-contributed metadata that is
+enriched with external references and additional data. The targeted metadata platforms and
+aggregators, or data sinks, are shown on the right and marked in green. The journals and university
+publishers collaborating with the project are listed at the bottom (see the full list of names and links
+at https://projects.tib.eu/optimeta/en/).
+The plugins are available as public beta releases and can be freely downloaded and installed from our
+public
+GitHub
+repositories
+at
+https://github.com/TIBHannover/optimetaCitations/
+and
+https://github.com/TIBHannover/optimetaGeo. The plugins will be improved based on feedback
+from our OPTIMETA project partners and the OJS community (users and developers) that is focused,
+in particular, on the user experience.
+9
+
+Enriching the scholarly metadata commons
+Fig. 2. Implementation example of The OPTIMETA Way: OPTIMETA citation plugin and geoplugin
+for OJS and the connected external data sources, data sinks, and collaboration partners.
+3.2.2 Citation plugin
+The citation plugin aims to gather machine-readable citation metadata during the publication process
+with the goal of publishing the metadata in open bibliographic data sources. The process is integrated
+into the existing OJS workflow for submitting publications. When the author begins a submission to
+the journal, OJS provides a special metadata field for references. We assume that this field is then
+filled by either the author or the journal editors. The citations are entered into OJS in a raw format,
+i.e., in a freely modifiable text field (Fig. 3).
+Fig. 3. Example of a raw citation.
+The citation plugin parses the raw references and extracts the Digital Object Identifiers (DOI) if
+present. After extracting the DOIs, a look-up algorithm enriches the metadata based on external and
+open bibliographic data sources in a semantically structured format. Currently, the citation plugin
+queries the open APIs of Crossref and OpenAlex, which both provide good coverage and promising
+metadata quality. We have chosen to collect metadata from external sources rather than parsing the
+citations with parsing tools as we found the results we queried based on the DOI from Crossref and
+OpenAlex were, in general, more complete and accurate. Developing a custom parsing service or
+integrating an already existing tool would add unnecessary complexity to the plugin. All of these
+steps are triggered manually with a single click of a button. Therefore, the additional time needed for
+10
+
+data source
+data sink
+OPTIMETA
+PKP
+OpenAlex
+OJS
+Project
+3.2.1.x
+Crossref
+citation
+3.3.x
+plugin
+DataCite
+references
+WIKIDATA
+|OpenAlex
+user
+geoplugin
+time period and
+ORKG
+location 
+search portal
+partner
+ulb.Q
+jKi
+zhb:
+GeoNames
+journals
+PUEISHING
+heiJOURNALS
+OPTiMETA lmplementation: Software, data sources, and data sinksopen citations and spatiotemporal metadata. Research Ideas and Outcomes 7: e66264. https://doi.org/10.3897
+/ rio.7.e 66264Enriching the scholarly metadata commons
+the enrichment of citation metadata comprehends only a few minutes or less, which is neglectable
+compared to the amount of time consumed for the overall research and publishing process as shown
+in Fig. 1.
+The reliance on DOIs for finding the full reference information is a current limitation of the plugin.
+The alternative would be to query Crossref with the full reference was not implemented because of
+the limitations of non-membership access to the API and because the reliable SimpleTextQuery form
+(https://apps.crossref.org/SimpleTextQuery) is provided for manual use only. We are not aware of a
+comparable full reference query feature for OpenAlex. In these circumstances, we decided that given
+the resulting metadata is of higher quality and less manual interaction is required, having a higher
+degree of automation outweighed the drawback of missing references that do not have a DOI. For
+new submissions, the plugin will encourage authors to add missing DOIs during the submission
+process, thus ensuring a reasonably high level of metadata quality and completeness. As OpenAlex
+harvests from Crossref, having both services as data sources for the plugin may seem superfluous.
+However, this redundancy avoids being reliant on one specific service into the future and, as
+OpenAlex harvests other data sources as well, using both sources is likely to provide additional and
+potentially more complete information.
+After these steps have been completed, the enriched results extracted from the external open APIs are
+presented to the author for review. The review can be done by simply checking whether the results
+are correct. The various parts of the citations can also be edited manually (Fig. 4). The now
+semantically structured metadata including title, authors with their corresponding author identifier
+(ORCID iD), etc. are then stored in the OJS database.
+Fig. 4. Example of a semantically structured citation which can be edited manually.
+After the enrichment process, the citations can be deposited with external services either manually,
+by clicking the deposit button, or through an automated process managed by the OJS scheduler. The
+first workflow is currently implemented for OpenCitations. The combined metadata are structured
+according to Massari and Heibi (2022) and submitted as an issue to a specified GitHub repository of
+OpenCitations (https://github.com/GaziYucel/open_citations_croci_depot). The issue containing the
+metadata can then be processed and harvested by OpenCitations.
+11
+
+https://doi.org/10.3897/ric
+URN
+URL
+Christian Hauschke
+https://orcid.org/0000-000
+iD
+Daniel Nust
+https://orcid.org/0000-000
+iD
+Anette Cordts
+https://orcid.org/0000-000
+iD
+血
+Svantje Lilienthal
+https://orcid.org/0000-000
+iD
+血
+Autor hinzufugen
+OPTIMETA - Strengtheninc
+Research Ideas and Outcor
+2021
+Volume
+Issue
+First page
+Last pageEnriching the scholarly metadata commons
+The initial plugin versions focus on using DOIs as the supported publication identifier. In future
+releases, we are planning to support non-DOI identifiers. Furthermore, being able to import
+bibliographic metadata via common bibliographic standards like RIS and BibLaTeX would improve
+the user experience. The current focus of plugin development is to provide support during the
+publication process for one article. However, journal operators will, naturally, not only want to
+publish citation information for new articles, but also those from their back catalogue. To address this
+need, we plan to design a special overview page that will enable articles to be processed in batches.
+Ultimately, we are also aiming to link into more data sinks as this will enable the widest possible
+dissemination of citation metadata. For example, we are currently working toward implementation
+with Wikidata.
+3.2.3 Spatio-temporal metadata plugin
+The OPTIMETA geoplugin enables the collection and display of spatio-temporal metadata for
+individual research articles in OJS instances. The term "geo" was used in the name as it is more
+broadly understood and shorter than the more technical "spatio-temporal". Geospatial and geographic
+data usually include temporal aspects, in that answers to "where in space" questions are always
+connected to a "when in time" as well. The "geospatial" metadata are more dominant than the
+temporal metadata in the plugin for different reasons. First, the display of geographical features on a
+map is more visual and, thus, more interesting than one or several time periods shown as numbers.
+Second, the novelty and power of geospatial metadata are higher than those of temporal metadata
+because textual descriptions or classifications of time, such as "in the year 2022" or "during the
+cretaceous period" are more readily picked up through text searches, whereas spatial relations are
+harder capture through text descriptions and very difficult to pick up using text searches.
+The geoplugin, in particular, extends the submission workflow. Due to the simple and intuitive way of
+entering geo-spatial metadata, the provision of temporal and spatial metadata can be carried out
+quickly within a few minutes. Authors are asked to provide temporal metadata in the form of a
+simple time period with a start and end date. The date can be entered manually, e.g., by putting in the
+dates separated by a hyphen into the form field, "2021-01-01 - 2022-02-02", or with an interactive
+calendar pop-up as shown in Fig. 5. The time period is a simple string and can be used to model
+temporal uncertainty and allows for imprecision where needed or not known. For example, a history
+paper may document a hegemony's duration as "753 - 1234", while the observation of a new species
+may require a precise date such as "2022-08-08 - 2022-08-09".
+12
+
+Enriching the scholarly metadata commons
+Fig. 5. Screenshot of the submission form showing the form field and popup for input of a time
+period in the lower third of the image.
+Next, authors are asked to provide spatial metadata using an interactive map as shown in Fig. 6.
+Authors can add multiple geometries of various types, i.e., points, polylines, rectangles, and
+polygons. These can be used to adequately represent spatial features related to articles, e.g., places of
+residence of a historic person, animal tracks, remotely observed areas, or a herd's territory,
+respectively. An author may choose to quickly add a coarse rectangle providing rather imprecise data
+or to zoom into the map and enter detailed individual data points, either as time permits or the
+submission demands. The author is assisted in the creation of this data by two background layers: an
+open data street map by OpenStreetMap and free-to-use aerial imagery tiles provided by Esri. In this
+way, both natural features and human-built structures can provide orientation. While creating the
+geometries forming the detailed spatial metadata, the geoplugin constantly queries the Geonames
+gazetteer
+service
+(https://www.geonames.org/)
+using
+coordinates
+from
+the
+geometries
+to
+automatically derive the bounding rectangle or bounding box of the smallest encompassing
+13
+
+Abstract *
+心
+B
+u
+三
+三
+x
+X2
+&
+<>
+List of Contributors
+Add Contributor
+Name
+E-mail
+Role
+Primary Contact
+In Browse Lists
+admin admin
+[email protected]
+Journal manager
+Additional Refinements
+Keywords
+Add additional information for your submission. Press 'enter' after each term.
+Time and location
+Temporal Properties
+Define the temporal properties of the articles content by specifying the begin and end dates. The input is possible via the text field as well as via the calendar view. Click the input field below
+this text to open the calendar. Press "Apply" to save the calender setting. Press "clear" to remove the current data. Textual input must match the following format and be confirmed by
+clicking "Apply" (not Enter!): begin and end are seperated by a hyphen surrounded by spaces; dates are given as "YYYY-MM-DD", whereby "YYYY" stands for years, "MM" for months (with a
+leading zero), and "DD" for days (with a leading zero)
+<
+Aug 2022
+Sep 2022
+>
+Su
+Mo
+Tu
+We
+Th
+Fr
+Sa
+Su
+Mo
+Tu
+We
+Th
+Fr
+Sa
+n the control on the left side between polyline, polygon, rectangle and point. By the icons
+31
+1
+2
+3
+4
+5
+6
+28
+29
+30
+31
+1
+2
+3
+eft corner it is possible to zoom in and out. In the upper right corner you can change the
+gery ("Esri World Imagery" layer). You can also set whether the geometric shape(s) and the
+7
+8
+9
+10
+11
+12
+13
+4
+5
+9
+7
+8
+9
+10
+tailed description in the next paragraph of this form. Below the layer control you will find a
+14
+15 
+16
+17
+18
+19
+20
+11
+12 
+13
+14
+15 
+16
+17
+ent locations that match your input. By clicking on a suggestion it will be accepted and a
+ectangle can also be deleted/ edited by clicking on the trash-/ edit icon on the left side. You
+21
+22
+23
+24 
+25
+26
+27
+18
+19
+20
+21
+22 
+23
+24
+28
+29
+30
+31
+1
+3
+25
+26
+27
+28
+29
+30
+4
+5
+6
+7
+8
+9
+10
+2
+3
+4
+5
+6
+7
+8
+2022-08-01 - 2022-08-10
+Clear
+ApplyEnriching the scholarly metadata commons
+administrative area. The administrative unit is given in a form field below the map. The data for
+administrative units varies widely for different countries around the world and is, arguably, most
+useful for countries in the global north.
+Fig. 6. Screenshot of the submission form showing the interactive map for collecting spatial
+metadata, in this case a point geometry in City of Münster and a polygon around the city of Hanover
+in blue; these geometries are enclosed in the administrative unit "Earth, World, Germany" whose
+bounding rectangle is shown on the map in black.
+The information collected during the submission can then be reviewed during the editorial process.
+Ultimately, the data is stored as plain text in the OJS database in GeoJSON (https://geojson.org/)
+format. A single FeatureCollection includes the geometries and a short provenance statement
+indicating who created the data or where it was derived from. We chose GeoJSON for this purpose
+despite the limitations resulting from its inability to handle different coordinate reference systems
+because of its wide usage and simplicity. For the purpose of discovering research articles on a global
+scale, the accuracy of several metres of the coordinate reference system "World Geodetic System
+1984" (WGS 84) is entirely sufficient, especially considering that most geometries are manually
+created on an interactive map.
+14
+
+This article covers location(s) or area(s) shown on the map below.
+Define the location of the articles content by one or more geometric shape(s). You can choose in the control on the left side between polyline, polygon, rectangle and point. By the icons
+below you can edit or delete the created geometric shape(s). With the "_" and "+" in the upper left corner it is possible to zoom in and out. In the upper right corner you can change the
+background maps. You can choose between a street map ("openStreetMap" layer) and aerial imagery ("Esri World Imagery" layer). You can also set whether the geometric shape(s) and the
+administrative unit should be displayed or not. For the administrative unit you will find a more detailed description in the next paragraph of this form. Below the layer control you will find a
+search for the map. You can search for locations and by hitting "Enter" you will be offered different locations that match your input. By clicking on a suggestion it will be accepted and a
+matching rectangle, describing the boundaries of the location, will be displayed on the map. This rectangle can also be deleted/ edited by clicking on the trash-/ edit icon on the left side. You
+can choose a coarser level of detail to protect sensitive locations.
+Vorpommern
+woje
++
+Hamburg
+ar
+jewodztwo
+Grudziadz
+na
+Bremerhaven
+zach
+dniopomorskie
+Groningen
+Coldenburg
+wojewodztwo
+przow
+kujawsko-
+Bremen
+Assen
+G
+Wiel
+kopolski
+pomorskie
+Niedersachsen
+Berlin
+olle
+vich
+Haarlem
+Wolfsburg
+wojewodztwo
+Plack
+abruck
+Nederl
+Hannover
+Potsdam
+wodztwo
+nd
+wielkopolskie
+buskie
+Polska
+Sachsen
+War
+DenHaag
+Munster
+Bielefeld
+Anhalt
+Arn
+hem
+Zielona
+Kalisz
+Middelburg
+Cottbus
+Gora
+wojewodztwo
+0
+Eindhove
+Nordrhein
+Leipzig
+Chosebuz
+fodzkie
+sh-sea
+Brugge
+Westfalen
+Kassel:Deutschland
+Voanderen
+Dusseldorf
+unkerque
+Siegen
+Thuringen
+wojewodztwo
+贝
+achen
+Sachsen
+Librec
+dolnos/gskie
+Czestochowa
+Lille
+A
+Bonn
+Belgie/
+wojewo
+Chemnitz
+wojewodztwo
+swietokr
+Belgique
+Koblenz
+Hessen
+rSeverozapad
+opolskie
+Belgien
+Praha
+Severovychod
+aKatowice
+Hauts-de
+France
+Frankfurtram
+Ostrava
+wojewodztwo
+Letzebuerg
+Main
+Wurzburg
+Cesko
+molopolskie
+Plzen
+StredniMoravd
+Rouen
+Mannheim
+Nurnberg
+jihozdpad
+Zilina
+Reims
+Jihowychod
+100 km
+Saarbrcken
+Paris
+Bayern
+50 mi
+Karisruhe
+Grar
+Ingolstadt
+Ceske
+Baden-wurttemberg
+Budejovic
+Leaflet I Map data: @ OpenStreetMap contributors
+Note that the temporal and spatial metadata will be published under the following license: CC-o
+Administrative Unit
+On basis of your input in the map, administrative units are proposed according to your input on the map. Each time you update the inputs, the coverage information gets new calculated and
+updated correspondingly. You are able to delete administrative unit(s) by the red "x". If you hover over the administrative unit(s) the superior hierarchy of administrative unit(s) is displayed if
+available. Besides you can add further administrative units. You are only able to insert a further administrative unit if it fits to the already given hierarchy of administrative unit(s), and the given
+geometric shape(s) in the map. If you begin to insert, there are some suggestions you can accept by clicking, but nevertheless you can input your own administrative unit by hitting "Enter".
+The administrative unit (in black) which is the lowest common denominator for all geometric shape(s) is shown in the map. The administrative unit is not editable or deletable in the map, but
+here via the input field.
+Earth x
+Europe ×
+Federal Republic of Germany ×
+Save and continue
+CancelEnriching the scholarly metadata commons
+Fig. 7. Screenshot of the article landing page with spatio-temporal metadata; this article has multiple
+polygons covering parts of Brazil with a matching textual description below the map, but there is no
+time period; the column on the right contains the download button for metadata in GeoJSON format.
+15
+
+OPTIMETA
+Current
+Archives
+About -
+Map
+Q Search
+Home / Archives / Vol. 5 No. 1 (2022): Ausgabe 1- Mai 2022 / Articles
+Linguistic Analysis of the Newspaper Discourse in Brazil: The Older
+People and cOVID-19
+Author Author
+Published
+https: / Lorcid.org/0000-0002-1825-0097
+2022-05-19
+Keywords: coVID-19, older people, newspaper discourse
+Issue
+Vol. 5 No. 1 (2022): Ausgabe 1 - Mai
+2022
+Abstract
+ Section
+Articles
+Brazil has been one of the countries most affected the coVID-19 pandemic.
+The measures taken by the Brazilian government to contain the spread of
+the virus have been the subject of national and international criticism.
+Download geospatial metadata
+Among these measures is the topic of
+ protection and isolation ofolder adults, which has been subject of
+目GeoJSON
+About GeojsoN
+discussion in Germany. Based on a discourse led by analytical and linguistic
+Geodata license: CC-0
+ approach this paper analyses the media and public perception of the older
+ generation concerning the coronavirus.
+Using both quantitative and qualitative methods of corpus analysis, we
+investigate the question of whether the older people are
+ associated with the COVID-19 virus in a particular form and whether they
+are exposed to specific discrimination as a population
+group. The results of the data gathered from March to July 2020 show that
+older adults are often described as being vulnerable
+and belonging to the risk group in Brazilian media.
+Time and location
+This article covers location(s) or area(s) shown on the map below. 
+Manaus
+Q
+Goiani
+Cru
+BeloHo
+500 km
+laSierr
+500mi
+ParaguayLeafet Map data: @ OpensStreetMap contibutors
+The administrative units enclosing the article's places are: Earth, South
+America, Brazil ?
+Temporal and spatial metadata are published under the following license: CC-0Enriching the scholarly metadata commons
+Fig. 8. Screenshot of an article landing page with publication information and spatio-temporal
+metadata; above the map the time period of interest is given, the geometries describing this article's
+content are several polylines representing travel routes.
+The spatio-temporal metadata is then published together with the article on various pages: the article
+landing page (see Fig. 7 and 8), the landing page for the issue (see Fig. 9), and on a separate page for
+the journal itself. The article landing page also contains a download button providing easy access to
+the spatio-temporal metadata in GeoJSON format. The journal landing page features synchronised
+highlighting as shown in Fig. 9 if the user holds the cursor over a geometric feature on the map. Both
+the feature on the map and the corresponding article in the list above is highlighted in red. Clicking
+on the issue or journal map opens a small popup window (not shown) containing the publication
+metadata (author, title, etc.) and a link to the article landing page. Below each map, there is a clear
+16
+
+OPTIMETA
+Current
+Archives
+About -
+Map
+Q Search
+Home / Archives / Vol. 5 No. 1 (2022): Ausgabe 1 - Mai 2022 / Articles
+Herrschaft vom Pferderucken - Reisekonigtum zur Zeit Heinrichs
+IV.
+Author Author
+Published
+https. .Lorcid.org/0000-0002-1825-0097
+2022-05-19
+Issue
+Vol. 5 No. 1 (2022): Ausgabe 1 - Mai
+Abstract
+2022
+Die Hauptstadt und das Land gehoren fur moderne Menschen untrennbar
+ Section
+zusammen.
+Articles
+Eines geht ohne das andere nicht. Bereits in der Grundschule lernen wir die
+ entsprechenden Begriffspaare auswendig. In Hauptstadten wird Politik
+ gemacht, sie sind die Schaltzentrale ihres Landes und der Ort, an dem das
+Download geospatial metadata
+Parlament, viele wichtige Ministerien sowie auslandische Konsulate und
+Botschaften zu finden sind; meistens in einem eigenen Regierungsviertel.
+目GeoJSON
+About GeoJSON
+Fur uns ist die Hauptstadt heute selbstverstandlich ein Synonym fur das
+Zentrum der Macht und fur diejenigen, die politische Entscheidungen
+Geodata license: CC-0
+treffen. Demnach ,entscheidet Berlin", man fragt, .was Washington
+ eigentlich denkt", .,welche Zusicherungen Moskau gemacht hat" und was
+, Peking dazu sagt".
+Time and location
+This article covers a time period from 1053 to 1105. ?
+This article covers location(s) or area(s) shown on the map below. ?
+ield
+Groningen
+Hamburg
+Szczecir
+Berlin
+gham
+Nederland
+London
+tschland
+Belgie/
+Wroctaw
+Dresden
+Belgique/
+Belgien
+Kra
+nnes
+Munchen
+Osterreich
+antes
+de Loire
+Schweiz
+France
+Suisse/Svizzeral
+svizra
+100 mi
+The administrative units enclosing the article's places are: ?
+Temporal and spatial metadata are published under the following license: CC-0Enriching the scholarly metadata commons
+statement about the licence of the spatio-temporal data, to which authors will need to agree to while
+creating the metadata. The licence is fixed to a public domain licence, CC-0, to ensure the broadest
+possible usage.
+Fig. 9. Screenshot of the issue view in the public demo journal, see
+https://service.tib.eu/optimeta/index.php/optimeta/issue/view/1. The standard OJS theme is extended
+with a "Times & locations" section below the list of articles of the issue. The mouse cursor over the
+spatial feature on the map at the bottom triggers a highlighting of the geometry and corresponding
+article in the list above in red.
+The pages shown above, all target human users, but the spatio-temporal metadata are also included in
+the HTML website of the article landing page in machine-readable form. These metadata fields
+17
+
+OPTIMETA
+Current
+Archives
+About 
+Map
+Q Search
+Home / Archives / Vol. 5 No. 1 (2022): Ausgabe 1 - Mai 2022
+Vol. 5 No. 1 (2022): Ausgabe 1 - Mai 2022
+Published: 2022-05-18
+Articles
+Linguistic Analysis of the Newspaper Discourse in Brazil:TheolderPeople and covID-19
+ Author Author
+Using textual volunteered geographic information to model nature-based activities: A case study from Aotearoa
+New Zealand
+Optimeta Admin
+Herrschaft vom Pferderucken - Reisekonigtum zur Zeit Heinrichs IV.
+ Author Author
+The Pottery of Mount Zion: An Overview from Islamic to Iron Age Periods
+ Author Author
+Erstnachweis des Eiparasitoiden Trissolcus basalis (Wollaston, 1858) in Osterreich (Hymenoptera: Scelionidae)
+Optimeta Admin
+Revisiting Conditional Typology for Bangla
+ Author Author
+Times & locations
+2000 m
+Leaflet I Map data: @ OpenStre
+eetMap contributors
+Temporal and spatial metadata are published under the following license: CC-0Enriching the scholarly metadata commons
+enable scraping and harvesting through other services. Fig. 10 shows selected values as displayed in
+the HTML header of a test article, each defined by a name and, if available, a well-defined scheme.
+Alongside other publication metadata, the spatial metadata is included in several forms and schemas
+including the Dublin Core fields DC.SpatialCoverage and DC.Coverage. The former is included as a
+textual encoding of the full GeoJSON record (line 10 in Fig. 10), the latter (line 20) as a textual
+representation of the administrative units starting with the largest units and working down to the
+more generic field, geo.placename, which contains the smallest available administrative unit. In the
+example provided, this is a country name, but it can also be more specific, for example, the name of a
+town. Finally, the bounding rectangle of the smallest administrative unit is given in the fields ISO
+19139 in an XML-encoding of the geographic bounding box according to the ISO 19139 standard
+and DC.box using a simple list of the coordinates of the four cardinal directions limiting the rectangle
+separated by semicolons. The temporal metadata is stored in the field
+DC.temporal and
+DC.PeriodOfTime, both using textual representations of a time period as defined by ISO8601.
+The initial development phase focused on the collection of metadata during submission and the
+display of spatial metadata. Later phases will focus on the development of a more sophisticated and
+interactive display of the temporal metadata, specifically, putting the time period(s) of papers on a
+common timeline for specific issues and whole journals, adding support for multiple time periods,
+increasing the range of historic date formats supported (BC, time frames of millions of years, etc.),
+building in a function for entering coordinates directly, support for personalised reference datasets
+(related to a journal’s themes/topics, e.g., biospheres, habitats) for use in spatial metadata creation,
+and deriving spatio-temporal metadata semi-automatically, e.g., by retrieving information from data
+deposits or examining data files in supplementary materials.
+Fig. 10. Screenshot of the source code of the article landing page showing selected HTML meta
+attributes given in the HTML header, including different representations of spatial and temporal
+metadata.
+18
+
+1 <!DOCTYPE html>
+<html xml:lang="en-Us" lang="en-Us"><head>
+3 <meta http-equiv="content-type" content="text/html; charset=UTF-8">
+4 <meta charset="utf-8">
+5 <meta name="viewport" content="width=device-width, initial-scale=l.o">
+<title>Test 3: Three l Journal of Optimal Geolocations</title>
+9 <meta name="DC.temp0ral" scheme="Is08601" content="2022-06-27/2022-06-30">
+ll <meta name="geo.placename" content="Italian Republic">
+12 <meta name="DC.box" content="name=Italian Republic; northlimit=47.091783741544; southlimit=35.49285259236; westlimit=6.6266
+ <meta name="Is0 1913g" content="&lt;gmd:EX_GeographicBoundingBox&gt;&lt;gmd:westBoundLongitude&gt;&lt;gco:Decimal&gt;6.6266
+l4 <meta name="DC.Period0fTime" scheme="IS08601" content="2022-06-27/2022-06-30">
+<meta name="citation journal title" content="Journal of Optimal Geolocations">
+<meta name="citation author" content="c Contributor">
+<meta name="citation title" content="Test 3: Three">
+<link rel="schema.Dc" href="http://purl.org/dc/elements/l.l/">
+<meta name="DC.Coverage" xml:lang="en" content="Earth, Europe, Italian Republic">
+<meta name="DC.Creator.PersonalName" content="C Contributor">
+<meta name="DC.Title" content="Test 3: Three">
+ <meta name="DC.Type" content="Text.Serial.Journal">
+<meta name="DC.Type.articleType" content="Articles">Enriching the scholarly metadata commons
+3.3
+Enriching the scholarly metadata commons
+We conceptualise the scholarly metadata commons as a special subset of the knowledge commons
+(Hess and Ostrom 2006; Mansell 2013), in which an openly licenced and, thus, collectively owned
+aggregation of scholarly metadata is governed and shared among the community of interested
+scholarly and related stakeholders. This commons has various manifestations that present data in a
+user-friendly interface, in the form of websites or APIs, and enable both the contribution and
+extraction of data. Wikidata is a widely known example of such an interface.
+We make use of and contribute to the Scholarly Metadata Commons through both plugins:
+1. Citations plugin: With this plugin, we expand the open data pool for research information by
+providing enriched and user-verified metadata, collected and distributed at the time of
+publication, through open APIs. We also incorporate sources not currently included in the
+standard scientometric data sources because of their language or because they are not
+supported by big publishing houses.
+2. Spatio-temporal metadata plugin: Through this plugin, we enable new use cases such as
+location-based assessments of research activities and location-based research discovery, based
+on, for example, (1) questions about the geographical area being studied and (2) new
+transdisciplinary connections between research outputs based on time periods and areas of
+interest beyond commonly used keywords and full-text search.
+4
+Discussion
+The OPTIMETA Way described above provides three important contributions, which we implement
+here as exemplary with the presented plugins. The first contribution is the enlargement of the
+Scholarly Metadata Commons with metadata generated during the publication process. The built-in
+mechanisms for looking up existing metadata and the following import of persistent identifiers, such
+as ORCID iDs, enable the creation of strongly linked research information and its subsequent
+exportation into existing data sinks. While non-English publications play an important role in the
+academic world (Kulczycki et al. 2020; Liu 2017; Nazarovets and Mryglod 2021), their metadata are
+not currently equally represented in the major citation databases (Tennant 2020; Vera-Baceta et al.
+2019).
+The second contribution is, to allow for many more Open Access journals in citation databases and
+other services built upon the scholarly metadata commons. Currently, OJS is being used by more than
+25,000 journals from 155 countries (the majority being from the Global South) publishing in 56
+languages (Khanna and Willinsky 2022). Using our plugin will lower the barrier for independent
+journals to contribute to open bibliographic metadata considerably, albeit currently only if the
+journals use OJS. While this is a large step towards a solution, we cannot yet eliminate the problem
+entirely. Therefore, we hope that The OPTIMETA Way will be implemented in other publication
+platforms in the future.
+19
+
+Enriching the scholarly metadata commons
+The third contribution is the expansion of the scholarly metadata commons through the inclusion
+of spatio-temporal metadata, which facilitates the use of open metadata for new use cases. As Niers
+and Nüst (2020) explain, spatio-temporal metadata can be used to detect biases in the geographic
+coverage of research, for example, when research in a given field focuses heavily on one region
+overlooking other areas that may be no less interesting in the process. Spatio-temporal metadata can
+also help recognise connections between research works and improve the understanding of
+geographical and time-based relations within an area of study. Furthermore, visualisations, especially
+in the form of maps, can support the transfer of research content and the need for research as a whole.
+To date, the availability of spatio-temporal metadata has remained low and with the release of
+geoplugin, we will contribute a new component to the ecosystem of open scholarly publishing.
+Furthermore, the geoplugin will enable new use cases in location-data-based assessment of research
+activities: (1) answering questions about the area that has been investigated, e.g., to demonstrate a
+specific coverage or distribution of research locations and (2) detecting potentially valuable
+transdisciplinary connections between research outputs based on time periods and areas of interest
+that go beyond commonly used keywords and full-text search, e.g., connecting historical works on
+social questions in central Europe with current research on health. In the future this metadata can be
+used to build platforms for timely notifications about publications based on user-defined
+spatio-temporal interests, i.e., so that users or systems can be notified of new publications that cover
+an area of particular interest to an assessment scheme. Intentionally imprecise coordinates can be
+used to preserve the privacy of human subjects or hide protected entities.
+The integrative power of spatial relationships between research articles has already been
+acknowledged by others. However, none of the existing solutions follow The OPTIMETA Way and
+are, therefore, too complex, not integrated into the publishing workflow, or do not contribute to the
+open scholarly metadata commons. For example, the JournalMap (https://www.journalmap.org/; Karl
+et al. 2013) shows research paper locations and publication metadata (title, abstract, etc.) for
+map-based discovery. However, JournalMap is limited to point geometries and while there is an API
+and
+some
+collaboration
+with
+publishers
+(https://www.journalmap.org/publishers;
+https://web.archive.org/web/20161016000907/https://newsroom.taylorandfrancisgroup.com/news/pre
+ss-release/taylor-francis-journal-map-partnership#.WALFJmF_o88), the data is not fully open. The
+announcement that a data download option is "coming soon" has been on the website since its
+inception (see https://web.archive.org/web/20130615020154/https://www.journalmap.org/downloads)
+and
+the
+licence
+is
+defined
+as
+Creative
+Commons
+Attribution
+Share-Alike
+(CC-BY-SA,
+https://www.journalmap.org/developer/documentation/1-0), but the terms of use then limit the licence
+terms
+considerably
+and
+prohibit
+commercial
+use
+of
+the
+data
+(https://www.journalmap.org/terms-of-use). The website does offer some advanced filtering options,
+including additional thematic filtering options. However, the commercial options advertised on the
+website work against our understanding of knowledge advancement. Second, Kmoch et al. (2018)
+analysed articles from geoscientific journals to automatically derive spatial metadata from the
+unstructured information in articles' bibliographic metadata. The extracted data was then published in
+a public geospatial catalogue service. However, this approach required considerable technological
+knowledge and lacked human quality checks, as not all data was checked by the most suitable
+experts. Therefore, despite being a valid approach for dealing with the fact that spatio-temporal
+20
+
+Enriching the scholarly metadata commons
+metadata was not collected in the past, it is neither a complete solution nor in line with The
+OPTIMETA Way. Garzón and Nüst (2021b) took a similar approach with the tool geoextent, which
+they used to create a discovery index based on geospatial metadata for generic research data
+repositories. They used a brute-force approach to retrieve spatial extents from as many geospatial file
+formats as possible. This could be an intermediary approach to enrich article metadata if the articles
+properly cite the data used, though human verification would likely be needed as datasets may be
+cited for many different reasons. An implementation of the search portal (cf. Fig. 2) that collects
+spatio-temporal metadata from multiple journals, the OPTIMAP, is currently under development (see
+https://optimap.science/ and https://github.com/ifgi/optimetaPortal).
+The advantages offered by the availability of open citation information are undeniable. Peroni
+and Shotton (2020) provide an extensive list of beneficiary stakeholders: researchers who do not
+belong to "the elite club of research universities that can afford subscription access to the commercial
+citation indexes WoS and Scopus", bibliometricians, who want to provide research data on their
+research, librarians, funders, research managers, and much more. A key value of the citation plugin is
+that it allows a large set of Open Access journals to share their authors' publications in the open
+research commons, regardless of language or subject area and whether it is a publisher-led or
+independent scholar-led journal. The often shamefully overlooked long-tail of academic research will
+thus become visible and be given the opportunity to be properly integrated, especially the often
+overlooked non-English literature (Lazarev and Nazarovets 2018).
+With respect to science communication and assessment, Krüger (2020) describes how the social
+distrust of science is to be countered by a performativity-measuring quantification of research output
+and associated metadata and indicators. She argues, convincingly, that bibliometric infrastructures
+and applications have their own ideas about how research can be understood through their use.  By
+expanding the scope and range of what we have in terms of metadata, The OPTIMETA Way cannot
+fully prevent this, but it can address the extent of the problem by limiting the distorted perception of
+what can be observed, measured, assessed, and considered knowledge through its digital
+representation in metadata. In this way, the improved availability of spatio-temporal and citation
+metadata means that research assessment can be carried out more quickly, easily, transparently, and
+responsibly. Desiderata in terms of metadata could be machine-actionable descriptions of research
+problems, methods, connections to external entities like funding IDs or funders, identifiers for
+physical samples, or identifiers for instruments.
+We estimate the risk of unethical use of the plugins is low and do not see particular potential for the
+misuse of spatio-temporal data. Even in the worst-case scenario, while intentionally defective
+metadata reduces discoverability, it does not impact other means of identifying publications. It is true
+that, in relation to citation metadata, the plugins do lower the barrier to depositing falsified citation
+information when irresponsible research assessment methods such as citation counts are of interest to
+a malevolent party. However, citation metadata can only be misused if both author and the handling
+editor, who is encouraged to check the citation information before publication, have malicious intent.
+Furthermore, the deposition in public databases does not happen anonymously and, once identified,
+any misuse can be rolled back and accounts can be blocked from uploading further data.
+21
+
+Enriching the scholarly metadata commons
+In the future, the implementations of The OPTIMETA Way presented here could be extended with
+new features and improved usability based on the experiences reported by journals from different
+disciplines. Regarding technology, we imagine more sophisticated methods, such as machine learning
+approaches or the extraction of information from PDFs and data files, could be integrated into the
+OPTIMETA
+plugins
+to
+increase
+the usability and extent of the metadata. For example,
+acknowledgements such as funding bodies and grant IDs, author contributions (e.g., based on an
+acknowledgement section using CRediT statements), or subject classifications, could be collected in
+a similar, semi-automatic way and publicly deposited according to The OPTIMETA Way. However,
+to protect the quality of the data, validation by a human expert should not be omitted. The scope of
+the enhancement with metadata can be widened to include preprints, monographs, and edited
+collections, although semantically meaningful attributes to metadata fields will be required to
+distinguish non-reviewed research outputs from reviewed ones. To support preprints and books, the
+citation plugin and geoplugin can be ported to PKP's preprint platform in addition to other book
+publishing platforms.
+The shift in granularity and speed that can be expected due to more open and also more pressing
+research as societal challenges are tackled in the future will require even more timely, validated
+research metadata for effective communication. Research is increasingly being published in stages
+(e.g.,
+Octopus,
+https://www.octopus.ac/about)
+and
+as
+individual
+building
+blocks
+(e.g.,
+idea/text/interpretation, code/software, data), rather than as a polished textual artefact years after the
+issue might already be resolved. Researchers will more regularly share their results accessibly on free
+infrastructures and peer review practices will adapt, e.g., with overlay journals (Brown 2010; Rousi
+and Laakso 2022). However, these technical challenges are small when compared to the
+organisational challenges of ensuring the long-term maintenance of the plugins we have developed.
+While the current funding facilitated the development of stable plugins and provided for them to be
+sent to select collaboration partners for evaluation, and while more and more programs funding core
+research software are being founded (e.g., https://chanzuckerberg.com/eoss/), we are still facing a
+chicken-and-egg problem. For broad uptake, journals require a commitment to long-term software
+maintenance, while funding to maintain the plugins and improve them is acquired more easily when
+broad usage can be demonstrated.
+The cultural shift towards Open Access and FAIR research information housed in open
+infrastructures (Hauschke et al. 2021a; Hendricks et al. 2021) will happen at different speeds in
+different countries and disciplines and result in the coexistence of a variety of platforms. This is an
+advantage over today’s centralised system and the power large publishers have over it, but it is also a
+challenge as these services will have to be able to connect and exchange metadata. Therefore, it is our
+hope that The OPTIMETA Way will be transferred to other elements within the academic open
+infrastructure, so that targeted, novel, and even small scale metadata attributes can be collected from
+the most knowledgeable party, with minimal impact on existing workflows, and shared broadly,
+openly, and quickly for the advancement of knowledge.
+22
+
+Enriching the scholarly metadata commons
+5
+Conflict of Interest
+The authors declare that the research was conducted in the absence of any commercial or financial
+relationships that could be construed as a potential conflict of interest.
+6
+Author Contributions
+The authors contributed equally to this work. Development of the described plugins was conducted
+by Daniel Nüst (spatio-temporal metadata plugin) and Gazi Yücel (citation plugin).
+7
+Funding
+The authors acknowledge the financial support by the Federal Ministry of Education and Research of
+Germany (BMBF) in the framework of OPTIMETA (grant numbers 16TOA028A and 16TOA028B).
+8
+Acknowledgements
+The authors would like to thank our project partners for the continuous discussions on how to
+improve the OPTIMETA Way. In particular, special thanks go to Open Citations and PKP for their
+supportive engagement in technical discussions. We thank Tom Niers for developing the first
+prototype of the spatio-temporal metadata plugin, and Svantje Lilienthal for contributions to the early
+conceptual discussions on the citations plugin. We thank Julie Davies from the academic editing
+service of the University of Münster for her support in revising the manuscript.
+9
+References
+Bijsterbosch, Magchiel/Dunning, Alastair/Jansen, Darco/Haring, Max/Rijcke, Sarah de/Vanderfeesten, Maurice (2022).
+Seven Guiding Principles for Open Research Information. https://doi.org/10.5281/zenodo.6074943.
+Brembs, Björn (2021). Tweet on Elsevier's publishing costs. Available online at
+https://twitter.com/brembs/status/1440942528023470081.
+Brown, Josh (2010). An Introduction to Overlay Journals. Available online at https://discovery.ucl.ac.uk/id/eprint/19081/.
+Cagan, Ross (2013). The San Francisco Declaration on Research Assessment. Disease models & mechanisms 6 (4),
+869–870. https://doi.org/10.1242/dmm.012955.
+Conlon, Michael/Woods, Andrew/Triggs, Graham/O'Flinn, Ralph/Javed, Muhammad/Blake, Jim/Gross,
+Benjamin/Ahmad, Qazi/Ali, Sabih/Barber, Martin/Elsborg, Don/Fofack, Kitio/Hauschke, Christian/Ilik,
+Violeta/Khan, Huda/Lawless, Ted/Levernier, Jacob/Lowe, Brian/Martin, Jose/McKay, Steve/Porter,
+Simon/Walther, Tatiana/White, Marijane/Wolff, Stefan/Younes, Rebecca (2019). VIVO: a system for research
+discovery. Journal of Open Source Software 4 (39), 1182. https://doi.org/10.21105/joss.01182.
+euroCRIS (2018). 14th International Conference on Current Research Information Systems.
+https://doi.org/10.25798/2y3h-k877.
+Federal Geographic Data Committee. ISO 191** Suite of Geospatial Metadata Standards. Available online at
+https://www.fgdc.gov/metadata/iso-suite-of-geospatial-metadata-standards.
+Franceschi-Bicchierai, Lorenzo (2022). Academic Journal Claims it Fingerprints PDFs for ‘Ransomware,’ Not
+Surveillance. Available online at
+https://www.vice.com/en/article/4aw48g/academic-journal-claims-it-fingerprints-pdfs-for-ransomware-not-surveill
+ance.
+23
+
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+26
+

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+Online Learning Based Mobile Robot
+Controller Adaptation for Slip Reduction
+Huidong Gao ∗ Rui Zhou ∗ Masayoshi Tomizuka ∗ Zhuo Xu ∗
+∗ Department of Mechanical Engineering, University of California,
+Berkeley, CA 94720 USA (e-mail: {hgao9, ruizhouzr, tomizuka,
+zhuoxu}@berkeley.edu)
+Abstract:
+Slip is a very common phenomena present in wheeled mobile robotic systems. It has undesirable
+consequences such as wasting energy and impeding system stability. To tackle the challenge
+of mobile robot trajectory tracking under slippery conditions, we propose a hierarchical
+framework that learns and adapts gains of the tracking controllers simultaneously online.
+Concretely, a reinforcement learning (RL) module is used to auto-tune parameters in a lateral
+predictive controller and a longitudinal speed PID controller. Experiments show the necessity of
+simultaneous gain tuning, and have demonstrated that our online framework outperforms the
+best baseline controller using fixed gains. By utilizing online gain adaptation, our framework
+achieves robust tracking performance by rejecting slip and reducing tracking errors when the
+mobile robot travels through various terrains.
+Keywords: Trajectory Tracking, Slip Rejection, Reinforcement Learning, Hierarchical Control
+1. INTRODUCTION
+1.1 Background and Motivation
+Mobile robots are used in various industrial applications
+such as manufacturing, process and aerospace. They often
+run on slippery terrains, or routes with rapid cornering,
+which induces skidding and slipping. Excessive slip may
+cause motion instability, undermine maneuverability and
+lead to possible collisions, thus should be prevented.
+To mitigate slip, many works try to identify slip parame-
+ters or terrain states, and design simple control laws with
+robot kinematic models, e.g. Pico et al. [2022] and Kim
+and Lee [2016]. Sebastian and Ben-Tzvi [2019] and Wang
+and Zhai [2020] further choose to model slip as distur-
+bance in the kinematics model and estimate by observers.
+However, state vectors are of high order and matrix inverse
+calculations could be massive. Another line of work focuses
+on wheel dynamics with traction forces. Tian et al. [2009]
+choose to use the Magic formula to derive the relationship
+between traction force and slip ratio, while Nandy et al.
+[2011] formulates a detailed slip dynamics with certain
+switching conditions. Although dynamic models consider
+forces in addition to kinematics models, they usually re-
+quire system identification for different scenarios and have
+poor generalization abilities.
+There are also other works utilizing reinforcement learning
+to directly learn a policy; such as in Xu et al. [2018], Tang
+et al. [2019], Chang et al. [2020], Cai et al. [2020], Xu et al.
+[2021]. However, end-to-end RL approaches require consid-
+erable training time and pose challenges in explainability.
+Instead of end-to-end RL, Carlucho et al. [2019], Gao et al.
+[2022] uses RL only to optimize controllers, but the results
+are highly dependent on action space discretization.
+1.2 Contributions
+Our work focuses on trajectory tracking control of mo-
+bile robots under slippery conditions. We follow a similar
+approach as in Carlucho et al. [2019], and propose to
+use a hierarchical framework that optimizes gains for the
+tracking controllers online. An RL module is used to tune
+gains in a lateral predictive Stanley controller and a longi-
+tudinal speed PID controller simultaneously for regulation.
+By dividing the control part into longitudinal and lateral
+control modules and tuning gains directly, we are able to
+improve lateral and speed tracking errors in a straight-
+forward way. By using a higher level RL module, we are
+able to tune multiple low-level controllers simultaneously
+in real-time. Furthermore, the RL module is only able to
+determine the conservativeness in the controllers, and thus
+the entire framework is more explainable than an end-to-
+end RL controller.
+The contributions of our work can be summarized as
+follows: 1) We propose an hierarchical framework that
+actively optimizes controllers to slip conditions through
+RL gain-tuning. 2) We reason the necessity of simultaneous
+online gain tuning through experiments. 3) We demon-
+strate that our adaptive framework outperforms the best
+fixed-gain baselines by 6.6% and 12.7% for average lateral
+error and max lateral error by simulation.
+2. METHODOLOGY
+2.1 Problem Overview
+Fig. 1 illustrates our tracking problem layout. The robot’s
+goal is to travel from xstart, following a predefined trajec-
+tory to reach xend. Here we define certain terms to describe
+the robot’s motion and the tracking state. We use lateral
+arXiv:2301.13283v1  [cs.RO]  30 Jan 2023
+
+displacement error e to represent the closest tracking error
+relative to the reference trajectory (in unit m). ∆θ is the
+yaw error, which is the difference between reference yaw
+θref and actual yaw θ (in unit rad). ∆v is the speed error,
+which is the difference between absolute values of reference
+velocity vref and actual velocity v (in unit m/s).
+The tracking task can then be formulated as a Markov
+Decision Process defined by M = (S, A, T , R, γ). S repre-
+sents the state space; A is the action space; T (s
+′|s, a) is the
+state transition model; R(s, a) is the reward function; and
+γ ∈ [0, 1) is the discount factor. The RL formulation aims
+to learn a policy π(a|S). The agent then follows the policy
+π, obtains an observation st at time t and performs an ac-
+tion at. It then receives from the environment a reward Rt
+and a new observation st+1, and π is updated accordingly.
+The final trained model gives an action selection policy
+π that maximizes the expectation of a discounted sum of
+rewards E[�T
+t=1 γt−1Rt].
+Fig. 1. Schematic diagram of the problem setup.
+The state S has 5 variables: e, ∆θ, ∆v, ∆vc vs actual,
+and ∆ωc vs actual. e, ∆θ, and ∆v are as discussed in the
+beginning of the section. ∆vc vs actual (in unit m/s)and
+∆ωc vs actual (in unit rad/s) represent the difference be-
+tween actual body velocities and body velocities calculated
+from wheel velocity commands(shown in Eqn. 1 and 2).
+Intuitively, a large value of ∆vc vs actual or ∆ωc vs actual
+indicates the robot is slipping more severely as the wheel
+velocity commands are not fully transferred to actual body
+velocities.
+v = (ωR + ωL) · R
+2
+(1)
+ω = (ωR − ωL) · R
+b
+(2)
+Eqn. 1-2: transformation from wheel commands to
+calculated body linear and angular velocities. ωR and ωL
+are right and left wheel angular velocity commands, R is
+wheel radius, and b is distance between wheels.
+The action A is [v, ω], which are linear and angular
+velocity commands. Notice the final input wheel velocity
+commands are calculated from reversing equations 1-2.
+The low level controller executes the commands and gets a
+reward at this step. The reward for each step is defined in
+Eqn. 3. Here we penalize e, ∆θ and ∆v, with coefficients
+Rdist, Rang and Rspeed. The cumulative reward is defined
+as �T
+t=1 γt−1Rt, where Rt is the step reward.
+Rt(st, at) = Rdist · e2 + Rang · ∆θ2 + Rspeed · ∆v2
+(3)
+2.2 Proposed Framework
+We propose to utilize reinforcement learning to actively
+tune parameters in lateral and longitudinal control mod-
+ules on a differential drive TurtleBot. The proposed frame-
+work consists of a RL-based high-level module, a lateral
+control module, a longitudinal control module, a low-
+level tracking controller, and the robot. The framework
+is visualized in Fig. 2. The RL module takes observed
+robot states obs; reference trajectory xref, and outputs
+gains Kstanley and Kspeed. The two control modules use
+the gains accordingly and output acceleration command α
+and steering angle command δ, and then transfer them into
+linear and angular velocity commands [v, ω]. The low level
+controller then executes the command on the robot and
+feeds the directly observed states back into the RL module
+to calculate for step rewards, and the policy is updated
+accordingly. The loop stops when the positional error of
+the robot and final goal position is within a threshold.
+The entire framework realizes our MDP formulation, and
+is trained end-to-end using an RL algorithm.
+Fig. 2. Proposed framework.
+2.3 Lateral Control: Predictive Stanley
+We tackle our trajectory tracking problem by breaking
+it up into longitudinal and lateral control problems. The
+longitudinal controller is responsible for regulating the
+robot’s speed while the lateral controller aims to reduce
+the lateral error during path tracking.
+The proposed lateral control approach utilizes a version of
+Predictive Stanley controller, which is built on the basic
+Stanley controller. The basic Stanley controller is divided
+into three regions: saturated low region, saturated high
+region, and nominal region. ψ is the heading of the vehicle
+with respect to the heading of the trajectory at the point of
+the projected shortest distance to the vehicle position, e(t)
+is the lateral error, v is current speed, and K = Kstanley is
+the controller gain. See Fig. 3 for reference. The steering
+angle command is given by:
+δ(t) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ψ(t) + arctan( Ke(t)
+v(t) ),
+|ψ(t) + arctan( Ke(t)
+v(t) )| < δ(max)
+δ(max),
+ψ(t) + arctan( Ke(t)
+v(t) ) >= δ(max)
+−δ(max),
+ψ(t) + arctan( Ke(t)
+v(t) ) <= −δ(max)
+(4)
+As discussed in AbdElmoniem et al. [2020], the proposed
+predictive Stanley control approach introduces a third
+input, which is a developed array of future vehicle states,
+propagated along the vehicle track, denoted as P1, P2... PN
+
+0
+0
+Xend
+e
+re
+Vref
+x
+XstartLateral control
+nley
+Predictive
+re
+Stanley
+Low Level
+RL
+Robot
+Controller
+Longitudinal contro
+PID
+speed
+Vref
+obsFig. 3. Predictive Stanley Representation.
+as shown in Fig. 3. At each future state, the corresponding
+δ is calculated based on current epi. The final steering
+angle command is calculated by augmenting the output
+of each basic Stanley controller at each state to eliminate
+the error along the path not only at the reference point,
+as shown in Eqn. 5 and 6. Consequently, the predictive
+Stanley controller is able to deal with the sudden changes
+in the heading angle of the trajectory by having this
+preview capability.
+δ(t) =
+N
+�
+i=0
+pi[ψi(t) + arctan(Kepi(t)
+v(t)
+)]
+(5)
+pi = p2
+i−1 for i = 2...N
+(6)
+Eqn. 5-6. The final steering command. pi is the
+weight, which represents how each controller contributes
+in determining the final value of the steering angle. Here
+we set N = 2 and p1 = 0.2.
+We show the advantage of our predictive Stanley controller
+over the basic Stanley controller by running them on a
+TurtleBot with the same trajectory. The visualization in
+Fig. 4 clearly shows that the predictive Stanley controller
+is able to adjust for abrupt turns. See detailed comparison
+in AbdElmoniem et al. [2020].
+2.4 Longitudinal Control: PID
+We use a simple proportional control for speed regulation.
+The acceleration command becomes:
+α(t) = Kspeed(vref − v(t))
+(7)
+With the steering and acceleration commands, we can
+deduce the robot’s linear and angular velocity commands
+[v, ω] using Eqn. 8 and 9, which are then executed by the
+low level controller.
+vcommand = v(t) + α(t) · ∆T
+(8)
+ωcommand = δ(t)
+∆T
+(9)
+2.5 Reinforcement Learning module
+The RL module in Fig. 2 consists of actor and critic neural
+network layers. The entire framework in Fig. 2 utilizes soft
+actor-critic (SAC) during training.
+Fig. 4. Predictive Stanley vs. Basic Stanley Con-
+troller. The average lateral error for trajectory
+A for predictive and basic Stanley controllers are
+0.0147m and 0.0374m, respectively; for trajectory B
+are 0.0077m and 0.0347m, respectively.
+3. EXPERIMENTS
+Our experiments were designed and conducted in order to
+answer the following questions:
+(1) Is simultaneous gain tuning necessary?
+(2) Is online gain tuning better than fixing the gains
+throughout the trajectory?
+(3) How to interpret our framework’s output?
+To answer these questions, we carry out simulated ex-
+periments using PyBullet by Coumans and Bai [2016–
+2021], with a TurtleBot waffle-pi model. To evaluate our
+framework, we propose to use a set of long-term and short-
+term metrics. Long-term metrics focus on measuring the
+performance throughout the entire trajectory, while short-
+term metrics focus on the short-time performance while
+the robot is slipping. Here we define slipping condition as
+those robot body states satisfying ∆vc vs actual > |0.7|m/s
+or ∆ωc vs actual > |3|rad/s.
+For long-term criteria, we define average episodic reward
+r, average lateral error e, average speed error ∆v, and
+average RMS of change in low level control command
+∆u(which measures command stability). u = [ωL, ωR],
+which denotes left and right wheel velocity control action
+commands, and is calculated based on [v, ω], using Eqn. 1
+and 2. The long-term metrics are calculated with respect
+to the entire trajectory.
+r =
+1
+Ttraj
+Ttraj
+�
+i=0
+ri
+(10)
+e =
+1
+Ttraj
+Ttraj
+�
+i=0
+ei
+(11)
+
+e.
+p2
+Xend
+p1
+0
+V
+y
+Xstart
+>x8
+8
+trajectory
+trajectory
+7
+target
+7
+target
+6
+6
+5
+5
+4
+4
+3
+3
+2
+2
+1
+1
+0
+0
+-1
+2
+4
+16
+-1
+0
+8
+2
+4
+6
+8
+Trajectory A
+8
+8
+trajectory
+trajectory
+7
+target
+7
+target
+6
+6
+5
+5
+4
+4
+3
+3
+2
+2
+1
+1
+0
+0
+-1
+2
+-1
+4
+6
+8
+0
+2
+4
+6
+8
+Trajectory B∆v =
+1
+Ttraj
+Ttraj
+�
+i=0
+|vi − vref|,
+(12)
+∆u =
+1
+Ttraj
+Ttraj
+�
+i=1
+∥ui+1 − ui∥,
+(13)
+For short-term criteria, we define max lateral error
+throughout the trajectory emax, average lateral error dur-
+ing slipping(Tslip) eslip, average speed error during slipping
+∆vslip, and average RMS of change in low level control
+action during slipping ∆uslip.
+emax = max{ei}Ttraj
+0
+(14)
+eslip =
+1
+Tslip
+Tslip
+�
+i=0
+ei
+(15)
+∆vslip =
+1
+Tslip
+Tslip
+�
+i=0
+|vi − vref|,
+(16)
+∆uslip =
+1
+Tslip
+Tslip
+�
+i=1
+∥ui+1 − ui∥,
+(17)
+3.1 Simulation environment setup and training
+Fig. 5 shows bird-eye view simulation renderings of three
+example setups. Blue color represents high frictional areas
+with frictional coefficient µ=0.9, and red represents low
+frictional areas with µ=0.01. The red patches are of size
+1 m by 1 m. The green trajectory is generated using a
+spline generator by Sakai et al. [2018]. The planner takes
+n number of 2D points and generates a smooth trajectory
+connecting all the given points.
+To randomize the trajectory, we choose to use 5 random
+points for curve generation. Each point is Uniform[1, 2] m
+away from the previous point, with Uniform[−0.5π, 0.5π]
+rad angle from the previous point. The initial point [x, y]
+position follows distribution: x = Uniform[1, 2] m, y =
+Uniform[3.5, 4.5] m.
+To randomize the ground configuration, the red patches
+are randomly generated for each new trajectory, and we
+set 30% of the total area to be red.
+We use SAC algorithm to train the entire framework. The
+policy network and the value network in the SAC are fully-
+connected two-layer neural networks of size 64. Learning
+rate is 0.0006, γ is set to be 0.99. The coefficients Rdist,
+Rang and Rspeed are set to be -20, -1, -1, respectively. The
+entire framework is trained until convergence.
+3.2 Is simultaneous gain tuning necessary?
+We propose to utilize RL to tune the two gains simulta-
+neously, rather than having two separate frameworks to
+determine each. To verify the necessity of simultaneous
+gain tuning, we vary Kstanley and Kspeed from 0.5 to 5.0
+with 0.5 increments, and plot heatmaps for each of the
+Fig. 5. Simulation Rendering. Bird-eye view image with
+annotations. White dot is the robot, green line is the
+reference trajectory, and red dot is the goal position.
+criteria discussed previously (Fig. 6 and 7). Each point
+on the heatmap represents the result of using a specific
+combination of Kstanley and Kspeed. Each point result is
+calculated by averaging the results of running 100 pre-
+generated random trajectories with random ground setups.
+It can be shown that for each criteria, the best result
+happens when considering Kstanley and Kspeed together.
+For example, for e, fixing Kstanley to be 2.5 will result in a
+best Kspeed of 3.5, but fixing Kstanley to be 5.0 will result
+in a best Kspeed of 2.0. Therefore the gains have to be
+tuned simultaneously in order to achieve the best results.
+For different criteria, the optimum happens at different
+combinations of Kstanley and Kspeed, because the criteria
+are focused on different aspects. For instance, to stabilize
+command and reduce ∆u, e may be compromised because
+commands need to be tuned less abruptly.
+Fig. 6. Parameter sweeping results for long-term
+metrics
+3.3 Is online gain tuning better than fixed parameters?
+We propose to tune the gains online throughout the
+entire trajectory rather than using fixed gains. To verify
+this, we use a baseline model. The baseline model uses
+the same framework as in Fig. 2, but without the RL
+module. Instead of varying the two gains online, the
+baseline model utilizes fixed best gains found by offline
+parameter sweeping in Kstanley and Kspeed. The best gain
+combinations of baseline model for each metric is shown
+in the second column in Parameter Sweeping in Tables
+1 and 2. We run the same 100 pre-generated random
+
+Ir
+e, sign flipped
+0.5
+0.5
+1.0
+1.0
+0.02
+0.15
+1.5 -
+1.5
+0.03
+2.0
+0.20
+2.0
+2.5
+2.5
+0.04
+0.25
+3.0
+3.0
+0.05
+3.5
+3.5
+0.30
+4.0
+4.0
+0.06
+4.5
+0.35
+4.5
+5.0
+5.0
+0.07
+0.40
+0.51.01.52.02.53.03.54.04.55.0
+0.51.01.52.02.53.03.54.04.55.0
+Kspeed
+Av, sign flipped
+Au, sign flipped
+0.5
+0.10
+0.5
+1.0
+1.0 -
+10.0
+0.15
+1.5
+1.5
+12.5
+2.0
+0.20
+2.0
+15.0
+2.5
+2.5
+0.25
+3.0
+3.0
+17.5
+3.5
+0.30
+3.5
+20.0
+4.0
+4.0
+0.35
+22.5
+4.5
+4.5
+5.0 
+25.0
+0.40
+5.0
+0.51.01.52.02.53.03.54.04.55.0
+0.51.01.52.02.53.03.54.04.55.0
+Kspeed
+KspeedFig. 7. Parameter sweeping results for short-term
+metrics
+trajectories for the baseline model and our trained model,
+and log the results in the two tables. It can be shown that
+our framework is able to improve e, emax, eslip by 6.6%,
+12.7%, and 4.7%, respectively.
+One thing to notice is that the best results for each metric
+happens at different gain combinations with the baseline
+parameter-sweeping model. For example e has best results
+when setting Kstanley = 2.0 and Kspeed = 0.5, but for
+∆u it’s Kstanley = 0.5 and Kspeed = 3.5, which means
+the baseline model will perform worse if using the same
+combination of gains for all metrics. However our model
+is still able to beat the best of each baseline model metric
+with a universal trained policy, in lateral error metrics and
+∆u metric.
+Table 1. Long-term metrics comparison.
+The second column in Parameter Sweeping
+indicates at what value of gains the best metric
+result was obtained.
+Metrics
+Parameter Sweeping
+Proposed Framework Improvement
+(−)r
+0.110 ± 0.118 Kstanley = 1.5
+Kspeed = 3.5
+0.084 ± 0.125
+23.6%
+e
+0.015 ± 0.010 Kstanley = 2.0
+Kspeed = 0.5
+0.014 ± 0.017
+6.6%
+∆v
+0.096 ± 0.042 Kstanley = 0.5
+Kspeed = 3.5
+0.097 ± 0.058
+-1.0%
+∆u
+8.320 ± 2.040 Kstanley = 0.5
+Kspeed = 3.5
+5.917 ± 2.165
+28.9%
+Table 2. Short-term metrics comparison.
+Metrics
+Parameter Sweeping
+Proposed Framework Improvement
+emax
+0.079 ± 0.055 Kstanley = 2.0
+Kspeed = 0.5
+0.069 ± 0.066
+12.7%
+eslip
+0.021 ± 0.013 Kstanley = 2.0
+Kspeed = 0.5
+0.020 ± 0.032
+4.7%
+∆vslip 0.295 ± 0.082 Kstanley = 4.5
+Kspeed = 5.0
+0.42 ± 0.087
+-42.3%
+∆uslip
+18.9 ± 5.93
+Kstanley = 1.0
+Kspeed = 3.5
+21.27 ± 4.76
+-12.5%
+3.4 Explainability of our framework output
+We visualize two setup results in Fig. 8 and 9. The
+upper figures show the baseline model with best gain
+combinations, and the lower figures show our model.
+In the first setup, the robot using the baseline model failed
+to reach the end and got stuck when first enters the patch
+area, while our model is able to succeed. A closer look in
+the right figure reveals that our model reduces Kstanley
+when the robot enters the low frictional area and detects
+slip. It makes sense as when slip happens, heavy steering
+will not help much and could worsen the slip. A good
+tuning on Kstanley and Kspeed helps the robot to control
+the slip and succeed in the tracking task in this case. And
+it is up to the RL module to decide when and how much
+to tune the two gains. Also, the RL module does not have
+prior knowledge of the location of low-frictional area, and
+it is able to tune the gains based on current tracking status
+and reduce lateral error successfully.
+Similar observation can be made in the second setup.
+Both models succeeded in the task, but the baseline model
+induced more lateral error when the robot entered low-
+frictional area, because it did not lower Kstanley accord-
+ingly.
+Fig. 8. Trajectory Comparison 1. The left figures show
+the bird-view map. The red patches correspond to
+the low frictional area. The right figures show how
+the gains evolve versus time. The blue dotted line
+indicates when the robot is on the low frictional patch.
+A value of 1 indicates the robot is on patch.
+4. DISCUSSION
+The experiments conducted show that simultaneous and
+online tuning of gains are necessary for mobile robot
+trajectory tracking under slippery conditions. Our model is
+able to tune the gains in lateral and longitudinal controls,
+and beat the baseline model in terms of lateral error
+metrics.
+To reason the need of simultaneous gain tuning, con-
+sider when the robot is slipping or trajectory contains
+a large curvature. Both acceleration and steering com-
+mands should be considered to achieve an optimal tracking
+performance. For instance, with a large curvature, speed
+regulation can be relaxed while the steering command
+
+emax, sign flipped
+eslip, sign flipped
+0.5
+0.08
+0.5
+0.02
+1.0
+1.0
+0.10
+0.03
+1.5
+1.5
+0.12
+2.0
+2.0
+0.04
+2.5
+0.14
+2.5
+0.05
+3.0
+3.0
+0.16
+3.5
+3.5
+0.06
+0.18
+4.0
+4.0
+0.07
+4.5
+0.20
+4.5
+5.0
+0.22
+5.0
+0.08
+0.51.01.52.02.53.03.54.04.55.0
+0.51.01.52.02.53.03.54.04.55.0
+Kspeed
+Kspeed
+AVslip, sign flipped
+Auslip, sign flipped
+0.5
+0.30
+0.5
+-18
+1.0
+1.0
+-20
+0.35
+1.5 
+1.5
+-22
+2.0 -
+-0.40
+2.0
+24
+2.5 
+2.5
+0.45
+3.0
+3.0
+26
+3.5 
+0.50
+3.5
+28
+4.0
+4.0
+30
+0.55
+4.5 -
+4.5
+32
+5.0
+0.60
+5.0
+34
+0.51.01.52.02.53.03.54.04.55.0
+0.51.01.52.02.53.03.54.04.55.010.0
+target
+Kstanley
+trajectory
+7.5
+Kspeed
+on patch
+5.0
+6
+2.5
+0.0
+2.5
+3 
+5.0
+2
+7.5
+10.0
+2
+3
+4
+5
+6
+200
+400
+600
+800
+1000
+1200
+=
+10.0
+target
+Kstaniey
+trajectory
+7.5
+Kspeed
+on patch
+6
+5.0
+2.5
+0.0
+2.5
+3
+5.0 
+2
+7.5
+10.0
+2
+3
+4
+5
+6
+0
+25
+50
+75
+100
+125
+150
+175
+Proposed FrameworkFig. 9. Trajectory Comparison 2. Using online tuning
+instead of fixing gains alleviates deviation when the
+robot enters the slippery area. e for baseline and pro-
+posed framework are 0.0247 and 0.0112, respectively.
+emax are 0.0624 and 0.0336, respectively.
+needs to have a bigger gain. Or when robot is slipping,
+both gains might need to be adjusted, as seen in Fig. 8 and
+9. The magnitude of commands needs to be determined
+simultaneously, and the RL module in our framework
+decides the magnitudes of both gains at each timestep.
+Also notice for some metrics such as ∆u and ∆uslip,
+Kspeed tuning dominates the performance. One reason is
+that speed regulation impacts the wheel velocity change
+more than angular regulation, because sometimes the
+curvature is not very abrupt for steering to change much,
+but speed has to be regulated all the time.
+We showed our framework is able to improve e, emax,
+eslip by 6.6%, 12.7%, and 4.7%, respectively. However
+the command and speed stability during slipping were
+compromised a bit. It makes sense as the robot tries to
+relax other constraints in order to reduce the lateral error.
+In many mobile robot slipping scenarios such as in a
+factory setting, the most important aspect is to reduce
+the lateral error because deviations could lead to robot
+hitting unwanted objects and causing harms. Therefore a
+lower stability in speed and command are acceptable.
+5. CONCLUSION AND FUTURE WORK
+To reduce slip for mobile robots, we propose a hierarchical
+framework that utilizes an RL module to adapt gains
+for the tracking controllers simultaneously online. We
+demonstrated the necessity of simultaneous gain tuning,
+and showed that our online framework outperforms the
+best baseline model using fixed gains, especially in terms
+of long and short term lateral errors.
+REFERENCES
+AbdElmoniem, A., Osama, A., Abdelaziz, M., and Maged,
+S.A. (2020). A path-tracking algorithm using predictive
+stanley lateral controller. International Journal of Ad-
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+Cai, P., Mei, X., Tai, L., Sun, Y., and Liu, M. (2020).
+High-speed autonomous drifting with deep reinforce-
+ment learning. IEEE Robotics and Automation Letters,
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+Carlucho, I., De Paula, M., and Acosta, G.G. (2019).
+Double q-pid algorithm for mobile robot control. Expert
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+Chang, H., Xu, Z., and Tomizuka, M. (2020).
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+attribute network: Decomposing reinforcement learn-
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+Coumans, E. and Bai, Y. (2016–2021). Pybullet, a python
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+machine learning. http://pybullet.org.
+Gao, H., Zhou, R., Tomizuka, M., and Xu, Z. (2022). Re-
+inforcement learning based online parameter adaptation
+for model predictive tracking control under slippery con-
+dition. In 2022 American Control Conference (ACC),
+2675–2682. IEEE.
+Kim, J. and Lee, J. (2016).
+A kinematic-based rough
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+exploration rover.
+In 2016 IEEE/RSJ International
+Conference on Intelligent Robots and Systems (IROS),
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+R.,
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+T.,
+Chakraborty, G., and Kumar, C. (2011). Detailed slip
+dynamics for nonholonomic mobile robotic system. In
+2011 IEEE International conference on mechatronics
+and automation, 519–524. IEEE.
+Pico, N., Jung, H.r., Medrano, J., Abayebas, M., Kim,
+D.Y., Hwang, J.H., and Moon, H. (2022).
+Climbing
+control of autonomous mobile robot with estimation of
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+Mechanical Science and Technology, 36(2), 959–968.
+Sakai, A., Ingram, D., Dinius, J., Chawla, K., Raffin,
+A., and Paques, A. (2018). Pythonrobotics: a python
+code collection of robotics algorithms. arXiv preprint
+arXiv:1808.10703.
+Sebastian, B. and Ben-Tzvi, P. (2019). Active disturbance
+rejection control for handling slip in tracked vehicle
+locomotion. Journal of Mechanisms and Robotics, 11(2),
+021003.
+Tang, C., Xu, Z., and Tomizuka, M. (2019). Disturbance-
+observer-based tracking controller for neural network
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+IEEE Transactions on Intel-
+ligent Transportation Systems, 21(9), 3961–3972.
+Tian, Y., Sidek, N., and Sarkar, N. (2009).
+Modeling
+and control of a nonholonomic wheeled mobile robot
+with wheel slip dynamics. In 2009 IEEE Symposium on
+Computational Intelligence in Control and Automation,
+7–14. IEEE.
+Wang, S. and Zhai, J. (2020).
+A trajectory tracking
+method for wheeled mobile robots based on disturbance
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+and Systems, 18(8), 2165–2169.
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+Zero-shot
+deep reinforcement learning driving policy transfer for
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+tion Systems (ITSC), 2865–2871. IEEE.
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+contact-aware online context inference for generalizable
+non-planar pushing. In 2021 IEEE/RSJ International
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+176–182. IEEE.
+
+10.0
+target
+Kstanley
+7
+trajectory
+7.5
+Kspeed
+onpatch
+5.0
+6
+2.5
+5.
+0.0
+2.5
+4 -
+5.0
+3
+7.5
+-10.0
+2
+3
+5
+6
+0
+100
+200
+300
+400
+500
+600
+700
+K
+stanley
+= 2.0,K
+= 0.5
+speed
+10.0
+target
+Kstanley
+7
+trajectory
+7.5
+Kspeed
+on patch
+5.0
+6
+2.5
+5
+0.0 -
+2.5
+4-
+5.0 -
+7.5
+10.0
+4
+5
+6
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Proposed Framework

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+FOUR WAYS TO RECOVER THE SYMBOL OF A
+NON-BINARY LOCALIZATION OPERATOR
+SIMON HALVDANSSON
+Abstract. We present a set of results on how the symbol of a localization operator can
+be recovered from spectral information, the image of white noise or the image of an or-
+thonormal basis. This extends earlier results which have been limited to the case where
+the symbol is a binary mask. Moreover, we present some numerical aspects of the different
+methods and discuss their performance.
+1. Introduction and main results
+Arguably the main tool of time-frequency analysis is the short-time Fourier transform,
+defined for a signal ψ ∈ L2(Rd) and window g ∈ L2(Rd) as
+Vgψ(x, ω) =
+�
+Rd ψ(t)g(t − x)e−2πiωt dt
+where the variables x, ω ∈ Rd are referred to as the time and frequency, respectively. A
+standard result [26] states that this mapping can be inverted so that the signal ψ can be
+recovered from Vgψ weakly as
+ψ =
+�
+R2d Vgψ(x, ω)g(t − x)e2πiωt dx dω.
+Using a function f : R2d → R, usually referred to as the symbol or mask, we can weigh
+this reconstruction so that certain frequencies and time intervals have more or less priority.
+Formally, this happens via the localization operator
+Ag
+f : ψ �→
+�
+R2d f(x, ω)Vgψ(x, ω)g(t − x)e2πiωt dx dω.
+(1)
+Such operators have applications in signal analysis [34, 39, 41, 46], acoustics [8, 31, 47],
+pseudo-differential operators [28, 32], physics [7, 25] and operator theory [23, 28, 38] among
+others and their properties have been deeply studied [10, 13, 22, 49]. Abreu and D¨orfler
+[2] first considered the inverse problem of recovering the symbol f from the localization
+operator Ag
+f through various measurements related to Ag
+f and this work has been continued
+in [2, 3, 37, 42]. All these investigations have been focused on the case where f is a binary
+mask, i.e., f : R2d → {0, 1}. The main contribution of this article is showing corresponding
+results for more general classes of f as well as developing novel approaches to recovering f
+which have not been considered before.
+There are several reasons to consider the case of non-binary symbols. If we view the
+inverse problem as a calibration, it is reasonable that imperfections in the system may cause
+the corresponding symbol to deviate from the intended binary design. Symbol discontinuities
+can also cause audible artifacts known as musical noise in the audio setting [8] and it is
+Date: January 2023.
+Keywords: Localization operator, Inverse problem, Operator identification, Symbol recovery.
+1
+arXiv:2301.11618v1  [math.FA]  27 Jan 2023
+
+2
+SIMON HALVDANSSON
+therefore beneficial to design those systems with non-binary symbols in the first place.
+In some audio filtering contexts where binary masks are currently used such as in [31], a
+non-binary value is associated to each time-frequency pair and a mask is then constructed
+by thresholding. This approach, while straight-forward, is unlikely to be optimal which has
+motivated the use of non-binary masks [8]. Moreover, localization operators can be identified
+with function-operator convolutions and Gabor-Toeplitz operators [38] and inverting the
+symbol to operator mappings is of independent theoretical interest in these settings.
+Below we state all of our main results before having established all the relevant notation.
+In particular we formulate some results with function-operator convolutions f ⋆ S, Wigner
+transforms W(ϕ), the modulation spaces M1 and Cohen’s class distributions QS(ψ). These
+are all detailed in Section 2 but hopefully the general idea of the theorems should be clear.
+If not the reader can return to the formulations after finishing the preliminaries section.
+Our first set of results shows how a smooth positive symbol can be approximated and
+how the error shrinks as we increase the smoothness of f. In particular, our estimator for
+f2, the average observed spectrogram, is given by
+ρ(z) = 1
+K
+K
+�
+k=1
+|Vϕ(Ag
+fNk)(z)|2
+(2)
+where (Nk)K
+k=1 are K realizations of (complex) white noise and ϕ is our reconstruction
+window which does not necessarily have to coincide with g. This construction is from [42]
+in the binary case. The notion and interpretation of the white noise will be made precise in
+Section 2.3. We will show how, as K → ∞, this estimator converges with high probability
+to
+ϑ(z) =
+�
+m
+λ2
+m|Vϕhm(z)|2
+(3)
+where (λm)m and (hm)m are the eigenvalues and eigenfunctions of Ag
+f. Using the framework
+of quantum harmonic analysis and asymptotics of products of localization operators, we will
+show how ϑ in turn is a good approximation of f2.
+Theorem 1.1. Let f be a real-valued, bounded, integrable function with bounded derivative
+and bounded variation, ρ the average observed spectrogram (2) with white noise variance σ2
+and g, ϕ ∈ S(Rd) with ∥g∥L2 = ∥ϕ∥L2 = 1. Then there exists a constant C > 1 such that
+P
+�����
+ρ(z)
+σ2 − ϑ(z)
+���� > t
+�
+≤ 3 exp
+�
+−CK min
+�
+t2
+ϑ(z)2 ,
+t
+ϑ(z)
+��
+,
+(4)
+��ϑ − f2 ∗ |Vϕg|2��
+L∞ ≤ ∥f∥L∞
+�
+� �
+|α|=1
+∥∂αf∥L∞
+�
+� ∥g∥4
+M1,
+(5)
+��f2 ∗ |Vϕg|2 − f2��
+L1 ≤
+��
+R2d
+��(∇f2)(z)
+�� dz
+� ��
+R2d |z||Vϕg(z)|2 dz
+�
+.
+(6)
+Through some deeper analysis, we can replace the three rather disparate estimates above
+by one L1 estimate.
+
+FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+3
+Theorem 1.2. Let f ∈ Cd+2
+c
+(R2d) be real-valued, ρ be given by (2) with white noise variance
+σ2, g, ϕ ∈ S(Rd) with ∥g∥L2 = ∥ϕ∥L2 = 1 and define
+B1 = A
+�
+��∥K∥L2 +
+�
+�
+2d
+�
+j=1
+��∂d+1
+j
+K
+��2
+L2
+�
+�
+1/2�
+�� ,
+B2 =
+��
+R2d
+��(∇f2)(z)
+�� dz
+� ��
+R2d |z||Vϕg(z)|2 dz
+�
+where
+K(y, z) = f(y)
+�
+� �
+|α|=1
+� 1
+0
+∂αf(y + t(z − y)) dt(z − y)
+�
+� Vgg(y − z)
+and A is a constant independent of f and g. Then there exists a C > 0 such that
+P
+��
+R2d
+����
+ρ(z)
+σ2 − f(z)2
+���� dz > B1 + B2 + t
+�
+≤ ∥f∥2
+L2
+t
+√
+K
+� 3√π
+2
+√
+C
+erf
+�√
+CK
+�
++
+3
+C
+√
+K
+e−CK
+�
+.
+The above theorem should be read as “the L1 estimation error is bounded by B1 + B2
+with high probability provided K is large enough”. Note also that the quantity B1 is finite
+which is proved in Lemma 3.4 below.
+Next we discuss two approaches which rely on spectral data about the operator. These
+are also stated for mixed-state localization operators, introduced and defined in Section 2.2.1
+below, as this stronger result follows directly from our methods. For the readers convenience,
+we also state the corresponding statement for the “pure” localization operators discussed
+above.
+Theorem 1.3. Let f ∈ L1(R2d) be real-valued and with bounded variation and S, T ∈ S1
+be positive with tr(S) = tr(T) = 1. Then if f ⋆ S = �
+m λm(hm ⊗ hm),
+�����
+N
+�
+m=1
+λmQT (hm) − f
+�����
+L1
+≤
+∞
+�
+m=N+1
+|λm| + Var(f)
+�
+R2d |z|(S ⋆ ˇT)(z) dz.
+In particular, if ϕ ∈ L2(Rd) with ∥ϕ∥ = 1 and Ag
+f = �
+m λm(hm ⊗ hm), then
+�����
+N
+�
+m=1
+λm|Vϕ(hm)|2 − f
+�����
+L1
+≤
+∞
+�
+m=N+1
+|λm| + Var(f)
+�
+R2d |z||Vϕg(z)|2 dz.
+Moreover, in the N = ∞ case, if T = S,
+∞
+�
+m=1
+λmQS(hm)(z) = f ∗ (S ⋆ ˇS)(z)
+which can be deconvolved if the Fourier transform F(S ⋆ ˇS) is zero free as
+f = F−1
+�F(f ∗ (S ⋆ ˇS))
+F(S ⋆ ˇS)
+�
+.
+While the above result depends on an approximation T of S, we are able to sidestep this
+in the following theorem.
+
+4
+SIMON HALVDANSSON
+Theorem 1.4. Let f ∈ L1(R2d) be real-valued with bounded variation and S ∈ S1 be
+positive, then if f ⋆ S = �
+m λm(hm ⊗ hm) and S = �
+n sn(ϕn ⊗ ϕn),
+�
+m
+λmW(hm)(z) = f ∗
+�
+n
+snW(ϕn)(z)
+where W(ϕ) is the Wigner transform of ϕ. In particular, if S = g ⊗ g so that f ⋆ S = Ag
+f,
+�
+m
+λmW(hm)(z) = f ∗ W(g)(z).
+Moreover, if the window functions (ϕn)n are in the Schwartz space, the convergence is in
+L1 with the error bounds
+�����
+�
+m
+λmW(hm) − f
+�����
+L1
+≤ Var(f)
+�
+R2d |z|
+�����
+�
+n
+snW(ϕn)(z)
+����� dz
+and the corresponding statement holds for the rank-one case S = g ⊗ g.
+Lastly, we discuss an approach based on noting that adding up all the spectrograms of
+an orthonormal basis yields the function which is identically one in a manner which can be
+likened to a tiling of phase space via a partition of unity. If we then apply our localization
+operator with symbol f to each basis element, this tiling should only make a contribution
+proportional to the size of f2. This intuition turns out to be correct and is quantified in the
+following theorem.
+Theorem 1.5. Let f ∈ Cd+2
+c
+(R2d) be real-valued and g, ϕ ∈ S(R2d) with ∥g∥L2 = ∥ϕ∥L2 = 1.
+Define
+B1 = A
+�
+��∥K∥L2 +
+�
+�
+2d
+�
+j=1
+��∂d+1
+j
+K
+��2
+L2
+�
+�
+1/2�
+�� ,
+B2 =
+��
+R2d
+��(∇f2)(z)
+�� dz
+� ��
+R2d |z||Vϕg(z)|2 dz
+�
+where
+K(y, z) = f(y)
+�
+� �
+|α|=1
+� 1
+0
+∂αf(y + t(z − y)) dt(z − y)
+�
+� Vgg(y − z)
+and A is a constant independent of f and g. Then for any orthonormal basis {en}n of
+L2(Rd),
+�����
+�
+n
+|Vϕ(Ag
+fen)|2 − f2
+�����
+L1
+≤ B1 + B2.
+Notational conventions. The Schatten p-class of operators with singular values in ℓp will
+be denoted by Sp while the larger class of bounded operators on L2(Rd) will be denoted
+by L(L2). The adjoint of such an operator A will be denoted by A∗ and we will write
+ˇA = PAP where P is the parity operator P : f(t) �→ f(−t). For the open ball centered at
+z with radius r we will write Br(z). For a function f of several variables, we will write ∂n
+j f
+for the n:th derivative in the j:th variable. We will also use a multiindex α = (α1, . . . , αd)
+for the derivative ∂αf = ∂α1 · · · ∂αdf and denote by Cn the set of functions f for which
+∂αf is continuous for all α ∈ Nd with |α| ≤ n. The associated subspace Cn
+c will specify
+
+FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+5
+those functions which have compact support. For the Schwartz functions on Rd we will
+write S(Rd). Indicator functions of sets Ω will be denoted by χΩ. Inner products with no
+subscript will always refer to L2(Rd) inner products so that ⟨·, ·⟩ = ⟨·, ·⟩L2(Rd). For matrices
+of size N × M with entries in C, we will write CN×M.
+2. Preliminaries
+2.1. Time-frequency analysis. We highlight some important facts from time-frequency
+analysis, which we will have use for, in a compact form. For a more complete introduction
+the reader is referred to [26, 49].
+2.1.1. Short-time Fourier transform. One of the main ideas underlying time-frequency anal-
+ysis is that real world signals are correlated in time and frequency. To make this dependence
+clear, the short-time Fourier transform (STFT) allows us to analyse the frequency content
+of some signal ψ around a specific time by weighing ψ by some window function which is
+well-localized in time. It is defined as
+Vgψ(z) = ⟨ψ, π(z)g⟩ =
+�
+Rd ψ(t)g(t − x)e−2πiωt dt.
+where the time-frequency shift π(z) is a unitary, square integrable representation acting as
+π(z)ψ(t) = π(x, ω)ψ(t) = e2πiωtψ(t − x) and the point z = (x, ω) ∈ R2d is said to belong
+to phase space. A non-trivial result known as Moyal’s formula states that we can compute
+inner products of two signals using their associated short-time Fourier transforms as
+⟨Vg1ψ1, Vg2ψ2⟩L2(R2d) = ⟨ψ1, ψ2⟩⟨g1, g2⟩.
+In particular, the mapping Vg : L2(Rd) → L2(R2d) is an isometry provided ∥g∥L2 = 1. As a
+consequence of the above relation, we have the reconstruction formula from the introduction
+which tells us how we can get a signal back from its STFT
+ψ =
+�
+R2d Vϕψ(z)π(z)g dz.
+(7)
+Often in application, the square modulus |Vgψ|2 of the STFT, the spectrogram, is used as
+it is real-valued, non-negative and represents the energy distribution of the signal. From
+the experimental side, it is often the spectrogram, not the STFT, that is measured in
+applications such as in ptychography [29] and X-ray crystallography.
+2.1.2. Localization operators. Provided we want to localize the support of a signal ψ in phase
+space, an obvious idea is to limit the reconstruction in (7) to some set Ω ⊂ R2d. By the
+uncertainty principle, this is impossible but works up to a small error depending on the size
+of Ω. Generalizing this idea, we can add a weighing factor, or symbol, f ∈ L1(R2d) to the
+reconstruction which tells us how much we want to reconstruct different parts of the phase
+space representation of ψ. Formally we write the application of the localization operator
+Ag
+f to ψ as
+Ag
+fψ(t) =
+�
+R2d f(z)Vgψ(z)π(z)g(t) dz.
+(8)
+One can use different window functions g1, g2 for the STFT and the reconstruction above,
+resulting in a non self-adjoint localization operator. We will however not attempt to treat
+this case in this paper as we use the self-adjointness extensively. Localization operators were
+originally investigated by I. Daubechies [14, 15].
+
+6
+SIMON HALVDANSSON
+2.1.3. Modulation spaces. Feichtinger’s algebra M1(Rd), originally introduced in [20], is
+defined as the set of tempered distributions f such that Vgf ∈ L1(R2d) where g is a Schwartz
+window function. It is a special case of the modulation spaces [19, 20] which are defined by
+integrability properties of short-time Fourier transforms and have the convenient property
+that they are independent of the window function g used.
+2.1.4. Cohen’s class of time-frequency distributions. There are several quadratic time-frequency
+distributions with similar properties to the spectrogram. Those which fulfill some basic de-
+sirable properties are commonly referred to as Cohen’s class distributions [9] and include
+the spectrogram as a special case. They can all be written as
+QΦ(ψ) = Φ ∗ W(ψ)
+where Ψ is a tempered distribution and W(ψ) = W(ψ, ψ) is the Wigner transform of ψ,
+defined as
+W(ψ, φ)(x, ω) =
+�
+Rd ψ(t + x/2)φ(t − x/2)e−2πiω·t dt.
+2.2. Quantum harmonic analysis. The theory of quantum harmonic analysis, first de-
+veloped by Werner in [48], will play a central role in our proofs. Its main components are
+definitions of convolutions between functions and operators and pairs of operators, taking
+the form
+f ⋆ S =
+�
+R2d f(z)π(z)Sπ(z)∗ dz,
+T ⋆ S(z) = tr
+�
+Tπ(z) ˇSπ(z)∗�
+.
+(9)
+where the first integral should be interpreted as a Bochner integral and ˇS = PSP. As we
+will see below, both of these definitions satisfy versions of Young’s inequality which we will
+make use of. However, we first compute the prototypical function-operator and operator-
+operator convolutions since these serve as our main motivation for using the framework of
+quantum harmonic analysis.
+Example 2.1. The function-operator convolutions f ⋆ (ϕ ⊗ ϕ) for f ∈ L1(R2d) and ϕ ∈
+L2(Rd) is precisely the localization operator Aϕ
+f . Indeed,
+f ⋆ (ϕ ⊗ ϕ)ψ =
+�
+R2d f(z)π(z)(ϕ ⊗ ϕ)π(z)∗ψ dz
+=
+�
+R2d f(z)⟨π(z)∗ψ, ϕ⟩π(z)ϕ dz
+=
+�
+R2d f(z)Vϕψ(z)π(z)ϕ dz = Aϕ
+f ψ.
+Example 2.2. The simplest case of operator-operator convolutions reduces down to the
+spectrogram. Indeed, for ψ, φ ∈ L2(Rd) we have that
+(ψ ⊗ ψ) ⋆ (φ ⊗ φ)ˇ(z) = tr
+�
+(ψ ⊗ ψ)π(z)(φ ⊗ φ)π(z)∗�
+=
+�
+n
+�
+(ψ ⊗ ψ)π(z)(φ ⊗ φ)π(z)∗en, en
+�
+=
+�
+n
+⟨en, π(z)φ⟩⟨π(z)φ, ψ⟩⟨ψ, en⟩
+= |⟨ψ, π(z)φ⟩|2 = |Vφψ(z)|2
+where (en)n was an arbitrary orthonormal basis used to compute the trace.
+
+FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+7
+Many properties of function-operator and operator-operator convolutions are analogous
+to the corresponding statements for classical function-function convolutions as we will see
+below.
+Just as the integral is replaced by the trace in the definition of operator-operator convo-
+lutions (9), when measuring the size of operators, we will use the Schatten p-norms which
+are defined as
+∥A∥Sp = tr(|A|p)1/p
+where |A| =
+√
+A∗A is the absolute value of A. These norms induce the Schatten p-classes
+of operators, the most notable of which are the trace-class operators S1 and the Hilbert-
+Schmidt operators S2. As these operators are compact, they have a spectral decomposition
+of the form
+A =
+�
+n
+an(ψn ⊗ φn)
+where (ψn)n and (φn)n are orthonormal bases, ψ ⊗ φ is the rank-one operator acting as
+f �→ ⟨f, φ⟩ψ and (an)n is in ℓp if A ∈ Sp.
+Next we collect some basic properties of these convolutions, the proofs of which can be
+found in [35].
+Proposition 2.3. Let f, g ∈ L1(R2d), S ∈ Sp, T ∈ Sq for 1 ≤ p, q ≤ ∞ with 1
+p + 1
+q = 1 and
+R ∈ S1. Then
+(i) (f ⋆ S)∗ = ¯f ⋆ S∗,
+(ii) (f ⋆ R) ⋆ T = f ∗ (R ⋆ T),
+(iii) (f ∗ g) ⋆ S = f ⋆ (g ⋆ S),
+(iv) ∥f ⋆ S∥Sp ≤ ∥f∥L1∥S∥Sp,
+(v) ∥h ⋆ R∥Sp ≤ ∥h∥Lp∥R∥S1,
+(vi) ∥S ⋆ R∥Lp ≤ ∥S∥Sp∥R∥S1.
+In the next subsections, we dive deeper into some topics in quantum harmonic analysis
+which will be of use. For a more thorough introduction with more motivation and results,
+the reader is referred to [35].
+2.2.1. Mixed-state localization operators. Localization operators reconstruct a function with
+respect to a single window or pair of windows in the non self-adjoint case. This construction
+has been generalized to multiple windows by considering the function-operator convolution
+f ⋆ S which can be seen as a weighted sum of localization operators [36]. Indeed, if S =
+�
+n sn(ϕn ⊗ ϕn), then
+f ⋆ S = f ⋆
+�
+n
+sn(ϕn ⊗ ϕn) =
+�
+n
+snAϕn
+f .
+Later on in our results, we will need for our (mixed-state) localization operators to be self-
+adjoint. In view of Proposition 2.3 (i), this requires the symbol f to be real-valued and the
+window operator S to be self-adjoint.
+2.2.2. Fourier-Wigner transform. Another central tool of quantum harmonic analysis is the
+Fourier-Wigner transform, mapping operators to functions, defined for S ∈ S1 as
+FW (S)(z) = e−πixω tr(π(−z)S).
+Our interest in the Fourier-Wigner transform is primarily based on its convolution properties
+which mirror those of the classical Fourier transform.
+To state the relevant result, we
+
+8
+SIMON HALVDANSSON
+first need to define the symplectic Fourier transform which essentially is a rotated two
+dimensional Fourier transform
+Fσ(f)(z) =
+�
+R2d f(z′)e−2πiσ(z,z′) dz′
+where z = (x, ω), z′ = (x′, ω′) and σ(z, z′) = ωx′ − ω′x is the standard symplectic form. We
+can now state the result which is analogous to the classical convolution theorem.
+Proposition 2.4. Let f ∈ L1(R2d) and S, T ∈ S1. Then
+FW (f ⋆ S) = Fσ(f)FW (S),
+Fσ(T ⋆ S) = FW (T)FW (S).
+2.2.3. Weyl quantization. A quantization procedure provides a mapping between functions
+and operators. One such example is the mapping f �→ f ⋆ S which is the mapping we hope
+to invert in this paper. In time-frequency analysis and quantum harmonic analysis, we often
+make use of Weyl quantization which can be defined weakly as
+⟨Lfψ, φ⟩ = ⟨f, W(φ, ψ)⟩
+where we refer to the mapping f �→ Lf as the Weyl transform. For the inverse mapping,
+meaning the function associated to the operator S, we write aS and call it the Weyl symbol
+of S.
+Weyl quantization has a particular nice formulation in quantum harmonic analysis where
+it can be written as
+aS = Fσ(FW (S)).
+In particular, it can be shown that aψ⊗φ = W(ψ, φ). It also holds that Weyl quantization
+is compatible with the convolutions of quantum harmonic analysis in the sense that
+T ⋆ S = aT ∗ aS,
+af⋆S = f ∗ aS
+(10)
+for T, S ∈ S1 and f ∈ L1(R2d).
+2.2.4. Cohen’s class as operator-operator convolutions. The class of quadratic time-frequency
+distributions discussed in Section 2.1.4 has a convenient formulation in quantum harmonic
+analysis using the Weyl quantization relations (10) above. By letting ˇS be the Weyl quan-
+tization of the tempered distribution Φ defining QΦ and using that aψ⊗ψ = W(ψ), we get
+that
+QΦ(ψ) = QS(ψ) = (ψ ⊗ ψ) ⋆ ˇS.
+(11)
+This point of view makes it particularly easy to deduce properties of of Cohen’s class dis-
+tributions such as bounding Lp norms or characterizing positivity.
+2.3. Functional analytic and probabilistic aspects of white noise. The core of our
+approach to symbol recovery using white noise is computing spectrograms of random noise.
+This is inspired by recent theoretical work in [5, 24, 30] and others. For further details we
+refer the reader to the discussion in [42] as we follow their proof strategy. The two main
+result which we need are stated in [42] and we also state them for the sake of completeness.
+The first is a version of the Hanson-Wright inequality [4, 43].
+
+FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+9
+Theorem 2.5 ([42, Theorem 3.1]). Let X be an m-dimensional complex Gaussian random
+variable with X ∼ CN(0, Σ) and A ∈ Cm×m Hermitian. Then there exists an universal
+constant Chw > 0 such that for every t > 0,
+P
+�
+|⟨AX, X⟩ − E{⟨AX, X⟩}| > t
+�
+≤ 2 exp
+�
+−Chw min
+�
+t2
+∥Σ∥2s∥A∥2
+F
+,
+t
+∥Σ∥s∥A∥s
+��
+where ∥ · ∥s and ∥ · ∥F are the spectral and Frobenius norms, respectively.
+Secondly, the next lemma gives a constructive way to deal with the application of an
+operator to white noise.
+Lemma 2.6 ([42, Lemma 4.2]). Let g be a Schwartz function with ∥g∥L2 = 1 and N a
+realization of complex white noise with variance σ2. Then there exists a sequence (αm)m
+where αm ∼ CN(0, σ2) of independent complex normal variables such that almost surely,
+Ag
+f(N) =
+�
+m
+λmαmhm
+where Ag
+f =
+�
+m
+λm(hm ⊗ hm)
+with almost sure absolute convergence in L2(R2d).
+We also remark that if our white noise is not complex but rather real valued, all results
+will still hold but with possibly larger constants. See [42, Section 2.2] for a discussion on
+this.
+2.4. Approximate identities. The variation of a function f ∈ L1(R2d) is defined as
+Var(f) = sup
+��
+R2d f(z) div φ(z) dz : φ ∈ C1
+c (R2d, R2d), ∥φ∥∞ ≤ 1
+�
+and in the special case where f ∈ C1(R2d), it can be written as
+Var(f) =
+�
+R2d |∇f(z)| dz.
+We say that functions f with Var(f) < ∞ have bounded variation. In the case where f is the
+indicator function of some compact subset Ω ⊂ R2d with smooth boundary, the variation of
+f is equal to the Haussdorff measure of the boundary [16].
+In what follows, we will want to measure how much a function is changed when it is
+convolved by some kernel. The next lemma quantifies this using the concept of variation
+introduced above.
+Lemma 2.7 ([3, Lemma 3.2]). Let ψ ∈ L1(R2d) have bounded variation and φ ∈ L1(R2d)
+with
+�
+R2d φ(z) dz = 1, then
+∥ψ ∗ φ − ψ∥L1 ≤ Var(ψ)
+�
+R2d |z||φ(z)| dz.
+In the following, we will sometimes refer to φ as the blurring kernel.
+3. Recovery via white noise
+White noise approaches in time frequency analysis has recently received attention in [1,
+42]. The idea underlying our approach is that a spectrogram of white noise is approximately
+constant and that we therefore should be able to get approximations of the symbol f by
+looking at how a localization operator with symbol f changes the spectrogram. Our tech-
+niques are based on measuring how close the white noise is to being constant and how close
+multiplying by f is to applying the localization operator.
+
+10
+SIMON HALVDANSSON
+3.1. Generalities. For the first part of Theorem 1.1, we will need a version of [42, Lemma
+5.1] with unknown variance and non-binary masks. Both modifications are minor but we
+give a detailed proof for the sake of completeness.
+Lemma 3.1. Let all variables be as in Theorem 1.1. Then there exists C > 0 such that for
+every z ∈ R2d,
+P
+�����
+ρ(z)
+σ2 − ϑ(z)
+���� > t
+�
+≤ 3 exp
+�
+−CK min
+�
+t2
+ϑ(z)2 ,
+t
+ϑ(z)
+��
+.
+Proof. We will first prove that
+P
+���ρ(z) − σ2ϑ(z)
+�� > t
+�
+≤ 3 exp
+�
+−CK min
+�
+t2
+σ4ϑ(z)2 ,
+t
+σ2ϑ(z)
+��
+from which the result follows upon multiplying t by σ2 on both sides. The entire proof is
+focused on setting up so that Theorem 2.5 can be applied.
+Fix L ∈ N and define the complex Gaussian random vector
+Xm = α⌈m/L⌉
+mod (m−1,L)+1
+for m = 1, . . . , KL where each α is CN(0, σ2) distributed. We can write this more implicitly
+as
+X = (α1
+1, . . . , α1
+L, α2
+1, . . . , α2
+L, . . . , αK
+1 , . . . , αK
+L ) ∼ CN(0, σ2IKL).
+We also define the matrix-valued function M : R2d → CL×L as
+M(z)ℓ,m := λℓλmVϕhℓ(z)Vϕhm(z)
+where (λm)m and (hm)m are the eigenvalues and eigenfunctions of Ag
+f. Note that for each
+z ∈ R2d, the resulting matrix is Hermitian as the eigenvalues are real valued.
+Using M as a building block, we define the following block-diagonal Hermitian matrix
+M : R2d → CKL×KL,
+M(z) = 1
+K
+�
+�
+�
+M(z)
+0
+...
+0
+M(z)
+�
+�
+� ∈ CKL×KL.
+Now let PL denote the projection onto the space spanned by the first L eigenfunctions of
+Ag
+f. It then follows that
+ρL(z) := 1
+K
+K
+�
+k=1
+|Vϕ(PLAg
+fNk)(z)|2 = ⟨M(z)X, X⟩.
+
+FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+11
+We now compute the remaining quantities in the statement of Theorem 2.5, starting with
+E
+�
+⟨M(z)X, X⟩
+�
+. By the independence of the αk
+m’s and Lemma 2.6,
+E(ρL(z)) = E
+�
+1
+K
+K
+�
+k=1
+��Vϕ
+�
+PLAg
+fNk
+�
+(z)
+��2
+�
+= 1
+K
+K
+�
+k=1
+L
+�
+ℓ=1
+L
+�
+m=1
+λℓλmE
+�
+αk
+ℓ αkm
+�
+Vϕhℓ(z)Vϕhm(z)
+=
+L
+�
+m=1
+λ2
+mσ2|Vϕhm(z)|2 =: σ2ϑL(z).
+Next we estimate the Frobenius and spectral norms of M(z) from above as
+∥M(z)∥2
+F = 1
+K ∥M(z)∥2
+F = 1
+K
+L
+�
+ℓ=1
+L
+�
+m=1
+��λℓλmVϕhℓ(z)Vϕhm(z)
+��2
+= 1
+K
+� L
+�
+m=1
+|λmVϕhm(z)|2
+�2
+≤ 1
+K
+� ∞
+�
+m=1
+|λmVϕhm(z)|2
+�2
+= 1
+K ϑ(z)2
+and
+∥M(z)∥s = 1
+K ∥M(z)∥s = 1
+K
+sup
+∥X∥2=1
+�
+�
+L
+�
+m=1
+�����
+L
+�
+ℓ=1
+(M(z))m,ℓXℓ
+�����
+2�
+�
+1/2
+≤
+� L
+�
+m=1
+L
+�
+ℓ=1
+��λℓλmVϕhℓ(z)Vϕhm(z)
+��2
+�1/2
+= ϑL(z) ≤ ϑ(z).
+We now apply Theorem 2.5 with ∥Σ∥s = ∥σ2IKL∥s = σ2 and the above estimates to obtain
+P(|ρL(z) − σ2ϑL(z)| ≥ t) ≤ 2 exp
+�
+−Chw
+64 K min
+�
+t2
+σ4ϑ(z)2 ,
+t
+σ2ϑ(z)
+��
+.
+(12)
+To lift this result to the full L = ∞ setting, we define the following three error terms
+R1(z) = 1
+K
+K
+�
+k=1
+∞
+�
+ℓ=L+1
+L
+�
+m=1
+λℓλmαk
+ℓ αkmVϕhℓ(z)Vϕhm(z),
+R2(z) = 1
+K
+K
+�
+k=1
+L
+�
+ℓ=1
+∞
+�
+m=L+1
+λℓλmαk
+ℓ αkmVϕhℓ(z)Vϕhm(z),
+R3(z) = 1
+K
+K
+�
+k=1
+∞
+�
+ℓ=L+1
+∞
+�
+m=L+1
+λℓλmαk
+ℓ αkmVϕhℓ(z)Vϕhm(z)
+
+12
+SIMON HALVDANSSON
+and make the following crude estimate
+P(|ρ(z) − σ2ϑ(z)| ≥ t) ≤ P
+�
+|ρL(z) + R1(z) + R2(z) + R3(z) − σ2ϑ(z)| ≥ t
+�
+≤ P
+�
+|ρL(z) − σ2ϑ(z)| + |R1(z)| + |R2(z)| + |R3(z)| ≥ t
+�
+≤ P
+�
+|ρL(z) − σ2ϑ(z)| ≥ t
+4
+�
++
+3
+�
+s=1
+P
+�
+|Rs(z)| ≥ t
+4
+�
+.
+To bound this, we first choose L so large that |σ2ϑ(z)−σ2ϑL(z)| ≤ t
+8 which is possible since
+the sum defining ϑ(z) is uniformly convergent. We then have the estimate
+P
+�
+|ρL(z) − σ2ϑ(z)| ≥ t
+4
+�
+≤ P
+�
+|ρL(z) − σ2ϑL(z)| ≥ t
+8
+�
+≤ 2 · exp
+�
+−Chw
+64 K min
+�
+t2
+σ4ϑ(z)2 ,
+t
+σ2ϑ(z)
+��
+.
+by (12) above. If we can fold estimates of R1, R2 and R3 into this form we will have finished
+the proof. We deal with R1 in detail and remark that R2 and R3 can be treated with similar
+methods. First note that
+|R1(z)| ≤ 1
+K
+K
+�
+k=1
+∞
+�
+ℓ=1
+∞
+�
+m=L+1
+|λℓλmαk
+ℓ αk
+m|
+≤ 1
+K
+K
+�
+k=1
+�
+∞
+�
+m=L+1
+|λm|
+�1/2 �
+∞
+�
+m=L+1
+|λm||αk
+m|2
+�1/2 � ∞
+�
+ℓ=1
+|αℓ|
+�1/2 � ∞
+�
+ℓ=1
+|λℓ||αk
+ℓ |2
+�1/2
+≤ 1
+K
+�
+∥f∥L1
+∞
+�
+m=L+1
+|λm|
+�1/2 K
+�
+k=1
+∞
+�
+ℓ=1
+|λℓ||αk
+ℓ |2.
+Hence,
+P
+�
+|R1(z)| ≥ t
+4
+�
+≤ P
+�
+�
+K
+�
+k=1
+∞
+�
+ℓ=1
+|λℓ||αk
+ℓ |2 ≥ Kt
+4
+�
+∥f∥L1
+∞
+�
+m=L+1
+|λm|
+�−1/2�
+�
+≤
+K
+�
+k=1
+P
+�
+�
+∞
+�
+ℓ=1
+|λℓ||αk
+ℓ |2 ≥ t
+4
+�
+∥f∥L1
+∞
+�
+m=L+1
+|λm|
+�−1/2�
+�
+≤ KP
+�
+�
+∞
+�
+ℓ=1
+|λℓ||α1
+ℓ|2 ≥ t
+4
+�
+∥f∥L1
+∞
+�
+m=L+1
+|λm|
+�−1/2�
+� .
+It is clear that the quantity on the right hand side goes to zero as L → ∞ and hence in
+particular, we can choose L large enough so that
+P
+�
+|R1(z)| ≥ t
+4
+�
+≤ 1
+3 exp
+�
+−Chw
+64 K min
+�
+t2
+σ4ϑ(z)2 ,
+t
+σ2ϑ(z)
+��
+.
+Upon giving R2 and R3 the same treatment, the result follows.
+□
+
+FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+13
+3.2. Proof of Theorem 1.1. With the generalities out of the way, we are ready to start
+treating the three error estimates in Theorem 1.1.
+For (5) we will need results on the
+asymptotics of products of localization operators, first discussed in [12].
+The following
+combines the main results of [12] with a special case of [10, Theorem 4 (i)] and [10, Lemma
+5].
+Lemma 3.2. Fix N ∈ N and let a ∈ L∞(R2d), ∂αb ∈ L∞(R2d) for α ∈ N2d with |α| = N
+and ϕ1, ϕ2, ϕ3, ϕ4 ∈ M1(Rd). Then
+Aϕ1,ϕ2
+a
+Aϕ3,ϕ4
+b
+=
+N−1
+�
+|α|=0
+(−1)|α|
+α!
+AΦα,ϕ2
+a∂αb + EN
+(13)
+where Φα is a window which depends on ϕ1, ϕ2, ϕ3, ϕ4 and EN is an error term which can
+be written as
+EN = V ∗
+ϕ2TVϕ1
+where Vϕ is the STFT and T is an integral operator with kernel K : R2d × R2d → C given
+by
+K(y, z) = a(y)N
+�
+|α|=N
+� 1
+0
+(1 − t)N∂αb(y + t(z − y)) dt(z − y)α
+α!
+⟨π(z)ϕ4, π(y)ϕ1⟩.
+Moreover, the norm of EN can be bounded as
+∥EN∥L(L2) ≤ ∥a∥L∞
+�
+� �
+|α|=N
+1
+α!∥∂αb∥L∞
+�
+� ∥ϕ1∥M1∥ϕ2∥M1∥ϕ3∥M1∥ϕ4∥M1.
+We are now ready to string together the estimates established above to finish the proof
+of Theorem 1.1.
+Proof of Theorem 1.1. The three estimates in the theorem follow by different arguments,
+the first one being an immediate corollary of Lemma 3.1 as stated above.
+For (5), we first claim that
+ϑ(z) =
+�
+Ag
+f
+�2 ⋆ (ϕ ⊗ ϕ)ˇ(z).
+(14)
+Indeed, as Ag
+f = �
+m λm(hm ⊗ hm), it follows that
+�
+Ag
+f
+�2 = �
+m λ2
+m(hm ⊗ hm). Hence
+(15)
+�
+Ag
+f
+�2 ⋆ (ϕ ⊗ ϕ)ˇ(z) =
+�
+m
+λ2
+m(hm ⊗ hm) ⋆ (ϕ ⊗ ϕ)ˇ(z)
+=
+�
+m
+λ2
+m|Vϕhm(z)|2 = ϑ(z)
+by Example 2.2. Now using Lemma 3.2 with a = b = f, ϕ1 = ϕ2 = ϕ3 = ϕ4 = g and N = 1,
+we get
+�
+Ag
+f
+�2 = Ag
+f2 + E1 = f2 ⋆ (g ⊗ g) + E1.
+
+14
+SIMON HALVDANSSON
+Plugging this into (15) and applying Example 2.2 and then Lemma 3.2 yields
+ϑ(z) =
+�
+f2 ⋆ (g ⊗ g) + E1
+�
+⋆ (ϕ ⊗ ϕ)ˇ(z)
+= f2 ∗ |Vϕg|2(z) + E1 ⋆ (ϕ ⊗ ϕ)ˇ(z)
+=⇒
+��ϑ − f2 ∗ |Vϕg|2��
+L∞ = ∥E1 ⋆ (ϕ ⊗ ϕ)ˇ∥L∞ ≤ ∥E1∥L(L2)∥ϕ∥2
+L2
+≤ ∥f∥L∞
+�
+� �
+|α|=1
+∥∂αf∥L∞
+�
+� ∥g∥4
+M1.
+Lastly for (6), applying Lemma 2.7 with ψ = f2 and φ = |Vϕg|2 yields the desired conclusion.
+□
+3.3. Proof of Theorem 1.2. For Theorem 1.2, much of the machinery from the proof
+of Theorem 1.1 can be reused but we will need an additional estimate on the localization
+operator product asymptotics and a lemma turning the estimate in Lemma 3.1 into an L1
+error which is similar to [42, Lemma 5.4].
+As a first step, we state a simplified version of [45, Theorem 2] adapted to a context in
+which we will soon need it.
+Lemma 3.3. Let T : L2(R2d) → L2(R2d) be an integral operator with kernel K : R2d×R2d →
+C that has compact support in the first variable. Then
+∥T∥S1 ≤ A
+�
+��∥K∥L2 +
+�
+�
+2d
+�
+j=1
+��∂d+1
+j
+K
+��2
+L2
+�
+�
+1/2�
+��
+where the constant A is independent of K.
+Armed with this lemma, we can bound the trace norm of the E1 error operator from
+Lemma 3.2 above.
+Lemma 3.4. Let f ∈ Cd+2
+c
+(R2d) and g ∈ S(Rd) with ∥g∥L2 = 1, then there exists a constant
+A independent of f, g such that
+∥E1∥S1 ≤ A
+�
+��∥K∥L2 +
+�
+�
+2d
+�
+j=1
+��∂d+1
+j
+K
+��2
+L2
+�
+�
+1/2�
+�� < ∞
+where
+K(y, z) = f(y)
+�
+� �
+|α|=1
+� 1
+0
+∂αf(y + t(z − y)) dt(z − y)
+�
+� Vgg(y − z).
+Proof. From Lemma 3.2 we know that the error E1 can be written as
+E1 = V ∗
+g TVg
+where T is an integral operator with kernel K from the formulation of the lemma.
+Since Vg is an isometry and ∥AB∥S1 ≤ ∥A∥S1∥B∥L(L2), we conclude that it suffices to
+bound the trace norm of T. The bound in the formulation follows directly upon applying
+Lemma 3.3.
+The finiteness of the error bound follows from the compact support and g ∈ S(Rd) via
+[26, Theorem 11.2.5].
+□
+
+FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+15
+Lastly we formulate the promised L1-estimate, based on Lemma 3.1. Its formulation and
+proof is similar to that of [42, Lemma 5.4].
+Lemma 3.5. Let f ∈ L1(R2d) ∩ L∞(R2d). Then
+P
+��
+R2d
+����
+ρ(z)
+σ2 − ϑ(z)
+���� dz ≥ γ
+�
+≤ ∥f∥2
+L2
+γ
+√
+K
+� 3√π
+2
+√
+C
+erf(
+√
+CK) +
+3
+√
+C
+e−
+√
+CK
+�
+.
+Proof. By Markov’s inequality applied to the random variable
+�
+R2d
+��� ρ(z)
+σ2 − ϑ(z)
+��� dz, we have
+P
+��
+R2d
+����
+ρ(z)
+σ2 − ϑ(z)
+���� dz ≥ γ
+�
+≤ 1
+γ E
+��
+R2d
+����
+ρ(z)
+σ2 − ϑ(z)
+���� dz
+�
+= 1
+γ
+�
+R2d E
+�����
+ρ(z)
+σ2 − ϑ(z)
+����
+�
+dz
+= 1
+γ
+�
+R2d
+� ∞
+0
+P
+�����
+ρ(z)
+σ2 − ϑ(z)
+���� ≥ t
+�
+dt dz.
+(16)
+Next we estimate the inner integral for each z ∈ R2d using Lemma 3.1 as
+� ∞
+0
+P
+�����
+ρ(z)
+σ2 − ϑ(z)
+���� ≥ t
+�
+dt ≤ 3
+� ∞
+0
+exp
+�
+−CK min
+�
+t2
+ϑ(z)2 ,
+t
+ϑ(z)
+��
+dt
+= 3
+� ϑ(z)
+0
+exp
+�
+−CKt2
+ϑ(z)2
+�
+dt + 3
+� ∞
+ϑ(z)
+exp
+�
+−CKt
+ϑ(z)
+�
+dt
+= ϑ(z)
+� 3√π
+2
+√
+CK
+erf
+�√
+CK
+�
++
+3
+CK e−CK
+�
+.
+When computing the integral of this over R2d we will need to compute the L1-norm of ϑ.
+By (14) and Proposition 2.3 (vi) with p = 1, it can be bounded as
+∥ϑ∥L1 =
+��(Ag
+f)2 ⋆ (ϕ ⊗ ϕ)
+��
+L1
+≤
+��(Ag
+f)2��
+S1∥ϕ ⊗ ϕ
+��
+S1 = ∥Ag
+f∥2
+S2 ≤ ∥f∥2
+L2∥ϕ ⊗ ϕ∥2
+S1 = ∥f∥2
+L2
+where we used Proposition 2.3 (v) with p = 2 for the second to last step. Plugging this back
+into (16) yields
+P
+��
+R2d
+����
+ρ(z)
+σ2 − ϑ(z)
+���� dz ≥ γ
+�
+≤ 1
+γ ∥ϑ∥L1
+� 3√π
+2
+√
+CK
+erf
+�√
+CK
+�
++
+3
+√
+CK
+e−
+√
+CK
+�
+≤ ∥f∥2
+L2
+γ
+√
+K
+� 3√π
+2
+√
+C
+erf
+�√
+CK
+�
++
+3
+C
+√
+K
+e−CK
+�
+as desired.
+□
+We are now ready to complete the proof of Theorem 1.2.
+Proof of Theorem 1.2. We first claim that
+��ϑ − f2 ∗ |Vϕg|2��
+L1 ≤ A
+�
+��∥K∥L2 +
+�
+�
+2d
+�
+j=1
+∥∂d+1
+j
+K∥2
+L2
+�
+�
+1/2�
+�� .
+
+16
+SIMON HALVDANSSON
+Indeed, this follows from Lemma 3.4 as
+��ϑ − f2 ∗ |Vϕg|2��
+L1 =
+���
+Ag
+f
+�2 ⋆ (ϕ ⊗ ϕ)ˇ− Ag
+f2 ⋆ (ϕ ⊗ ϕ)ˇ
+��
+L1
+= ∥E1 ⋆ (ϕ ⊗ ϕ)ˇ∥L1
+≤ ∥E1∥S1∥ϕ ⊗ ϕ∥S1
+≤ A
+�
+��∥K∥L2 +
+�
+�
+2d
+�
+j=1
+��∂d+1
+j
+K
+��2
+L2
+�
+�
+1/2�
+��
+where we used Proposition 2.3 (vi) for the second to last step.
+We now expand the left hand side in the
+�� ρ
+σ2 − f2��
+L1 > B1 + B2 + t inequality using the
+above and Lemma 2.7 with ψ = f2 and φ = |Vϕg|2 to find
+P
+���� ρ
+σ2 − f2���
+L1 > B1 + B2 + t
+�
+≤ P
+���� ρ
+σ2 − ϑ
+���
+L1 +
+��ϑ − f2 ∗ |Vϕg|2��
+L1 +
+��f2 ∗ |Vϕg|2 − f2��
+L1 > B1 + B2 + t
+�
+≤ P
+���� ρ
+σ2 − ϑ
+���
+L1 + B1 + B2 > B1 + B2 + t
+�
+= P
+���� ρ
+σ2 − ϑ
+���
+L1 > t
+�
+≤ ∥f∥2
+L2
+t
+√
+K
+� 3√π
+2
+√
+C
+erf(
+√
+CK) +
+3
+C
+√
+K
+e−CK
+�
+where we in the last step used Lemma 3.5.
+□
+Remark. Both Theorem 1.1 and Theorem 1.2 have clear analogues in the Cohen’s class case
+which we believe to hold true. Indeed, it is straight-forward to show that
+E
+�
+1
+K
+K
+�
+k=1
+QS(f ⋆ S(Nk))
+�
+−−−−→
+K→∞
+∞
+�
+m=1
+λ2
+mQS(hm)(z),
+but controlling the error estimates requires generalizing Lemma 3.1 to the non rank-one
+case.
+Remark. The quantity
+�
+R2d |z||Vϕg(z)|2 dz which appears in Theorem 1.1 and Theorem 1.2
+should be seen as punishing the case ϕ ̸= g i.e. the reconstruction window differing from
+the window function g.
+4. Recovery via spectral data
+In this section we discuss and prove the two recovery results (Theorem 1.3 and Theorem
+1.4) which are dependent on the eigenvalues and eigenfunctions of the localization operator.
+We will see that these methods can perform better than the others proposed in this paper
+for three separate reasons.
+• The initial estimates afforded by the methods have smaller errors than the white
+noise methods.
+• Perfect recovery is possible if we can divide on the Fourier side.
+• Optimization schemes can approximate perfect recovery.
+The main drawbacks of these techniques are the following.
+• Instability of eigenfunctions and eigenvectors of matrices.
+
+FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+17
+• Computing spectral data can be computationally expensive in high dimensional
+cases.
+• When perfect recovery is possible, we need to divide by functions which are almost
+zero.
+For the reasons listed above, the methods in this section serve as an excellent alternatives
+to the other proposed methods when we have more exact knowledge about the localization
+operator.
+4.1. Weighted accumulated Cohen’s class. The accumulated Cohen’s class, introduced
+in [37], is a generalization of accumulated spectrograms from [3] where it was used for symbol
+recovery for binary localization operators Ag
+χΩ. There a central idea was that the eigenvalues
+can be separated into two groups with the first ≈ |Ω| being close to 1, followed by a sharp
+“plunge region” after which the remaining eigenvalues are all close to 0.
+This fact was
+originally proved in [18]. Based on this fact, the quantity
+⌈|Ω|⌉
+�
+m=1
+|Vg(hm)(z)|2 ≈ χΩ(z)
+was defined as the accumulated spectrogram. Later on, [37] extended the concept to mixed-
+state localization operators f⋆S by replacing the spectrograms by Cohen’s class distributions
+by approaching the proof from a quantum harmonic analysis perspective.
+Much of the work in these papers are focused on showing that the accumulated Cohen’s
+class is close to the quantity
+�
+m
+λmQS(hm)(z)
+(17)
+by going into specifics on the decay of the eigenvalues.
+However, since computing the
+accumulated spectrogram already requires knowing the eigenfunctions, we (almost always)
+have exact knowledge of the eigenvalues and can bypass this approximation step and include
+the eigenvalues from the beginning. In this way, the error of the approximation can be
+decreased with no loss in performance or computation time. Moreover, we don’t require a
+priori knowledge of |Ω| to decide the number of eigenpairs to include. A consequence of this
+approach is that the resulting estimator also works well for non-binary localization operators
+whose eigenvalues do not follow the same 0 − 1 dichotomy. We refer to the quantity (17) as
+the weighted accumulated Cohen’s class to highlight the addition of the eigenvalues weights.
+Both [3] and [37] restricted their attention to the case where the window g or the operator
+window S was known a priori. We lift this restriction by introducing a reconstruction window
+ϕ or reconstruction operator window T which does not have to agree with the original
+window g or S in the same way as we did for the average observed spectrogram. As we will
+see in the proof below, the proper estimator then instead becomes �
+m λmQT (hm)(z).
+Proof of Theorem 1.3. The key observation for the proof is that �
+m λmQT (hm) = f∗(S⋆ ˇT).
+To see this, expand f ⋆ S in its singular value decomposition f ⋆ S = �
+m λm(hm ⊗ hm) and
+note that
+f ∗ (S ⋆ ˇT) = (f ⋆ S) ⋆ ˇT =
+��
+m
+λm(hm ⊗ hm)
+�
+⋆ ˇT
+=
+�
+m
+λm(hm ⊗ hm) ⋆ ˇT =
+�
+m
+λmQT (hm)
+
+18
+SIMON HALVDANSSON
+where we used (11) for the last step. We can now compute
+�����
+N
+�
+m=1
+λmQT (hm) − f
+�����
+L1
+≤
+�����
+N
+�
+m=1
+λmQT (hm) −
+∞
+�
+m=1
+λmQT (hm)
+�����
+L1
++
+��f ∗ (S ⋆ ˜S) − f
+��
+L1
+≤
+∞
+�
+m=N+1
+|λm|∥QT (hm)∥L1 +
+��f ∗ (S ⋆ ˇT) − f
+��
+L1
+≤
+∞
+�
+m=N+1
+|λm| + Var(f)
+�
+R2d |z|(S ⋆ ˇT)(z) dz
+where we used that ∥QT (hm)∥L1 ≤ 1 by Proposition 2.3 (vi) with p = 1 and the estimate
+in Lemma 2.7.
+The deconvolution strategy for the perfect N = ∞ reconstruction detailed in the theorem
+follows directly from the standard Fourier convolution theorem.
+□
+Remark. Ideally, we would want S ⋆ ˇT to be a Dirac delta to make the above reconstruction
+exact in the sense that �
+m λmQT (hm) = f without needing to employ classical deconvo-
+lution. The closest we can get to this is in the lattice setting where such a construction is
+possible which is discussed in [44, Section 6.1].
+The error incurred from S ⋆ ˇT ̸= δ0 is partially captured in the
+�
+R2d |z|(S ⋆ ˇT)(z) dz
+factor which simplifies to
+�
+R2d |z||Vϕg(z)|2 dz in the rank-one setting which we recognize
+from Section 3. In Section 6.1 we numerically investigate the consequences of this.
+The reader familiar with [37] will note that we essentially followed the exact same path
+for the proof as in that paper without restricting ourselves to indicator functions f = χΩ
+and allowing T ̸= S.
+The recovery procedure detailed above is clearly linear and hence it is easy to see that
+the recovery procedure is continuous.
+We mean this in the sense that if I is the map
+f ⋆ S �→ f ∗ (S ⋆ ˇT) and A ∈ S1 is a perturbation, then
+��I(f ⋆ S + εA) − I(f ⋆ S)
+��
+L1 = ε∥A ⋆ ˇT∥L1 ≤ ε∥A∥S1
+(18)
+by Proposition 2.3 (vi).
+Examples of functions with non-zero STFT was discussed in [27] and notably includes the
+standard Gaussian. We present an example of Theorem 1.3 complete with a deconvolution
+to recover the symbol exactly in Section 6.3.1 below.
+4.2. Weighted accumulated Wigner distribution. The approach in Theorem 1.4 is
+perhaps the simplest of those detailed in this paper once framed as just computing the
+Weyl symbol of the localization operator and comparing with f. Note also that there is
+no requirement for a reconstruction window in this situation as we only make constructions
+based on the spectral data of the localization operator.
+Proof of Theorem 1.4. We prove the full case where S is a positive trace-class operator and
+note that the special rank-one case follows.
+As discussed in Section 2.2.3, the Weyl symbol of the function operator convolution
+f ⋆ S is given by f ∗ aS where aS is the Weyl symbol of S. By the linearity of the Weyl
+symbol mapping S �→ aS, we can compute this as using the spectral decomposition of
+
+FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+19
+S = �
+n sn(ϕn ⊗ ϕn) and the fact that aϕ⊗ϕ = W(ϕ).
+�
+m
+λmW(hm) = af⋆S = f ∗ aS = f ∗
+�
+n
+snW(ϕn).
+In order for the sum in the left-hand side to converge in L1, we need for the Wigner distri-
+bution of each eigenfunction hm to be integrable. This is equivalent to hm ∈ M1(Rd) which
+follows from ϕn ∈ S(Rd) by [6, Theorem 4.1].
+The L1-error estimate now follows by applying Lemma 2.7 with ψ = f and blurring kernel
+φ = �
+n snW(ϕn).
+□
+Just as in Theorem 1.3, we can deconvolve �
+m λmW(hm) = f ∗ �
+n snW(ϕn) to recover
+f exactly provided the Fourier transform F (�
+n snW(ϕn)) is zero-free.
+Note that the �
+m λmW(hm) sum is easily seen to converge pointwise by the bound
+|W(hm)(z)| ≤ 2d∥hm∥2
+L2 while we need the extra condition on the window for L1-convergence.
+This is why we could not formulate Theorem 1.4 with partial sums as we did for Theorem
+1.3.
+The above argument can be taken another step to show that the reconstruction procedure
+is not stable as was the case for accumulated spectrograms as showed in (18). To see that
+the inverse mapping I : S1 → L1(R2d), f ⋆ S �→ �
+m λmW(hm) is not continuous, fix
+ψ ∈ L2(Rd) \ M1(Rd) and consider the perturbation operator A = ψ ⊗ ψ for which we have
+∥I(f ⋆ S + εA) − I(f ⋆ S)∥L1 = ε∥I(ψ ⊗ ψ)∥L1 = ε∥W(ψ)∥L1 = ∞.
+In Section 6.3.2 we provide an example showing the performance of the estimator �
+m λmW(hm).
+5. Recovery via plane tiling
+Using quantum harmonic analysis, it is easy to show that the spectrograms of an or-
+thonormal basis add up to the function which is identically 1. Indeed, using the relation
+1 ⋆ (ϕ ⊗ ϕ) = ∥ϕ∥2
+L2IL2 from [48, Proposition 3.2 (3)], we get for a normalized ϕ ∈ L2(Rd)
+that
+�
+n
+|Vϕ(en)|2 =
+�
+n
+(en ⊗ en) ⋆ (ϕ ⊗ ϕ)ˇ
+=
+��
+n
+(en ⊗ en)
+�
+⋆ (ϕ ⊗ ϕ)ˇ
+= I ⋆ (ϕ ⊗ ϕ)
+= 1 ∗ (ϕ ⊗ ϕ) ⋆ (ϕ ⊗ ϕ)ˇ= 1 ∗ |Vϕϕ|2 = 1.
+Intuitively, we should expect that those basis elements whose spectrograms are primarily
+supported outside f should lose most of their mass when we apply Ag
+f to them and the rest
+should remain intact or be scaled by something proportional to f. This is the motivation
+for the plane tiling approach which we prove below. The proof is rather straight-forward
+and we are able to inherit the main error estimate from Theorem 1.2 as the sum approaches
+the same quantity ϑ as the K → ∞ situation in that theorem.
+
+20
+SIMON HALVDANSSON
+Proof of Theorem 1.5. We first rework the estimator �
+n |Vϕ(Ag
+fen)(z)|2 into a more man-
+ageable form using the self-adjointness of Ag
+f and Example 2.2 as
+�
+n
+|Vϕ(Ag
+fen)(z)|2 =
+�
+n
+�
+Ag
+fen ⊗ Ag
+fen
+�
+⋆ (ϕ ⊗ ϕ)ˇ(z)
+= Ag
+f
+��
+n
+en ⊗ en
+�
+Ag
+f ⋆ (ϕ ⊗ ϕ)ˇ(z)
+=
+�
+Ag
+fIAg
+f
+�
+⋆ (ϕ ⊗ ϕ)ˇ(z)
+=
+�
+Ag
+f
+�2 ⋆ (ϕ ⊗ ϕ)ˇ(z) = ϑ(z)
+where ϑ is the same as in Section 3. The same analysis on the size of ∥ϑ − f2∥L1 from the
+proof of Theorem 1.2 again applies and yields the desired conclusion.
+□
+Remark. The above result can be extended to mixed-state localization operators as was
+done in Section 4 through some technical considerations. More specifically, it is possible to
+control the error ∥ �
+n QT ((f ⋆ S)en) − f2∥L1 if S and T are positive rank-one operators
+whose spectral decomposition consists of Schwartz functions. This is done by bounding the
+trace norm of the error operator in the expansion (f ⋆ S)2 = f2 ⋆ S + E1 in a similar way to
+how it was done in the proof of Theorem 1.2.
+In most cases, we should expect this deterministic method to perform worse than the white
+noise approach discussed earlier due to slow convergence, we require many eigenfunctions to
+tile a significant portion of the time-frequency plane. However, since we are free to choose
+the orthonormal basis, we are able to tailor it according to any information we have about
+f. For example, if we know that f is centered around a particular point z0 in phase space
+we can choose en = π(z0)hn as our basis elements where hn is the n:th Hermite function.
+As the Hermite functions are eigenfunctions of the unit disk [14], the collection {en}n will
+then spread out radially from z0. This is illustrated in Section 6.4 below.
+6. Numerical implementation
+In computer applications there is no continuum and the integral in the the localization
+operator definition (1) is replaced by a sum, most often over some lattice, yielding what is
+referred to as a Gabor multiplier [21]. While showing that results of this paper carry over to
+this setting is a non-trivial undertaking which we do not attempt, we settle for investigating
+the numerical behavior and draw only empirical conclusions. Do note however that many
+results on localization operators carry over to the Gabor multiplier setting [11, 17, 21].
+With the above considerations out of the way, we first present a visual overview of the
+approximation process detailed in the Theorem 1.1 and Theorem 1.2.
+
+FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+21
+ρ20
+ρ200
+ϑ
+f2 ∗ |Vgg|2
+f2
+Figure 1. All the intermediate steps in the white noise approximation detailed in
+Section 3 for ϕ = g. The subscript on ρ indicates the number of samples K used.
+Note in particular that the difference between ϑ and f2 ∗ |V g
+g | which corresponds to the
+error estimate in Lemma 3.4 is negligible in the above example.
+All of the figures presented in this section were generated using the Large Time/Frequency
+Analysis Toolbox (LTFAT) [40].
+6.1. Reconstruction window disparity. The reconstruction window ϕ and the window
+g do not have to coincide although them doing so decreases the size of the error estimates
+which involves the quantity
+�
+R2d |z||Vϕg(z)|2 dz
+which in [42] is referred to as a mutual correlation measure. These can be found in three of
+our five main theorems. To give an indication of the real world impact of the ϕ ̸= g case, we
+present following three examples for the white noise estimator. This example is analogous
+to [3, Figure 3] for the accumulated spectrogram.
+
+0.8
+3.5
+0.6
+3
+0.4
+Frequency (normalized)
+2.5
+0.2
+0
+2
+0.2
+1.5
+-0.4
+1
+-0.6
+0.5
+-0.8
+0
+100
+200
+300
+400
+500
+600
+700
+Time(samples)0.8
+3.5
+0.6
+3
+0.4
+Frequency (normalized)
+2.5
+0.2
+0
+2
+0.2
+1.5
+-0.4
+1
+-0.6
+0.5
+-0.8
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)0.8
+3.5
+0.6
+3
+0.4
+Frequency (normalized)
+2.5
+0.2
+0
+2
+0.2
+1.5
+-0.4
+1
+-0.6
+0.5
+-0.8
+0
+100
+200
+300
+400
+500
+600
+700
+Time(samples)0.8
+3.5
+0.6
+3
+0.4
+Frequency (normalized)
+2.5
+0.2
+0
+2
+0.2
+1.5
+-0.4
+1
+-0.6
+0.5
+-0.8
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)0.8
+3.5
+0.6
+3
+0.4
+Frequency (normalized)
+2.5
+0.2
+0
+2
+0.2
+1.5
+-0.4
+1
+-0.6
+0.5
+-0.8
+0
+100
+200
+300
+400
+500
+600
+700
+Time(samples)22
+SIMON HALVDANSSON
+ϕ1
+ϕ2
+g
+f2
+ρϕ1
+ρϕ2
+ρg
+f2
+ρϕ1
+ρϕ2
+ρg
+f2
+ρϕ1
+ρϕ2
+ρg
+Figure 2. Two reconstruction window ϕ1 and ϕ2, the window g associated to the
+localization operator Ag
+f and three different examples of white noise estimations by
+ρ with the reconstruction windows, all with K = 2000 samples of white noise.
+6.2. Noise estimation. Since our estimator for f in Theorem 1.1 and Theorem 1.2 is
+�
+ρ
+σ2 ,
+we need some knowledge of σ2 in order to be able to scale our estimate of f correctly. In
+the continuous setting, estimating σ2 is not meaningful as we have an infinite number of
+samples and the estimation is hence trivial. However in the discrete setting, this can be
+achieved using standard tools provided the input (Nk)K
+k=1 is known. Alternatively, if only
+
+0.8
+3.5
+0.6
+3
+0.4
+Frequency (normalized)
+2.5
+0.2
+0
+2
+0.2
+1.5
+-0.4
+1
+-0.6
+0.5
+-0.8
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)0.8
+3.5
+0.6
+3
+0.4 
+Frequency (normalized)
+2.5
+0.2
+0
+2
+-0.2
+1.5
+-0.4
+1
+-0.6
+0.5
+-0.8
+0
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)0.8
+3.5
+0.6
+3
+0.4
+Frequency (normalized)
+2.5
+0.2
+0
+2
+0.2
+1.5
+-0.4
+1
+-0.6
+0.5
+-0.8
+0
+100
+200
+300
+400
+500
+600
+700
+Time(samples)1
+0.8
+0.9
+0.6
+0.8
+0.4 
+0.7
+0.2
+0.6
+0
+0.5
+-0.2
+0.4
+-0.4
+0.3
+-0.6
+0.2 
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)1
+0.9
+0.8
+0.8
+0.6
+0.7
+0.4
+0.6
+0.2
+0.5
+0
+0.4 
+-0.2
+0.3
+-0.4
+0.2 
+-0.6
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)1
+0.8
+0.9
+0.6
+0.8
+0.4
+0.7
+0.2
+0.6
+0
+0.5
+-0.2
+0.4
+-0.4
+0.3
+-0.6
+0.2
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)1
+1
+0.8
+0.9
+0.6
+0.8
+0.4
+0.7
+0.2
+0.6
+0
+0.5
+-0.2
+0.4 
+-0.4
+0.3
+-0.6
+0.2 
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)7
+0.8
+0.9
+0.6
+0.8
+0.4 
+0.7
+0.2
+0.6
+0
+0.5
+0.2
+0.4
+-0.4
+0.3
+-0.6
+0.2
+-0.8
+0.1
+0
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)1
+1
+0.8
+0.9
+0.6
+0.8
+0.4
+Frequency (normalized)
+0.7
+0.2
+0.6
+0
+0.5
+-0.2
+0.4
+-0.4
+0.3
+-0.6
+0.2
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)7
+1
+0.8
+0.9
+0.6
+0.8
+0.4 
+0.7
+0.2 
+0.6
+0
+0.5
+-0.2
+0.4 
+-0.4
+0.3
+-0.6
+0.2
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)1
+0.8
+0.9
+0.6
+0.8
+0.4
+Frequency (normalized)
+0.7
+0.2
+0.6
+0
+0.5
+0.2
+0.4
+-0.4
+0.3
+-0.6
+0.2
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)0.12
+0.1
+0.08
+0.06
+0.04
+0.02
+0
+0
+100
+200
+300
+400
+500
+600
+700
+8000.3
+0.25
+0.2
+0.15
+0.1
+0.05
+0
+0
+100
+200
+300
+400
+500
+600
+700
+8000.25
+0.2
+0.15
+0.1
+0.05
+0
+0
+100
+200
+300
+400
+500
+600
+700
+8000.8
+3.5
+0.6
+3
+0.4
+Frequency (normalized)
+2.5
+0.2
+0
+2
+0.2
+1.5
+-0.4
+1
+-0.6
+0.5
+-0.8
+0
+100
+200
+300
+400
+500
+600
+700
+Time(samples)FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+23
+observations of the spectrograms of random noise are known, σ2 can be estimated as the
+average value of |Vϕ(N)|2 as can be seen from the calculations in the proof of Lemma 3.1.
+6.3. Spectral reconstruction examples. In this section we test the performance of the
+estimators from Theorem 1.3 and Theorem 1.4.
+6.3.1. Weighted accumulated spectrograms. In the following example, we set both the recon-
+struction and original window to be the standard Gaussian for convenience. Consequently,
+the blurring kernel or equivalently the impulse response of the system f �→ ρf is given by
+a two dimensional Gaussian and it is easy to get a rough idea of the original symbol. Due
+to simple blurring kernel and exact spectral information, we are also able to deconvolve the
+estimator to recover the original symbol exactly.
+Figure 3. A symbol, the associated accumulated spectrogram and the deconvolved
+estimate of the symbol.
+Note that by viewing the blurring kernel as an impulse response, it becomes clear that
+we can find it for deconvolution purposes by letting the symbol be a Dirac delta which is
+feasible in the finite setting.
+6.3.2. Weighted accumulated Wigner distributions. The Wigner-based approach in Theorem
+1.4 is notably more direct than the spectrogram approach in Theorem 1.3 in that there is no
+reconstruction window. Hence it is reasonable to expect a smaller error with this method.
+Indeed, this is what we see in the example below.
+Note that due to how the Wigner transform is implemented in LTFAT, we only recover
+the portion of the symbol with positive frequencies. As a consequence, we only use the
+upper half of the symbol from the weighted accumulated spectrogram example although
+still at the same resolution. Note also that the Wigner transform is inherently redundant in
+an oversampling sense and in our case has 200 times the resolution of the original symbol.
+
+Symbol
+2.5
+0.8
+0.6
+2
+0.4
+Frequency (normalized)
+0.2
+1.5
+0
+0.2
+-0.4
+-0.6
+0.5
+-0.8
+0
+50
+100
+150
+200
+250
+300
+350
+Time (samples)Weightedaccumultadspectrogram
+1
+2
+0.8
+1.8
+0.6
+1.6
+0.4 
+1.4
+0.2
+1.2
+0
+1
+-0.2 
+0.8
+-0.4
+0.6
+-0.6
+0.4
+-0.8
+0.2
+0
+50
+100
+150
+200
+250
+300
+350
+Time (samples)Recovered symbol
+2.5
+0.8
+0.6
+2
+0.4
+Frequency (normalized)
+0.2
+1.5
+0
+0.2
+1
+-0.4
+-0.6
+0.5
+-0.8
+0
+50
+100
+150
+200
+250
+300
+350
+Time (samples)24
+SIMON HALVDANSSON
+Figure 4. A symbol and the corresponding weighted accumulated Wigner estima-
+tor.
+Due to the aforementioned redundancy and positive frequency restriction, finding the
+system impulse response to perform a deconvolution is not as straight-forward as in the
+weighted accumulated spectrogram case. However, there is no direct reason to believe that
+this should be impossible in this setting but we leave the an implementation to future work.
+6.4. Plane tiling example. In this section we consider an example inspired by the dis-
+cussion at the end of Section 5. Specifically, we let the symbol f be a circle centered at
+z0 = (400, 0) and choose our orthonormal basis to be Hermite functions translated by π(z0).
+
+Positive frequency symbol
+2.5
+2
+4
+2
+6
+8
+1.5
+10
+12
+14
+16
+0.5
+18
+20
+5
+10
+15
+20
+25
+30
+35
+40Accumulated Wigner
+2.5
+50
+2
+100
+150
+1.5
+200
+250
+0.5
+300
+350
+0
+400
+50
+100
+150
+200
+250
+300
+350
+400FOUR WAYS TO RECOVER THE SYMBOL OF A NON-BINARY LOCALIZATION OPERATOR
+25
+N = 50
+N = 150
+N = 200
+Figure 5. Parallel views of the plane tiling estimator and the sum of spectrograms
+of the same orthornmal basis for different number of terms N. Note how the first
+pair of images agree as the support of the spectrograms of the first few basis elements
+are all approximately contained in the symbol.
+The number of basis elements needed to cover the symbol is dependent on the size of
+the symbol as each basis element spectrogram only can have total L1-energy 1 by Moyal’s
+identity.
+Since the support of a spectrogram is never compact [33], there will inevitably be some
+“spillage” from eigenfunctions whose core spectrogram support are far away from the sup-
+port of f. This is illustrated in the following figure depicting the sum of spectrograms of
+Hermite functions localized outside of where the majority of their spectrogram support is.
+The localization operator symbol in this example is the indicator functions of a square.
+
+Localized n = 50
+0.8
+0.9
+0.6
+0.8
+0.4
+0.7
+Frequency (normalized)
+0.2
+0.6
+0
+0.5
+0.2
+0.4
+-0.4
+0.3
+-0.6
+0.2
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)Non localizedn =50
+0.8
+0.9
+0.6
+0.8
+0.4
+0.7
+Frequency (normalized)
+0.2
+0.6
+0
+0.5
+0.2
+0.4
+-0.4
+0.3
+-0.6
+0.2
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)Localizedn=150
+0.8
+0.9
+0.6
+0.8
+0.4
+0.7
+Frequency (normalized)
+0.2
+0.6
+0
+0.5
+0.2
+0.4
+-0.4
+0.3
+-0.6
+0.2
+-0.8 
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)Non localized n =150
+0.8
+0.9
+0.6
+0.8
+0.4
+0.7
+Frequency (normalized)
+0.2
+0.6
+0
+0.5
+0.2
+0.4
+-0.4
+0.3
+-0.6
+0.2
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)Localized n = 200
+0.8
+0.9
+0.6
+0.8
+0.4 
+0.7
+Frequency (normalized)
+0.2
+0.6
+0
+0.5
+0.2
+0.4
+-0.4
+0.3
+-0.6
+0.2
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)Non localized n = 200
+0.8
+0.9
+0.6
+0.8
+0.4
+0.7
+Frequency (normalized)
+0.2
+0.6
+0
+0.5
+0.2
+0.4
+-0.4
+0.3
+-0.6
+0.2
+-0.8
+0.1
+0
+100
+200
+300
+400
+500
+600
+700
+Time (samples)26
+SIMON HALVDANSSON
+Figure 6. Sum of spectrograms of the first N Hermite functions localized outside
+the majority of their support in phase space for N = 30, 60, 90, 120. Note that the
+scale of the plot is on the order of 10−33.
+References
+[1]
+L. D. Abreu, “Local maxima of white noise spectrograms and Gaussian entire func-
+tions,” Journal of Fourier Analysis and Applications, vol. 28, no. 6, 2022. doi: 10.10
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+

HdAyT4oBgHgl3EQfrvlK/content/tmp_files/2301.00565v1.pdf.txt

@@ -0,0 +1,406 @@
+Using meaning instead of words to track topics
+Judicael POUMAY1 and Ashwin ITTOO1
+ULiege/HEC Liege, Rue Louvrex 14, 4000 Liege, Belgium {judicael.poumay,
+ashwin.ittoo}@uliege.be
+Abstract. The ability to monitor the evolution of topics over time is
+extremely valuable for businesses. Currently, all existing topic tracking
+methods use lexical information by matching word usage. However, no
+studies has ever experimented with the use of semantic information for
+tracking topics. Hence, we explore a novel semantic-based method using
+word embeddings. Our results show that a semantic-based approach to
+topic tracking is on par with the lexical approach but makes different
+mistakes. This suggest that both methods may complement each other.
+Keywords: Topic tracking · lexical · semantic · topic models
+1
+Introduction
+Buried within the voluminous amounts of texts available online are meaning-
+ful insights, which could help in supporting business decision-making activities.
+Topic modelling methods extracts latent topic in a corpus [4,10] and can be used
+to discover these insights. Examples of applications include fraud detection [11],
+understanding employee and customer satisfaction [8,7]. Extracted topics can be
+tracked over time to understand their evolution or discover emerging one. Hence,
+we focus on this task of topic tracking in which the goal is to link instances of
+the same topic that have been extracted at different time periods.
+Several methods for tracking topics have been proposed in the past [3,6,13,12,9].
+These methods use measures such as the JS divergence [13,12,9] or online topic
+models [3,6] which rely on lexical information to track topic across time.
+However, no studies has ever experimented with using semantic informa-
+tion to track topics over time. Intuitively, semantic based approaches could be
+promising as they do not rely on simple surface form and can capture concepts
+such as synonymy. For example, given a topic about ”AI”, across time we could
+observe that the term ”Machine Learning” has become more popular than ”AI”.
+However, a lexical approach to topic tracking would not be able to handle such
+lexical drift and to relate those words over time. Conversely, such lexical vari-
+ation would have been captured by a semantic approach. Moreover, topic-word
+distributions are unstable across multiple runs [1], i.e. the resulting top words
+of a topic tend to change significantly. This entails that the lexical information
+we rely upon to track topics is also unstable even if the overall semantic of the
+topic remains the same. Thus, a semantic-based approach may be more robust.
+arXiv:2301.00565v1  [cs.CL]  2 Jan 2023
+
+2
+J. POUMAY A. ITTOO
+Hence, our work aims at investigating on the use of semantic information for
+topic tracking and its comparison against lexical information. Therefore, as our
+main contribution, we propose a novel semantic topic tracking method known
+as Semantic Divergence (SD) based on word embeddings. As an ancillary con-
+tribution, we study the challenges of topic tracking in the context of hierarchical
+topic modelling.
+2
+Background and Related work
+2.1
+Topic Modelling
+LDA [4] is the first traditional topic model. At the core of LDA is a Bayesian
+generative model with two Dirichlet distributions, respectively for the document-
+topic distributions and for the topic-word distributions. These distributions are
+learnt and optimized via an inference procedure which enables topics to be ex-
+tracted. The main weakness of LDA is that it requires the user to specify a
+predefined number of topics to be extracted.
+More complex topic models have been proposed since LDA. In particular,
+HTMOT [10] was proposed to simultaneously model topic hierarchy and tem-
+porality. Specifically, HTMOT produces a topic tree in which the depth and the
+number of sub-topic for each branch is defined dynamically during training. Ad-
+ditionally, HTMOT models the temporality of topics enabling the extraction of
+topics that are lexically close but temporally distinct.
+2.2
+Topic Tracking
+Topic tracking is the task of monitoring the evolution of topics through time.
+It was initially defined in a pilot study [2] in 1998 as the continuous automatic
+classification of a stream of news stories into known or new topics.
+Currently, two general framework compete for topic tracking. The first stream
+is that of online topic models. which incorporate new data incrementally [3,6]. In
+[3], the authors propose Online-LDA, a version of LDA able to update itself with
+new documents without having to access to previously processed documents. In
+practice, Online-LDA assumes that time is divided in slices and at each slice
+an LDA model is trained using the previous slice as prior. They were able to
+show that their system can find emerging topics by artificially injecting new
+topic into the news stream. They performed their experiments on the NIPS
+and Reuters-21578 datasets. Similarly in [6], the authors propose a model that
+can dynamically decide the right number of topics in an online fashion. They
+performed their experiments on the the 20 Newsgroup and the TDT-2 datasets.
+The second stream is concerned with linking topics extracted independently
+at different time periods [13,12,9]. In [13], the authors use about 30,000 abstracts
+of papers in various journals from 2000 to 2015. They then applied LDA to each
+year independently and linked topics using the Jensen-Shannon Divergence (JS)
+to measure their similarity [5]. In [12] the authors applied a similar method on
+
+Using meaning instead of words to track topics
+3
+news articles. However, they differ in that while [13] simply links topics together,
+[12] clusters them. This means that once two topic have been linked they form
+a cluster and subsequent topics will be compared to the whole cluster and not
+just the preceding topic. Finally in [9], the authors also proposed a tracking
+method using the JS divergence applied to scientific papers. However, they do
+not constraint linkage to a one-to-one mapping which allows for the fusion and
+splitting of topics. All of the aforementioned paper evaluated their topic tracking
+method using a qualitative analysis that demonstrated the performance of their
+technique.
+We based our work on that second stream because it allows for better paral-
+lelization as time slices are processed independently.
+3
+Methodology
+In this section, we will present our methodology for topic tracking. We will start
+by describing our corpus and topic extraction method. Next, we will define our
+SD measure. Finally, we will present the topic tracking algorithm.
+3.1
+Topic extraction
+To perform our experiments, we crawled 10k articles from the Digital Trends 1
+archives from 2019 to 2020. This news website is mainly focused on technological
+news with topics such as hardware, space exploration and COVID-19. For all
+articles, we extracted the text, title, category and timestamp. We pre-possessed
+the corpus according to HTMOT [10].
+To extract topics hierarchies (see figure 1), we used the HTMOT topic model
+[10] . The extracted topics are represented by a list of words and a list of entities.
+Fig. 1. Example of a topic hierarchy
+We follow HTMOT [10] and only focus on the first and second level of topic
+extracted. Specifically, the authors observe that deeper topics becomes more
+esoteric making them harder to understand by annotators representing a general
+audience. Consequently, this makes it difficult to assess the correctness of tracked
+topics at deeper levels of the topic tree.
+1 https://www.digitaltrends.com/
+
+ROOT
+Space
+COVID-19
+Astronauts
+Astronomy
+Vaccines
+Tests4
+J. POUMAY A. ITTOO
+3.2
+Proposed Semantic Divergence measure
+We will now describe our novel topic tracking method, which departs from the
+JS divergence traditionally applied in previous studies. We name our method
+”Semantic divergence” or SD. It uses word embeddings to measure the distance
+between topics. Each topic will be assigned an embedding as the sum of the
+embeddings of the top words in that topic weighted by their probability. Then,
+the distance between two topics is computed as the cosine distance of their
+respective embedding. We will use FastText as the word embedding. FastText
+helps with rare and out of vocabulary words. This is essential considering our pre-
+processing step includes lemmatization which may produce incorrectly spelled
+words. Hence the embedding of a topic is defined as follows :
+emb(t) =
+�
+(w,p)∈t
+p ∗ FastText(w)
+(1)
+And the Semantic Divergence between two topics is defined as :
+SD(t1, t2) = cosine(emb(t1), emb(t2))
+(2)
+Where w is a word in a topic t and p is the probability of that word.
+3.3
+Topic Tracking Algorithm
+Finally, to track topics across time we applied HTMOT on our corpus. For
+each year (2019 and 2020), we obtained a corresponding topic tree. Then, we
+computed the distance between every topics across both years using either JS
+or SD. To do this we used the top 100 words and top 15 entities to represent
+each topic. Subsequently, we ranked order all computed pairs of topics and then
+iteratively selected the most similar pairs (lowest SD or JS score) such that each
+topic is paired only once. Finally, we used a pre-defined threshold to remove
+pairs with a poor score.
+Note that our approach does not take into account structural information.
+Indeed, tracking topics in the context of hierarchical topic modelling presents
+another interesting challenge : there exist many possible resulting trees that are
+equally correct. In one run, we may extract the topic of space whose sub-topics
+can be grouped into space exploration and astronomy. Conversely, in another
+run, we may extract space exploration and astronomy as separate topics with
+their own sub-topics. Hence, it is difficult to leverage the structural information
+contained in the topic trees to track topics as it cannot be expected to respect
+a specific conceptual taxonomy.
+4
+Results : JS vs SD
+In this section, we will discuss how our semantic based method compares with
+respect to the traditional lexical based method.
+
+Using meaning instead of words to track topics
+5
+First, we studied the overlap between the two methods, i.e. the number of
+pairs extracted by both. We discovered that, 111 pairs were extracted with JS
+with a threshold of <0.4, while 121 pairs were extracted with SD with a threshold
+of <0.1. These threshold were set through empirical observation but may depend
+on the dataset used. These 111-121 pairs can be grouped into three categories
+(see figure 2). 72 pairs were the same between the two methods (60-65% of the
+total pairs). For example, topics such as space and video games were easily paired
+across both years by both methods. This already indicates that our SD method
+is able to pair topic across time with performance similar to JS. This leaves
+39-49 pairs that are different across the two methods (35-40% of the total pairs)
+which we can evaluate. Out of those different pairs, we notice that in most cases
+one method (e.g. SD) would track/link a topic pair across both years, while the
+other method (e.g. JS) did not as the best possible pair was above the threshold.
+We are then left with 10 different pairs that can themselves be paired according
+to which 2019 or 2020 topic they share (see figure 2).
+Fig. 2. The pairs extracted by both methods can be grouped into three categories. The
+circle represent topics and their color represent years (2019 blue; 2020 yellow). The link
+color represent the method used (JS red; SD green). The three categories are : 1) The
+pairs extracted by both methods (72). 2) The pairs that differ but share a topic (10)
+E.g. JS extracted the pair 4-D while SD extracted 4-E. 3) The pairs of topics that were
+only linked with one method (29-39).
+To compare the performance of the two tracking methods, we decided to
+use a survey comparing these 10 pairs of topics extracted by both JS and SD.
+Precisely, for each question, given an initial topic, annotators were shown the JS
+and SD pairing and asked which is better. Additionally, we also asked annotators
+to provide a confidence score on a scale from 1 to 5. In total, we received 38
+answers coming from a small online community focused on answering surveys2.
+The survey can be found on github 3.
+Looking at the survey results (table 1), it can be seen that SD slightly outper-
+forms JS with 54% of annotators preferring the former to the latter. Moreover,
+we also note that the annotators were confident in their evaluation, with an
+average confidence score of 3.3. Interestingly, there is a lot of variability in the
+2 https://www.reddit.com/r/SampleSize/
+3 https://github.com/JudicaelPoumay/TopicTrackingPaper
+
+72 Common pairs
+10 Pairs sharing a topic
+29-39 Pairs only linked with one method
+39-49 Pairs that differ between the two methods6
+J. POUMAY A. ITTOO
+answers. Some topics were clearly better paired with one method or the other
+(Q3 and Q5) while for others, it wasn’t as clear (Q1, Q2 and Q4).
+Table 1. The ”chose SD” column corresponds to the % of annotators that chose the
+SD pair as the best pair.
+Questions Chose SD Confidence level
+Q1
+42.1% (22) 3.2
+Q2
+63.2% (14) 2.6
+Q3
+21.1% (30) 3.7
+Q4
+65.8% (13) 3.5
+Q5
+78.9% (8)
+3.5
+Average
+54%
+3.3
+For example, figure 3 corresponds to Q1. It shows how a 2019 topic has been
+paired with 2020 topics using JS and SD. First, we can notice that the distance
+recorded between the pairs is close to the threshold for both methods. Specifi-
+cally, 0.29 for the JS pair and 0.09 for the SD pair (threshold = 0.4 for JS and
+0.1 for SD). This makes sense as good pairs (pairs with low JS/SD values) are
+extracted by both methods. Second, the 2019 topic is about social media data
+security. Whereas the chosen 2020 topic is about :
+– Social media when paired with JS.
+– Data security when paired with SD.
+Hence, both pairing seems suitable, which could explain the indecisiveness of
+annotators. Specifically, 16 of them decided the SD pairing was better whereas
+22 of them decided the JS pairing was better. Their confidence level for this
+question was 3.2 out of 5.
+Fig. 3. A first example of different pairing between SD and JS on the same 2019 topic.
+Similarly, figure 4 corresponds to Q5 and shows how another 2019 topic has
+been paired based on the two methods. Here, the 2019 topic is about web security.
+Whereas the chosen 2020 topic is about :
+
+2020 topics
+2019 topic
+2020 topics
+issue
+company
+company
+datum
+issue
+social
+service
+datum
+platform
+account
+account
+app
+app
+SD
+user
+JS-
+ban
+@Apple
+@Facebook
+@Facebook
+@iPhone
+@U.S.
+@TikTok
+@Google
+@Twitter
+@U.S.
+@Epic
+@instagram
+@Twitter
+@AppStore
+@FTC
+@TrumpUsing meaning instead of words to track topics
+7
+– Data security when paired with JS.
+– Web security topic when paired with SD.
+Moreover, the topic chosen by SD is a sub-topic of the topic chosen by JS which
+demonstrates the difficulty in topic tracking in a hierarchical setting. Indeed,
+it can be difficult to differentiate a topic from its sub-topic, especially if that
+sub-topic dominates the others as parent topics are the sum of their sub-topics.
+In this case, annotators agreed more and 30 out of 38 decided the SD pair was
+better. Their confidence level for this question was 3.5 out of 5.
+Fig. 4. A second example of different pairing between SD and JS on the same 2019
+topic.
+Hence, we argue that JS and SD are two fundamentally different approaches
+and that both have their advantages. JS is lexically driven and may work best
+for linking topics which tend to have a stable and precise vocabulary such as
+in legal documents. On the other hand, SD is driven by semantics and may be
+more appropriate for linking topics that have a greater lexical variability. Greater
+lexical variability may be the result of lexical drift over time as terms change
+in popularity or informal texts which do not use a standard vocabulary such as
+tweets. Hence, we believe that SD not only competes but complements JS for
+topic tracking.
+5
+Conclusion
+In this paper, we presented a novel semantic-based topic tracking method (SD).
+We showed that its performance was comparable to that of the state of the
+art method (JS), which is lexically-based. This validates our hypothesis that
+semantic information is valuable for tracking topics.
+Moreover, we have discussed the challenges associated with tracking topics
+in a topic hierarchy. First, topics and their sub-topic can be difficult to differ-
+entiate, which makes topic tracking more challenging. Second, deeper topics are
+more esoteric and consequently it is harder to assess the quality of their track-
+ing. Finally, topic hierarchy may have many equally correct arrangements which
+makes it difficult to leverage structural information for topic tracking.
+
+2020 topics
+2019 topic
+2020 topics
+security
+security
+issue
+hacker
+account
+datum
+attack
+password
+service
+system
+email
+account
+vulnerability
+SD
+attack
+JS-
+app
+@Garmin
+@Equifax
+@Apple
+@Security
+@Marriott
+@iPhone
+@Signal
+@Yahoo
+@Google
+@Forbe
+@NSA
+@Epic
+@ZDNet
+@AWS
+@AppStore8
+J. POUMAY A. ITTOO
+We believe that our work would benefit future studies investigating hybrid
+methods for topic tracking, such as by integrating lexical and semantic informa-
+tion.
+References
+1. Agrawal, A., Fu, W., Menzies, T.: What is wrong with topic modeling? and how
+to fix it using search-based software engineering. Information and Software Tech-
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+and tracking pilot study final report (1998)
+3. AlSumait, L., Barbar´a, D., Domeniconi, C.: On-line lda: Adaptive topic models
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+pii/S0167923619301034
+9. Liu, H., Chen, Z., Tang, J., Zhou, Y., Liu, S.: Mapping the technology evolu-
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+(MSOTA 2016). pp. 49–54 (2016)
+

HdAyT4oBgHgl3EQfrvlK/content/tmp_files/load_file.txt

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+page_content='Using meaning instead of words to track topics  Judicael POUMAY1 and Ashwin ITTOO1  ULiege/HEC Liege, Rue Louvrex 14, 4000 Liege, Belgium {judicael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='poumay,  ashwin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='ittoo}@uliege.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='be  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' The ability to monitor the evolution of topics over time is  extremely valuable for businesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Currently, all existing topic tracking  methods use lexical information by matching word usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' However, no  studies has ever experimented with the use of semantic information for  tracking topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Hence, we explore a novel semantic-based method using  word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Our results show that a semantic-based approach to  topic tracking is on par with the lexical approach but makes different  mistakes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' This suggest that both methods may complement each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Keywords: Topic tracking · lexical · semantic · topic models  1  Introduction  Buried within the voluminous amounts of texts available online are meaning-  ful insights, which could help in supporting business decision-making activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Topic modelling methods extracts latent topic in a corpus [4,10] and can be used  to discover these insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Examples of applications include fraud detection [11],  understanding employee and customer satisfaction [8,7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Extracted topics can be  tracked over time to understand their evolution or discover emerging one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Hence,  we focus on this task of topic tracking in which the goal is to link instances of  the same topic that have been extracted at different time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Several methods for tracking topics have been proposed in the past [3,6,13,12,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  These methods use measures such as the JS divergence [13,12,9] or online topic  models [3,6] which rely on lexical information to track topic across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  However, no studies has ever experimented with using semantic informa-  tion to track topics over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Intuitively, semantic based approaches could be  promising as they do not rely on simple surface form and can capture concepts  such as synonymy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' For example, given a topic about ”AI”, across time we could  observe that the term ”Machine Learning” has become more popular than ”AI”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  However, a lexical approach to topic tracking would not be able to handle such  lexical drift and to relate those words over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Conversely, such lexical vari-  ation would have been captured by a semantic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Moreover, topic-word  distributions are unstable across multiple runs [1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' the resulting top words  of a topic tend to change significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' This entails that the lexical information  we rely upon to track topics is also unstable even if the overall semantic of the  topic remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Thus, a semantic-based approach may be more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='00565v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='CL]  2 Jan 2023  2  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' POUMAY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' ITTOO  Hence, our work aims at investigating on the use of semantic information for  topic tracking and its comparison against lexical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Therefore, as our  main contribution, we propose a novel semantic topic tracking method known  as Semantic Divergence (SD) based on word embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' As an ancillary con-  tribution, we study the challenges of topic tracking in the context of hierarchical  topic modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  2  Background and Related work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='1  Topic Modelling  LDA [4] is the first traditional topic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' At the core of LDA is a Bayesian  generative model with two Dirichlet distributions, respectively for the document-  topic distributions and for the topic-word distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' These distributions are  learnt and optimized via an inference procedure which enables topics to be ex-  tracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' The main weakness of LDA is that it requires the user to specify a  predefined number of topics to be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  More complex topic models have been proposed since LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' In particular,  HTMOT [10] was proposed to simultaneously model topic hierarchy and tem-  porality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Specifically, HTMOT produces a topic tree in which the depth and the  number of sub-topic for each branch is defined dynamically during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Ad-  ditionally, HTMOT models the temporality of topics enabling the extraction of  topics that are lexically close but temporally distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='2  Topic Tracking  Topic tracking is the task of monitoring the evolution of topics through time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  It was initially defined in a pilot study [2] in 1998 as the continuous automatic  classification of a stream of news stories into known or new topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Currently, two general framework compete for topic tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' The first stream  is that of online topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' which incorporate new data incrementally [3,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' In  [3], the authors propose Online-LDA, a version of LDA able to update itself with  new documents without having to access to previously processed documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' In  practice, Online-LDA assumes that time is divided in slices and at each slice  an LDA model is trained using the previous slice as prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' They were able to  show that their system can find emerging topics by artificially injecting new  topic into the news stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' They performed their experiments on the NIPS  and Reuters-21578 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Similarly in [6], the authors propose a model that  can dynamically decide the right number of topics in an online fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' They  performed their experiments on the the 20 Newsgroup and the TDT-2 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  The second stream is concerned with linking topics extracted independently  at different time periods [13,12,9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' In [13], the authors use about 30,000 abstracts  of papers in various journals from 2000 to 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' They then applied LDA to each  year independently and linked topics using the Jensen-Shannon Divergence (JS)  to measure their similarity [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' In [12] the authors applied a similar method on  Using meaning instead of words to track topics  3  news articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' However, they differ in that while [13] simply links topics together,  [12] clusters them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' This means that once two topic have been linked they form  a cluster and subsequent topics will be compared to the whole cluster and not  just the preceding topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Finally in [9], the authors also proposed a tracking  method using the JS divergence applied to scientific papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' However, they do  not constraint linkage to a one-to-one mapping which allows for the fusion and  splitting of topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' All of the aforementioned paper evaluated their topic tracking  method using a qualitative analysis that demonstrated the performance of their  technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  We based our work on that second stream because it allows for better paral-  lelization as time slices are processed independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  3  Methodology  In this section, we will present our methodology for topic tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' We will start  by describing our corpus and topic extraction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Next, we will define our  SD measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Finally, we will present the topic tracking algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='1  Topic extraction  To perform our experiments, we crawled 10k articles from the Digital Trends 1  archives from 2019 to 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' This news website is mainly focused on technological  news with topics such as hardware, space exploration and COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' For all  articles, we extracted the text, title, category and timestamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' We pre-possessed  the corpus according to HTMOT [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  To extract topics hierarchies (see figure 1), we used the HTMOT topic model  [10] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' The extracted topics are represented by a list of words and a list of entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Example of a topic hierarchy  We follow HTMOT [10] and only focus on the first and second level of topic  extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Specifically, the authors observe that deeper topics becomes more  esoteric making them harder to understand by annotators representing a general  audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Consequently, this makes it difficult to assess the correctness of tracked  topics at deeper levels of the topic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  1 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='digitaltrends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='com/  ROOT  Space  COVID-19  Astronauts  Astronomy  Vaccines  Tests4  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' POUMAY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' ITTOO  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='2  Proposed Semantic Divergence measure  We will now describe our novel topic tracking method, which departs from the  JS divergence traditionally applied in previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' We name our method  ”Semantic divergence” or SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' It uses word embeddings to measure the distance  between topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Each topic will be assigned an embedding as the sum of the  embeddings of the top words in that topic weighted by their probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Then,  the distance between two topics is computed as the cosine distance of their  respective embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' We will use FastText as the word embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' FastText  helps with rare and out of vocabulary words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' This is essential considering our pre-  processing step includes lemmatization which may produce incorrectly spelled  words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Hence the embedding of a topic is defined as follows :  emb(t) =  �  (w,p)∈t  p ∗ FastText(w)  (1)  And the Semantic Divergence between two topics is defined as :  SD(t1, t2) = cosine(emb(t1), emb(t2))  (2)  Where w is a word in a topic t and p is the probability of that word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='3  Topic Tracking Algorithm  Finally, to track topics across time we applied HTMOT on our corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' For  each year (2019 and 2020), we obtained a corresponding topic tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Then, we  computed the distance between every topics across both years using either JS  or SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' To do this we used the top 100 words and top 15 entities to represent  each topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Subsequently, we ranked order all computed pairs of topics and then  iteratively selected the most similar pairs (lowest SD or JS score) such that each  topic is paired only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Finally, we used a pre-defined threshold to remove  pairs with a poor score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Note that our approach does not take into account structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Indeed, tracking topics in the context of hierarchical topic modelling presents  another interesting challenge : there exist many possible resulting trees that are  equally correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' In one run, we may extract the topic of space whose sub-topics  can be grouped into space exploration and astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Conversely, in another  run, we may extract space exploration and astronomy as separate topics with  their own sub-topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Hence, it is difficult to leverage the structural information  contained in the topic trees to track topics as it cannot be expected to respect  a specific conceptual taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  4  Results : JS vs SD  In this section, we will discuss how our semantic based method compares with  respect to the traditional lexical based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Using meaning instead of words to track topics  5  First, we studied the overlap between the two methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' the number of  pairs extracted by both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' We discovered that, 111 pairs were extracted with JS  with a threshold of <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='4, while 121 pairs were extracted with SD with a threshold  of <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' These threshold were set through empirical observation but may depend  on the dataset used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' These 111-121 pairs can be grouped into three categories  (see figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' 72 pairs were the same between the two methods (60-65% of the  total pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' For example, topics such as space and video games were easily paired  across both years by both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' This already indicates that our SD method  is able to pair topic across time with performance similar to JS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' This leaves  39-49 pairs that are different across the two methods (35-40% of the total pairs)  which we can evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Out of those different pairs, we notice that in most cases  one method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' SD) would track/link a topic pair across both years, while the  other method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' JS) did not as the best possible pair was above the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  We are then left with 10 different pairs that can themselves be paired according  to which 2019 or 2020 topic they share (see figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' The pairs extracted by both methods can be grouped into three categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' The  circle represent topics and their color represent years (2019 blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' 2020 yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' The link  color represent the method used (JS red;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' SD green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' The three categories are : 1) The  pairs extracted by both methods (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' 2) The pairs that differ but share a topic (10)  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' JS extracted the pair 4-D while SD extracted 4-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' 3) The pairs of topics that were  only linked with one method (29-39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  To compare the performance of the two tracking methods, we decided to  use a survey comparing these 10 pairs of topics extracted by both JS and SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Precisely, for each question, given an initial topic, annotators were shown the JS  and SD pairing and asked which is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Additionally, we also asked annotators  to provide a confidence score on a scale from 1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' In total, we received 38  answers coming from a small online community focused on answering surveys2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  The survey can be found on github 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Looking at the survey results (table 1), it can be seen that SD slightly outper-  forms JS with 54% of annotators preferring the former to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Moreover,  we also note that the annotators were confident in their evaluation, with an  average confidence score of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Interestingly, there is a lot of variability in the  2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='reddit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='com/r/SampleSize/  3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='com/JudicaelPoumay/TopicTrackingPaper  72 Common pairs  10 Pairs sharing a topic  29-39 Pairs only linked with one method  39-49 Pairs that differ between the two methods6  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' POUMAY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' ITTOO  answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Some topics were clearly better paired with one method or the other  (Q3 and Q5) while for others, it wasn’t as clear (Q1, Q2 and Q4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' The ”chose SD” column corresponds to the % of annotators that chose the  SD pair as the best pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Questions Chose SD Confidence level  Q1  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='1% (22) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='2  Q2  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='2% (14) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='6  Q3  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='1% (30) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='7  Q4  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='8% (13) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='5  Q5  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='9% (8)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='5  Average  54%  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='3  For example, figure 3 corresponds to Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' It shows how a 2019 topic has been  paired with 2020 topics using JS and SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' First, we can notice that the distance  recorded between the pairs is close to the threshold for both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Specifi-  cally, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='29 for the JS pair and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='09 for the SD pair (threshold = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='4 for JS and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='1 for SD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' This makes sense as good pairs (pairs with low JS/SD values) are  extracted by both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Second, the 2019 topic is about social media data  security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Whereas the chosen 2020 topic is about :  – Social media when paired with JS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  – Data security when paired with SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Hence, both pairing seems suitable, which could explain the indecisiveness of  annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Specifically, 16 of them decided the SD pairing was better whereas  22 of them decided the JS pairing was better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Their confidence level for this  question was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='2 out of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' A first example of different pairing between SD and JS on the same 2019 topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Similarly, figure 4 corresponds to Q5 and shows how another 2019 topic has  been paired based on the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Here, the 2019 topic is about web security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Whereas the chosen 2020 topic is about :  2020 topics  2019 topic  2020 topics  issue  company  company  datum  issue  social  service  datum  platform  account  account  app  app  SD  user  JS-  ban  @Apple  @Facebook  @Facebook  @iPhone  @U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  @TikTok  @Google  @Twitter  @U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  @Epic  @instagram  @Twitter  @AppStore  @FTC  @TrumpUsing meaning instead of words to track topics  7  – Data security when paired with JS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  – Web security topic when paired with SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Moreover, the topic chosen by SD is a sub-topic of the topic chosen by JS which  demonstrates the difficulty in topic tracking in a hierarchical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Indeed,  it can be difficult to differentiate a topic from its sub-topic, especially if that  sub-topic dominates the others as parent topics are the sum of their sub-topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  In this case, annotators agreed more and 30 out of 38 decided the SD pair was  better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Their confidence level for this question was 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='5 out of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' A second example of different pairing between SD and JS on the same 2019  topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Hence, we argue that JS and SD are two fundamentally different approaches  and that both have their advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' JS is lexically driven and may work best  for linking topics which tend to have a stable and precise vocabulary such as  in legal documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' On the other hand, SD is driven by semantics and may be  more appropriate for linking topics that have a greater lexical variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Greater  lexical variability may be the result of lexical drift over time as terms change  in popularity or informal texts which do not use a standard vocabulary such as  tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Hence, we believe that SD not only competes but complements JS for  topic tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  5  Conclusion  In this paper, we presented a novel semantic-based topic tracking method (SD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  We showed that its performance was comparable to that of the state of the  art method (JS), which is lexically-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' This validates our hypothesis that  semantic information is valuable for tracking topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  Moreover, we have discussed the challenges associated with tracking topics  in a topic hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' First, topics and their sub-topic can be difficult to differ-  entiate, which makes topic tracking more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Second, deeper topics are  more esoteric and consequently it is harder to assess the quality of their track-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' Finally, topic hierarchy may have many equally correct arrangements which  makes it difficult to leverage structural information for topic tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content='  2020 topics  2019 topic  2020 topics  security  security  issue  hacker  account  datum  attack  password  service  system  email  account  vulnerability  SD  attack  JS-  app  @Garmin  @Equifax  @Apple  @Security  @Marriott  @iPhone  @Signal  @Yahoo  @Google  @Forbe  @NSA  @Epic  @ZDNet  @AWS  @AppStore8  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' POUMAY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
+page_content=' ITTOO  We believe that our work would benefit future studies investigating hybrid  methods for topic tracking, such as by integrating lexical and semantic informa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdAyT4oBgHgl3EQfrvlK/content/2301.00565v1.pdf'}
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+Stream-K: Work-centric Parallel Decomposition for
+Dense Matrix-Matrix Multiplication on the GPU
+Muhammad Osama
+[email protected]
+University of California, Davis
+Davis, California, USA
+Duane Merrill
+[email protected]
+NVIDIA Corporation
+Santa Clara, California, USA
+Cris Cecka
+[email protected]
+NVIDIA Corporation
+Santa Clara, California, USA
+Michael Garland
+[email protected]
+NVIDIA Corporation
+Santa Clara, California, USA
+John D. Owens
+[email protected]
+University of California, Davis
+Davis, California, USA
+Abstract
+We introduce Stream-K, a work-centric parallelization of
+matrix multiplication (GEMM) and related computations in
+dense linear algebra. Whereas contemporary decompositions
+are primarily tile-based, our method operates by partitioning
+an even share of the aggregate inner loop iterations among
+physical processing elements. This provides a near-perfect
+utilization of computing resources, regardless of how effi-
+ciently the output tiling for any given problem quantizes
+across the underlying processing elements.
+On GPU processors, our Stream-K parallelization of GEMM
+produces a peak speedup of up to 14× and 6.7×, and an av-
+erage performance response that is both higher and more
+consistent across 32,824 GEMM problem geometries than
+state-of-the-art math libraries such as CUTLASS and cuBLAS.
+Furthermore, we achieve this performance from a single tile
+size configuration per floating-point precision, whereas to-
+day’s math libraries employ complex kernel-selection heuris-
+tics to select from a large ensemble of kernel variants.
+Keywords: Matrix-Multiplication, GPU, Load-Balancing
+1
+Introduction
+General matrix-matrix product (GEMM), convolution, and
+other similar computations constitute the dominant work-
+loads in many deep learning and scientific computing ap-
+plications. High-performance processors such as GPUs, for
+example, are designed to achieve nearly 100% of their theoret-
+ical peak math throughput when computing GEMM. Doing
+so, however, requires a work decomposition that perfectly
+occupies the underlying physical cores. As we show, attain-
+ing such high levels of processor utilization across a broad
+landscape of problems shapes and sizes can be challenging.
+Classically, GEMM implementations block their compu-
+tation using a data-parallel tiling of the output matrix, as-
+signing the independent production of output tiles among
+concurrent threads (or thread groups) [1, 8, 14]. The work per
+Distribution Statement “A” (Approved for Public Release, Distribution Un-
+limited).
+output tile is regular, and tile production tends to dispatch
+across idle physical cores in “waves”. The overall workload
+is well-balanced and processor utilization is highest when
+there are many waves, i.e., the number of output tiles greatly
+oversubscribes the number of cores.
+However, such oversubscription has shrunk considerably
+as processors have grown in size. An increased core count
+will require fewer waves to produce a given tile count. Big-
+ger cores will compel larger matrix blocking factors, leading
+to fewer waves of larger tiles. In general, execution sched-
+ules with fewer waves are much more likely to suffer from
+quantization inefficiency, i.e., the processor underutilization
+that occurs when the number of output tiles is not an even
+multiple of the number of processor cores. When the last
+wave is partially full, the unused cores must wait for the
+remaining threads to execute millions (if not billions) of
+multiply-accumulate (MAC) instructions before they are able
+to execute any dependent work.
+Figure 1a illustrates such a scenario on a hypothetical GPU
+with four streaming multiprocessor cores (SMs). If we block a
+384×384×128 GEMM computation into nine 128×128 output
+tiles, a data-parallel decomposition cannot achieve more than
+75% of the processor’s rated throughput. This theoretical
+utilization ceiling can be improved to 90% by halving the
+tile size as shown in Figure 1b. However, the finer-grained
+blocking factor will be less cache and scratchpad efficient,
+and may preclude any practical performance improvement.
+Quantization inefficiency is a concern for increasingly
+wide processors such as GPUs, where ALUs-per-core and
+cores-per-processor both currently number in the hundreds.
+Consequently, many common GEMM-like workloads now
+exhibit a final, partially full wave that comprises a significant
+fraction of the total computation time.
+The current remedy employed by GPU-based math and
+deep learning libraries is to deploy an ensemble of tiling con-
+figurations. When the ideal blocking factor does not quan-
+tize well, the library chooses among tiling alternatives with
+smaller concurrent work volumes, such as those illustrated
+in Figure 1b and Figure 2a.
+arXiv:2301.03598v1  [cs.DS]  9 Jan 2023
+
+(a) Data parallel decomposition with grid size g=9 CTAs,
+large 128 × 128 × 128 CTA work volumes,
+and 75% processor utilization ceiling
+(b) Data parallel decomposition with grid size g=18 CTAs,
+smaller 128 × 64 × 128 CTA work volumes,
+and 90% processor utilization ceiling
+Figure 1. Data-parallel execution schedules for 384 × 384 × 128 GEMM across a hypothetical four-SM GPU.
+(a) Fixed-split decomposition with splitting factor s=2,
+grid size g=18 CTAs, smaller 128 × 128 × 64 CTA work volumes,
+and 90% quantization efficiency
+(b) Basic Stream-K decomposition with grid size g=4 CTAs,
+larger 128 × 128 × 288 CTA work volumes,
+and nearly 100% quantization efficiency
+Figure 2. Tile-splitting execution schedules for 384 × 384 × 128 GEMM across a hypothetical four-SM GPU.
+Tile-based ensembles, however, present performance and
+logistical challenges for math libraries seeking to deliver the
+best-achievable performance across diverse problem sizes
+and shapes. Distributable code size can be problematic for
+large ensembles. For example, NVIDIA’s cuBLAS library [15]
+is hundreds of megabytes, often providing more than twenty
+pre-compiled kernel specializations per architecture for a
+given API entry point. Large ensembles also require sophis-
+ticated selection heuristics. In our evaluation, we show these
+heuristics can struggle to consistently identify the optimal
+configuration for arbitrary problems.
+Unlike these tile-based methods, our Stream-K decompo-
+sition always distributes an even share (within one) of the
+aggregate multiply-accumulate loop iterations required by
+the GEMM computation across SMs. Because the instruction
+workload of a single MAC-loop iteration is far smaller than
+that of an entire output tile, any variance in core workload is
+practically negligible. Stream-K uses the ideal blocking factor
+regardless of problem shape, has communication overheads
+that scale with processor width (rather than output tiles),
+and compiles to a single kernel.
+We use an enormous corpus of 32,824 GEMM shapes and
+sizes to evaluate Stream-K, which we implemented within
+NVIDIA’s CUTLASS library [8]. In comparison with CUT-
+LASS’s data-parallel implementation of the same blocking
+factor, Stream-K provides a substantially higher performance
+2
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+2
+4
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+8
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+B
+个
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+1
+-
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+-
+-
+CTA- 1
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+14
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+-
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+个
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+8
+SM3
+7
+8
+6
+CTA3
+to
+(time)response across our landscape of GEMM problems, demon-
+strating up to 14× speedup on NVIDIA A100 GPUs.
+To highlight the practical challenges of ensemble-based so-
+lutions, we also evaluate NVIDIA’s cuBLAS library as well as
+an oracle-driven ensemble of data-parallel CUTLASS tilings.
+Relative to both ensembles, we show that our single-kernel
+Stream-K achieves both (1) higher average performance,
+and (2) higher performance consistency. Versus cuBLAS,
+Stream-K demonstrates up to 6.7× speedup and virtually no
+instances of slowdown for compute-bound problems.
+2
+Background
+General Matrix Multiplication (GEMM) is defined as the
+product C = 𝛼AB + 𝛽C where 𝛼 and 𝛽 are scalar values
+and A, B, and C are matrices. (For simplicity, we assume
+𝛼 = 1, 𝛽 = 0 throughout this paper.) We refer to the shape
+of a given GEMM problem by the volumetric extents of its
+computation. For example, a m×n×k GEMM consumes m×k
+and k × n input matrices A and B, respectively, performs
+m × n × k multiply-accumulate operations, and produces an
+m × n output matrix C.
+GEMM is a performance-critical subroutine in many large-
+scale engineering and scientific applications. It plays an im-
+portant role in matrix factorization methods such as LU, QR,
+and Cholesky decomposition. High-performance modeling
+and simulation applications in engineering, climate simu-
+lation, cosmology, quantum chemistry, and other scientific
+domains rely on these factorization methods.
+Matrix multiplication is also the fundamental building
+block of modern deep learning (DL) methods. The training
+of deep neural networks (DNNs) is often performed on mas-
+sive datasets across large distributed systems [13]. Many DL
+training and inference operations are cast as matrix multi-
+plications. For example, image recognition and computer vi-
+sion models rely on convolution, which can be implemented
+directly as the product of filter and image datasets [4]. Trans-
+former architectures, which have come to dominate natural
+language processing and other applications, are almost en-
+tirely limited by the performance of large matrix products.
+Early work on GPU matrix-matrix multiplication from
+Larsen and McAllister framed the computation as a multi-
+texture multiplication and blending operation [11]. The user-
+programmable shared memory provided by subsequent GPU
+architectures enabled higher-performing data parallel schemes
+with two levels of blocking (shared memory and registers)
+with tile sizes informed via extensive micro-benchmarking
+analysis [2, 14, 17, 19] and auto-tuning [5, 7, 12].
+The MAGMA GPU math library was perhaps the first to
+optimize for diverse GEMM problem shapes [9]. Their solu-
+tion applied a constrained set of tiling parameters to a tem-
+plated CUDA C++ code stencil, generating several hundred
+data-parallel variants per API primitive (e.g., hgemm_tt() for
+half-precision transpose-transpose GEMM). They evaluated
+these variants to distill a small ensemble of typically three
+to five kernels that collectively perform well across a diver-
+sity of problem shapes. Kernel selection and dispatch for a
+given problem was governed by size thresholds expressed
+via simple handwritten rules.
+Subsequent GPU math libraries have employed more so-
+phisticated code-generation and kernel-selection compo-
+nents. For example, the ISAAC project uses machine learning
+techniques to predict an optimal tiling and/or splitting pa-
+rameterization for a given GEMM shape, which can then
+be instantiated either online or offline via a PTX-level code
+generator [18].
+NVIDIA’s cuBLAS [15] library has provided an extended
+cublasGemmEx interface that allows the caller to select from
+among 24 different GEMM “algorithms”. Carefully trained
+heuristics choose between this large space of alternatives
+when using the default interface. These algorithms imple-
+ment a variety of different data-parallel and fixed-split vari-
+ants, and it is common for cuBLAS to have assembled each
+variant into its own architecture-specific kernel program for
+code optimization purposes. The cross product of GEMM
+API functionality, strategic variants, and microarchitecture
+has resulted in distributions that are increasingly enormous,
+exceeding hundreds of megabytes of executable code.
+Given the fast-paced and rapidly changing nature of con-
+temporary deep learning, recent work has focused on pro-
+gramming models for simplifying the expression and con-
+struction high performance kernels that alter or supplement
+the GEMM computation. The CUTLASS C++ library pro-
+vides data-movement and multiply-accumulation classes for
+composing custom GEMM-like computations at all levels
+of the GPU thread hierarchy [8]. Triton [19] is a domain-
+specific language for tensor programming centered on the
+expression, transformation, and optimization of block/tile
+concepts. Other domain-specific programming languages
+such as Halide [16] and TVM [3] separate the expression of
+pointwise operators from that of loop scheduling. Fireiron [6]
+further adds data movement constructs into the scheduling
+grammar.
+3
+Existing Work Decomposition Strategies
+Modern processors typically store A, B, and C in a large, slow,
+distant memory and have access to a small, fast, scratchpad
+or cache memory. A primary goal for any GEMM implemen-
+tation is to leverage these local storage resources so that the
+resulting implementation is computation-bound.
+3.1
+Sequential Cache-Blocked
+The classic cache-blocked formulation of GEMM divides its
+computational volume into blocks and chooses a traversal
+order that exposes memory locality. Algorithm 1 presents a
+simplified implementation comprising six loops. The inner-
+most three loops iterate within the blocking factors BLK_M,
+3
+
+BLK_N, and BLK_K, while the outermost three iterate across
+them. If the cache can capture one block from each of the
+three matrices, the resulting data reuse among those ele-
+ments will significantly reduce the number of last-level mem-
+ory accesses [10].
+Algorithm 1 Sequential cache-blocked GEMM.
+1: ▷ tile-processing outer loops
+2: for mm ← 0 to m step BLK_M do
+3:
+for nn ← 0 to n step BLK_N do
+4:
+▷ zero-initialize output tile
+5:
+for mmm ← mm to (mm + BLK_M) do
+6:
+for nnn ← nn to (nn + BLK_N) do
+7:
+C[mmm,nnn] ← 0
+8:
+end for
+9:
+end for
+10:
+▷ perform the MAC iterations for this tile
+11:
+for kk ← 0 to k step BLK_K do
+12:
+▷ MAC iteration (fully unrolled)
+13:
+for mmm ← mm to (mm + BLK_M) do
+14:
+for nnn ← nn to (nn + BLK_N) do
+15:
+for kkk ← kk to (kk + BLK_K) do
+16:
+C[mmm,nnn] ← C[mmm, nnn] +
+17:
+(A[mmm,kkk] × B[kkk,nnn])
+18:
+end for
+19:
+end for
+20:
+end for
+21:
+end for
+22:
+end for
+23: end for
+3.2
+Data-parallel
+As shown in Algorithm 2, the data-parallel GPU formulation
+of GEMM is decomposed across a grid of parallel thread
+blocks, or cooperative thread arrays (CTAs)1. The grid is sized
+such that each CTA produces its own (BLK_M × BLK_N)
+output tile.
+For exposition, the MacLoop() subroutine of Algorithm 3
+encapsulates the multiply-accumulate workloads that com-
+pute the values of the CTA’s output tile. It performs a se-
+quence of MAC-loop iterations in the accumulation domain,
+e.g., the k-axis for GEMM. Each MAC-loop iteration com-
+prises a per-thread volume of (BLK_M × BLK_N × BLK_K) /
+CTA_THREADS MAC operations. As the computation pro-
+ceeds, fragments of the input matrices are staged through
+the SM’s shared memory for local reuse among individual
+threads.
+Although this particular presentation of MacLoop() de-
+ploys one thread per output tile element, the sophisticated
+implementations in CUTLASS [8] and cuBLAS [8] will: (1)
+fully unroll the per-thread MAC-loop iteration; (2) imple-
+ment additional blocking at the warp and/or thread levels;
+and (3) orchestrate a software pipeline of shared memory
+data movement across MAC-loop iterations.
+1Blocks of GPU threads are coscheduled in CTAs, which virtualize the
+hardware’s streaming multiprocessor cores (SMs).
+Unfortunately, this classic data-parallel decomposition is
+liable to suffer from quantization inefficiency on modern
+GPUs, as illustrated in Figure 1. Although an ensemble of di-
+verse blocking factors may uncover opportunities for greater
+processor utilization, it is unlikely to facilitate perfect quan-
+tizations for arbitrary problem sizes. Furthermore, smaller
+blocking factors have two drawbacks: (1) fewer instructions
+per MAC-loop iteration for covering the latencies of global
+and shared memory transfers in pipelined implementations;
+and (2) a higher proportion of memory operations relative
+to MAC instructions, which may prevent them from being
+computation-bound.
+Algorithm 2 Data-parallel GPU GEMM.
+1: _shared_ accum[BLK_M,BLK_N]
+2: iters_per_tile ← ⌈k/BLK_K⌉
+3: ▷ instantiate one CTA per output tile
+4: fork CTA[x] in [ ⌈m/BLK_M⌉ × ⌈n/BLK_N⌉ ] do
+5:
+▷ perform the MAC iterations for this tile
+6:
+accum ← MacLoop(x, 0, iters_per_tile)
+7:
+▷ store accumulators to output tile
+8:
+StoreTile(C, x, accum)
+9: join
+Algorithm 3 CTA-wide MacLoop() subroutine for perform-
+ing a sequence of MAC-loop iterations.
+1: procedure MacLoop(tile_idx, iter_begin, iter_end)
+2:
+_shared_ accum[BLK_M,BLK_N]
+3:
+_shared_ frag_a[BLK_M,BLK_K]
+4:
+_shared_ frag_b[BLK_K,BLK_N]
+5:
+▷ determine output tile coordinates
+6:
+mm ← BLK_M × (tile_idx / ⌈m/BLK_M⌉)
+7:
+nn ← BLK_N × (tile_idx % ⌈m/BLK_M⌉)
+8:
+▷ zero-initialize local accumulator storage
+9:
+accum ← 0
+10:
+▷ perform the specified range of MAC iters for this tile
+11:
+for iter ← iter_begin to iter_end do
+12:
+kk ← iter × BLK_K
+13:
+▷ copy global matrix fragments to local storage
+14:
+frag_a ← LoadFragment(A, mm, kk)
+15:
+frag_b ← LoadFragment(B, kk, nn)
+16:
+fork THREAD[mmm,nnn] in [BLK_M, BLK_N] do
+17:
+▷ MAC iteration per thread (fully unrolled)
+18:
+for kkk ← 0 to BLK_K do
+19:
+accum[mmm, nnn] ← accum[mmm,nnn] +
+20:
+(frag_a[mmm,kkk] × frag_b[kkk,nnn])
+21:
+end for
+22:
+join
+23:
+end for
+24:
+return accum
+25: end procedure
+3.3
+Fixed-split
+Alternatively, the granularity of work assigned to each CTA
+can be reduced via parallelization across the accumulation
+dimension. For a given output tile, the associativity of addi-
+tion allows the iteration domain to be split among multiple
+4
+
+concurrent CTAs, followed by a dependent “fixup” step to
+reduce the partial sums computed by each CTA. We high-
+light this fixed-split approach in Algorithm 4, where each
+output tile is cooperatively produced by s CTAs. Notably,
+it functions identically to the data-parallel decomposition
+when the splitting factor s = 1.
+The fixed-split decomposition is also featured in CUTLASS
+and cuBLAS. The splitting factor is implemented as a runtime
+parameter, allowing a single kernel executable to support
+multiple work volumes while retaining the ideal blocking
+factors for optimal data sharing and latency hiding. How-
+ever, as illustrated in Figure 2a, the prospect of achieving a
+perfect quantization from a uniform tile-splitting is unlikely.
+Furthermore, the extra overheads of communication and
+synchronization scale with both the overall problem size as
+well as the splitting factor.
+Algorithm 4 Fixed-split GPU GEMM with splitting factor s.
+1: _shared_ accum[BLK_M,BLK_N]
+2: iters_per_tile ← ⌈k/BLK_K⌉
+3: iters_per_split ← ⌈iters_per_tile/s⌉
+4: ▷ instantiate s CTAs per output tile
+5: fork CTA[x,y] in [ ⌈m/BLK_M⌉ × ⌈n/BLK_N⌉, s] do
+6:
+▷ perform the range of MAC iterations for this split
+7:
+iter ← y × iters_per_split
+8:
+iter_end ← min(iters_per_tile, iter + iters_per_split)
+9:
+accum ← MacLoop(x, iter, iter_end)
+10:
+▷ consolidate partial-sums across CTAs
+11:
+if y ≠ 0 then
+12:
+▷ store accumulators to temporary global storage
+13:
+StorePartials(partials[x,y], accum)
+14:
+Signal(flags[x,y])
+15:
+else
+16:
+�� accumulate partial sums from other CTAs contributing to this
+tile
+17:
+for cta ← 1 to s do
+18:
+Wait(flags[x,cta])
+19:
+accum ← accum + LoadPartials(partials[x,cta])
+20:
+end for
+21:
+▷ store accumulators to output tile
+22:
+StoreTile(C, tile_id, accum)
+23:
+end if
+24: join
+4
+Our Stream-K Decomposition
+Our Stream-K decomposition is a tile-splitting parallelization
+in which the splitting seams are completely dissociated from
+the tiling structure itself. Although we employ familiar block-
+ing and tiling strategies for data reuse, we instead quantize
+the GEMM computation into MAC-loop iterations, i.e., small
+volumes of CTA-wide BLK_M × BLK_N × BLK_K work. As
+presented in Algorithm 5, Stream-K evenly partitions the
+GEMM’s aggregate workload of MAC-loop iterations across
+a constant-sized grid of g CTAs. Each CTA’s range of MAC-
+loop iterations is mapped contiguously into the m → n → k
+linearization of the GEMM shape, crossing output-tile bound-
+aries as it may.
+Algorithm 5 Basic Stream-K GPU GEMM with grid size g.
+1: _shared_ accum[BLK_M,BLK_N]
+2: iters_per_tile ← ⌈k/BLK_K⌉
+3: total_iters ← ⌈m/BLK_M⌉ × ⌈n/ BLK_N⌉ × iters_per_tile
+4: iters_per_cta ← ⌈total_iters / g⌉
+5: ▷ instantiate g CTAs
+6: fork CTA[x] in [g] do
+7:
+iter ← x × iters_per_cta
+8:
+iter_end ← iter + iters_per_cta
+9:
+▷ iteration-processing outer loop
+10:
+while iter < iter_end do
+11:
+tile_idx ← iter / iters_per_tile
+12:
+tile_iter ← tile_idx × iters_per_tile
+13:
+tile_iter_end ← tile_iter + iters_per_tile
+14:
+▷ perform the range of MAC iterations for this tile
+15:
+local_iter ← iter - tile_iter
+16:
+local_iter_end ←
+17:
+min(iter_end, tile_iter_end) - tile_iter
+18:
+accum ←
+19:
+MacLoop(tile_id, local_iter, local_iter_end)
+20:
+▷ consolidate partial-sums across CTAs
+21:
+tile_started ← iter = tile_iter
+22:
+tile_ended ← (iter_end ≥ tile_iter_end)
+23:
+if ¬tile_started then
+24:
+▷ store accum to temporary global storage
+25:
+StorePartials(partials[x], accum)
+26:
+Signal(flags[x])
+27:
+else
+28:
+▷ store accumulators to output tile
+29:
+if ¬tile_ended then
+30:
+▷ accumulate partial sums from other CTA contributing
+to this tile
+31:
+cta_end ← tile_iter_end / iters_per_tile
+32:
+for cta ← (x+1) in cta_end do
+33:
+Wait(flags[cta])
+34:
+accum ← accum
+35:
++ LoadPartials(partials[cta])
+36:
+end for
+37:
+end if
+38:
+StoreTile(C, tile_id, accum)
+39:
+end if
+40:
+iter ← tile_iter_end
+41:
+end while
+42: join
+Should a given CTA’s starting and/or ending iterations
+not coincide with tile boundaries (as is expected to be the
+common case), it must consolidate its partial results with
+those of the other CTA(s) also covering that tile. In this
+basic implementation, each output tile in C is written by the
+CTA that performed that tile’s k = 0 MAC-loop iteration.
+Before it can do so, however, it must accumulate any partial
+sums shared from other CTAs in temporary global storage.
+Notably, Stream-K’s communication, synchronization, and
+global storage overheads are independent of problem size,
+scaling instead with the number of CTAs g.
+A secondary benefit of Stream-K is that synchronization-
+waiting is likely negligible when the number of output tiles is
+greater than the number of CTAs. In this regime, each output
+tile is covered by at most two CTAs, and the tile-processing
+5
+
+skew ensures that the accumulating CTA will not need its
+peer contributions until well after those collaborators have
+finished producing them.
+Continuing our earlier example, Figure 2b illustrates the
+basic Stream-K execution schedule of the 384 × 384 × 128
+GEMM problem on a hypothetical four-SM GPU. To fully
+occupy the GPU, we launch g = 4 CTAs. Assuming BLK_M =
+128, BLK_N = 128, and BLK_K = 4, each CTA is tasked with
+a 128 × 128 × 288 work volume comprising 72 MAC-loop
+iterations. This results in a 100% quantization efficiency, as all
+four SMs will execute the same number of MAC instructions.
+Additionally, the work volume of a single MAC-loop iter-
+ation is 32× smaller than that of an entire output tile. Con-
+sequently, a 32-way fixed-split decomposition would also
+provide a 100% quantization efficiency, but at the expense
+of an 8× larger “fixup” overhead. Furthermore, Stream-K is
+better able to hide the latency of inter-CTA synchronization
+due to the temporal skew between writers and readers when
+sharing partial sums.
+Stream-K also generalizes to both fixed-split and data-
+parallel decompositions. When the grid size g is an even
+multiple of the number of output tiles, Stream-K functions
+exactly as the fixed-split decomposition. Similarly, when g
+equals the number of output tiles, Stream-K behaves identi-
+cally to the data-parallel decomposition. We take advantage
+of this generalization to create an optimized hybridization
+of the Stream-K decomposition in following section (5.2).
+5
+Implementation Details
+The work decomposition we introduced in the last section
+can be instantiated in a number of different ways to suit
+the needs of different hardware architectures and software
+library designs. Our implementation targets NVIDIA GPUs
+and is designed to be integrated into existing libraries like
+cuBLAS and CUTLASS. In this section, we describe how
+we configure the kernels we launch and introduce a hy-
+bridization scheme that helps ensure users achieve maxi-
+mum GEMM performance across the widest possible range
+of problem shapes.
+We also emphasize that these are truly internal implemen-
+tation details. They are completely transparent to the user
+of a BLAS-like library and do not alter the library’s interface.
+The only observable impact is the improved performance
+characteristics that we analyze in Section 6.
+5.1
+Kernel Configuration
+The tile size chosen for blocking the GEMM computation is,
+of course, a critical parameter controlling the performance of
+the GEMM kernel. For modern NVIDIA GPUs, appropriate
+tile sizes are determined by the shape of matrices supported
+by the GPU’s Tensor Cores. Based on extensive empirical
+experience, we selected the smallest CTA-wide tile size ca-
+pable of achieving 99% of the GPU’s peak TFLOP/s for very
+large GEMM volumes for each supported precision. For the
+NVIDIA A100 GPU used in our experiments, these sizes are
+64×64×16 for FP64 problems and 128×128×32 for FP16→32
+problems.
+Achieving maximal GEMM performance from Stream-K
+parallelization also requires some degree of dynamic problem-
+specific configuration. Before launching a kernel we choose
+a grid size likely to yield the best performance on the specific
+problem shape at hand. This is in contrast to ensemble-based
+approaches which accommodate diverse problem shapes
+through the static generation of many kernel variants based
+on workload decomposition and blocking factor.
+Our grid size selection heuristic is based on a simple an-
+alytical model that minimizes the cost of reading, writing,
+and accumulating partial sums while equally distributing
+the MAC-loop iterations per CTA. Details of this analytical
+model are provided in the supplementary material (Appen-
+dix A.1). Parameters to the model are trivially chosen with
+empirical measurements and need only be done once per
+target architecture. The resulting parameters can then be
+compiled statically into the library. Again, this is in con-
+trast to ensemble-based approaches that rely on potentially
+complex heuristics and machine learning models for kernel
+selection at run time.
+5.2
+Data-parallel Hybridization
+The basic Stream-K decomposition can, in certain cases, ex-
+hibit tile-processing skew that leads to potentially adverse
+effects on cache performance. When the number of output
+tiles t is not an even multiple of the grid size g, the starting
+k-offset for the first MAC-loop iteration in each CTA will
+be different. Depending on the sizes and shapes of the in-
+put matrices and blocking factors, this skew may preclude
+these fragments from seeing reuse across CTAs in the GPU’s
+cache structure. In Figure 3a, for example, the initial k-axis
+fragment offsets for each of the four CTAs will be k = 0,
+k = 32, k = 64, and k = 96, respectively. Furthermore, this
+32-element skew between CTAs will persist for the duration
+of the GEMM computation.
+Tile-processing skew is a direct consequence of Stream-K’s
+workload balancing strategy. However, we can take measures
+to limit its duration by applying Stream-K’s iteration bal-
+ancing to a smaller, tile-aligned region of the total iteration
+domain such that the remaining tiles can be produced in full,
+temporally aligned waves.
+The simplest hybrid scheme is the “data-parallel + one-tile
+Stream-K” schedule illustrated in Figure 3b. It applies itera-
+tion balancing only among the tiles otherwise remaining for
+a final, partially full data-parallel wave. The total number of
+full waves is w = ⌊t/p⌋, where t is the number of output tiles
+and p is the number of SM cores in the GPU. Consequently,
+each Stream-K CTA receives an even share of iterations that
+is less than one tile’s worth. Unfortunately, this strategy has
+6
+
+(a) Basic Stream-K
+(b) DP + one-tile SK
+(c) Two-tile SK + DP
+Figure 3. Basic Stream-K vs. hybrid execution schedules for 896 × 384 × 128 GEMM across a hypothetical four-SM GPU.
+little ability to hide the synchronization latency for the ex-
+change of partial sums when three or more CTAs cover the
+same tile. In these scenarios, the accumulating CTA may be
+forced to wait for the contributions of other CTAs to become
+visible, as all but the last will be completing their final it-
+erations at roughly the same time. Furthermore, the basic
+version of our scheme for aggregating partials is serialized
+within a single CTA, and thus will likely cause SM workload
+imbalance when the number of contributing CTAs per tile is
+large.
+We address these problems with our “two-tile Stream-K
++ data-parallel” hybrid schedule, illustrated in Figure 3c. It
+performs one fewer full data-parallel wave in exchange for
+each Stream-K CTA receiving more than one tile’s worth of
+iterations (but fewer than two). This provides much better
+latency hiding when w ≥ 2, and each accumulating CTA will
+only need to receive partials from one other contributing
+CTA. Otherwise, it behaves identically to the “DP + one tile
+SK” schedule. This hybrid approach results in both improved
+memory access patterns and latency hiding. It also shows
+the versatility of the generic Stream-K looping structure
+to implement different scheduling policies within the same
+kernel instance.
+6
+Performance Evaluation
+We have implemented our Stream-K decomposition using
+NVIDIA’s CUTLASS library of CUDA C++ template abstrac-
+tions for authoring GEMM-like computations. CUTLASS pro-
+vides the optimized equivalent of the CTA-wide MacLoop()
+subroutine in Algorithm 3, which performs blocking, tiling,
+and software-pipelined data movement that is analogous
+to the closed-source cuBLAS and cuDNN implementations.
+Our evaluation encompasses both (1) double-precision FP64
+GEMM, and (2) mixed-precision FP16→32 GEMM. For the
+latter, the input matrices A and B comprise half-precision
+FP16 values, yet the internal accumulation and output matrix
+C values are single-precision FP32.
+Figure 4. The test domain of 32,824 GEMM problem shapes
+and sizes used for performance evaluation.
+{m} = {128 . . . 8192}, {n} = {128 . . . 8192}, {k} = {128 . . . 8192}
+Hardware environment. Our test GPU is the NVIDIA
+A100, which contains 108 SM cores. For measurement sta-
+bility, we lock the power envelope at 400 W and SM clocks
+at 1005 MHz (∼71% of their dynamic peak). This establishes
+FP64 tensor-core peak throughput of 13.9 TFLOP/s, and
+mixed FP16→32 tensor-core peak throughput of 222.3 TFLOP/s.
+Dataset. Our test corpus intends to approximate the enor-
+mous breadth and scope of device-wide GEMM problems
+that GPU math kernel libraries are designed to accommodate.
+As shown in Figure 4, we evaluate 32,824 different problem
+sizes and shapes, log-sampled at random within a domain
+of m, n, and k matrix dimensions whose volume spans six
+orders of magnitude.
+7
+
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+(time)log(m)
+10
+10
+12
+12
+14
+14
+14
+14
+13
+13
+12
+12
+11
+11
+10
+10
+log(n)
+6
+9
+8
+8
+7
+8
+10
+log(k)
+10
+12
+12
+14
+14vs.
+CUTLASS
+64 × 64 × 16
+vs.
+cuBLAS
+vs.
+cuBLAS
+> 150 ops/B
+vs.
+CUTLASS
+oracle
+Average
+1.23×
+1.06×
+1.03×
+1.05×
+StdDev
+0.45
+0.10
+0.03
+0.09
+Min
+0.77×
+0.68×
+0.99×
+0.70×
+Max
+5.63×
+2.55×
+1.24×
+1.64×
+Table 1. Stream-K FP64 Relative Performance
+vs.
+CUTLASS
+128 × 128 × 32
+vs.
+cuBLAS
+vs.
+cuBLAS
+> 150 ops/B
+vs.
+CUTLASS
+oracle
+Average
+1.63×
+1.13×
+1.15×
+1.12×
+StdDev
+1.46
+0.45
+0.12
+0.37
+Min
+0.80×
+0.64×
+0.98×
+0.61×
+Max
+14.7×
+6.74×
+1.85×
+4.63×
+Table 2. Stream-K FP16→32 Relative Performance
+Methodology. For both GEMM precisions, we build a sin-
+gle Stream-K kernel that has been specialized per the guide-
+lines in the Section 5. Furthermore, these kernels implement
+our “two-tile Stream-K + data-parallel” hybrid decomposi-
+tion. Our evaluation compares each Stream-K kernel with:
+1. the default data-parallel CUTLASS kernel of the same
+blocking factor;
+2. the cuBLAS ensemble for that precision (CUDA 11.6);
+and
+3. an idealized oracle that will always select the highest
+performing data-parallel CUTLASS blocking factor to
+execute for a given GEMM instance.
+For FP64 problems, this oracle selects among the ensem-
+ble of {(32×32×16), (32×64×16), (64×64×16), (64×128×16),
+(128×128×16)} blocking factor specializations. For FP16→32,
+it selects among the ensemble of {(64×64×64), (64×128×32),
+(128×128×32), (128×256×32)} blocking factor specializations.
+These specific specializations are an open-sourced strict sub-
+sets alternative of the corresponding cuBLAS GEMM kernel
+ensembles.
+The “roofline” plots of Figure 6a and Figure 5a highlight
+the spread of performance produced by the singleton data-
+parallel CUTLASS kernels. They plot the percentage of FP64
+and FP16→32 processor utilization as a function of compu-
+tational intensity. Ideally, a GEMM implementation’s per-
+formance response would manifest as a narrow band that
+adheres tightly to the machine’s bandwidth- and compute-
+bound performance ceilings. Here, the data-parallel kernels
+exhibit a fairly large dynamic range for any given regime of
+arithmetic intensity. In contrast, the performance responses
+from the equivalent Stream-K kernels in Figure 6d and Fig-
+ure 5d are much tighter. These observations are corroborated
+by Table 1 and Table 2, which show the Stream-K kernels
+outperforming their data-parallel FP64 and FP16→32 equiv-
+alents by an average of 1.23× and 1.63×, respectively. For
+extreme strong-scaling scenarios where m × n is small and k
+is large, our Stream-K kernels demonstrate up to 5.63× and
+14.7 × speedup, respectively.
+The second columns of Table 1 and Table 2 compare our
+Stream-K performance with that of cuBLAS. On average,
+our FP64 and FP16→32 Stream-K GEMM kernels respec-
+tively deliver 6% and 13% greater throughput than their cor-
+responding cuBLAS ensembles, with peak improvement of
+2.55× and 6.74×. This is a significant improvement over the
+breadth of 32K GEMM problem shapes and sizes with 20×
+less executable code (a single kernel for each precision) than
+NVIDIA’s vendor GEMM library, cuBLAS.
+Furthermore, the contrast between the FP64 and FP16→32
+cuBLAS performance responses (Figure 6b and Figure 5b)
+versus those of our hypothetical CUTLASS oracle ensembles
+(Figure 6c and Figure 5c) reveal the difficulties of design-
+ing kernel selection heuristics that deliver consistently good
+performance. Despite having access to the same blocking
+factor specializations, cuBLAS exhibits substantially wider
+dynamic ranges than the idealized data-parallel CUTLASS
+oracle. The performance spreads of our Stream-K kernels
+are narrower still, achieving up to 4.6× the idealized oracle
+performance and underscoring their ability to achieve uti-
+lization levels that are simply not possible from tile-centric
+work decompositions.
+Finally, we observe regimes of small, bandwidth-bound
+problem shapes where our largish blocking factors do not
+compete well against cuBLAS. However, if we restrict our
+scope to the domain of compute-bound problems (i.e., FP64
+problems having compute intensity > 150 ops/byte and FP16
+→ 32 problems > 400 ops/byte), Figure 7a and Figure 7b
+demonstrate that our singleton Stream-K kernels achieve
+unilaterally higher performance than the cuBLAS ensembles.
+The “noisy” relative performance in the regimes below these
+thresholds is not surprising, as Stream-K is attempting to
+make memory-bound computations run faster by adding
+more memory workload. This suggests a few avenues for
+future work, namely separate cost-modeling for the memory-
+bound regime and/or the bundling of a second Stream-K
+kernel having smaller tile size into a two-kernel ensemble.
+7
+Conclusion
+We presented Stream-K, a novel parallel workload decomposi-
+tion technique for scheduling general matrix multiplication
+(GEMM) and similar computations on wide architectures
+such as GPUs. Unlike other tile-splitting techniques, the
+MAC-loop iteration is our unit of workload quantization
+across processor cores. This affords excellent strong scaling
+and workload balancing because its cost is (1) a constant with
+respect to the problem shape, and (2) substantially smaller
+than that of an entire output tile.
+8
+
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Tensor core utilization % 
+Arithmetic intensity (operations / byte)
+(a) CUTLASS FP16→32 data-parallel “roofline”
+performance (blocking factors = 128×128×32).
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Tensor core utilization % 
+Arithmetic intensity (operations / byte)
+(b) cuBLAS (ensemble)
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Tensor core utilization % 
+Arithmetic intensity (operations / byte)
+(c) Idealized CUTLASS oracle (ensemble)
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+Tensor core utilization % 
+Arithmetic intensity (operations / byte)
+(d) Stream-K (blocking factors = 128×128×32)
+Figure 5. FP16→FP32 GEMM “roofline” performance utilization landscapes on NVIDIA A100 across 32K GEMM problem
+shapes and sizes.
+Furthermore, Stream-K produces an O(p) number of split-
+ting seams that are bound by the number of processor cores.
+Consequently, the overheads of strong scaling and workload
+balancing scale with processor width rather than problem
+size. This is a welcome feature for many applications that
+cannot afford to allocate large amounts of temporary storage
+equivalent to the problem output.
+Finally, we evaluated our Stream-K approach across a
+broad spectrum of GEMM shapes and sizes. We showed that
+a single blocking configuration of Stream-K can (1) achieve
+levels of absolute performance that match and/or exceed that
+of NVIDIA’s cuBLAS library, even when the latter is oper-
+ating at near-peak processor utilization, and (2) do so with
+much higher levels of performance consistency. Addition-
+ally, Stream-K is an attractive option for library construction
+and maintenance, as it presents an opportunity to reduce
+distribution sizes by an order of magnitude and removes the
+need for complex handcoded heuristics or machine learn-
+ing models for kernel selection without compromising per-
+formance. Stream-K is open-sourced within CUTLASS 2.11
+(https://github.com/NVIDIA/cutlass) and the performance
+shown within this paper can be reproduced when compiled
+using CUDA 11.8.
+For future works, we identify cache-aware, tile-access
+patterns such as Morton Order, an avenue for optimization.
+We also believe that Stream-K decomposition could provide
+a similar improved performance response for other GEMM-
+like workloads that struggle with the same quantization
+inefficiencies.
+9
+
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Tensor core utilization % 
+Arithmetic intensity (operations / byte)
+(a) CUTLASS data-parallel
+(blocking factors = 64×64×16)
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Tensor core utilization % 
+Arithmetic intensity (operations / byte)
+(b) cuBLAS (ensemble)
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Tensor core utilization % 
+Arithmetic intensity (operations / byte)
+(c) Idealized CUTLASS oracle (ensemble)
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Tensor core utilization % 
+Arithmetic intensity (operations / byte)
+(d) Stream-K (blocking factors = 64×64×16)
+Figure 6. FP64 GEMM “roofline” performance utilization landscapes on NVIDIA A100 across 32K problem shapes and sizes.
+Acknowledgments
+This material is based upon work supported by Defense Ad-
+vanced Research Projects Agency (DARPA) under Contract
+No. HR0011-18-3-0007. Any opinions, findings and conclu-
+sions or recommendations expressed in this material are
+those of the author(s) and do not necessarily reflect the views
+of the U.S. Government. Distribution Statement “A” (Ap-
+proved for Public Release, Distribution Unlimited). We would
+like to acknowledge Louis Feng, Valentin Andrei, Zhongyi
+Lin and Serban D. Porumbescu for their feedback on early
+drafts of the paper.
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+11
+
+A
+Supplementary Material
+A.1
+Analytical Modeling for Stream-K
+Configuration
+In practice, it is not always advantageous to invoke the
+Stream-K decomposition with as many CTAs as can be ac-
+tively resident on the GPU. Because it is a tile-splitting ap-
+proach, it incurs fixup costs above and beyond the simple
+data-parallel decomposition. Consequently, the fundamental
+proposition is one of strong scaling: how much additional
+parallelism can be expressed before the extra overhead causes
+a negative return on investment. Depending on the problem
+shape, the optimal splitting could be enough to fill the entire
+processor (i.e., g ← p), no splitting at all (i.e., g ← t), or
+somewhere in between.
+To predict this inflection point, we present a simple ap-
+proach to model the runtime of Stream-K as a function of
+grid size g. In the absence of other work on the GPU, the
+runtime of the entire Stream-K schedule will be the same as
+that of one of its tile-outputting CTAs, which we formulate
+as follows:
+timeCTA(g) ←a + b(FixupPeers(g) > 1)
++ c(ItersPerCta(g)) + d (FixupPeers(g) – 1)
+where:
+ItersPerCta(g) ←
+� ⌈
+m
+BLK_M⌉ × ⌈
+n
+BLK_N⌉ × ⌈
+k
+BLK_K⌉
+g
+�
+FixupPeers(g) ←
+
+�
+k
+BLK_K
+�
+IterationsPerCta(g)
+
+This CTA runtime model comprises four components. The
+a workload encompasses the one-time, fixed-size costs in-
+curred by each CTA, e.g., the grid launch latency, the initial
+compulsory cache misses, the cost of writing the final out-
+put tile to C, etc. The second component b incorporates
+the conditional costs of outputting temporary partial sums
+for scenarios where the number of output tiles does not
+quantize perfectly across the processor. The third—the per-
+iteration workload c—represents the instruction and stall
+workload of each MAC-iteration. The final, per-collaborator
+workload d is the cost of reading and accumulating the par-
+tial sums from another CTA covering the same tile. The set
+of workload constants {a, b, c, d } will be unique to each
+combination of blocking factors, matrix data type, and GPU
+microarchitecture, and can be determined empirically via
+microbenchmarks.
+Figure 8 illustrates the behavior of our grid size selec-
+tion model as parameterized for fp16-precision GEMM on
+NVIDIA’s A100 GPU using blocking factors BLK_M = 128,
+BLK_N = 128, and BLK_K = 32. Specifically, we highlight
+(a) GEMM 256 × 3584 × 8192
+56 output tiles, 256 iterations per tile
+gbest ← 108 CTAs, 132/133 iterations per CTA
+(b) GEMM 1024 × 1024 × 1024
+64 output tiles, 32 iterations per tile
+gbest ← 64 CTAs, 32 iterations per CTA
+(c) GEMM 128 × 128 × 16384
+1 output tile, 512 iterations per tile
+gbest ← 8 CTAs, 64 iterations per CTA
+Figure 8. Modeled Stream-K performance on NVIDIA A100
+(108 SMs) for BLK_M=128, BLK_N=128, BLK_K=32
+12
+
+Figure 9. Strong-scaling comparison of data-parallel and
+Stream-K execution schedules for 128 × 128 × 384 GEMM
+across a hypothetical four-SM GPU. Data-parallel causes the
+enormous k-dimension to be sequentially processed within
+single CTA, whereas Stream-K is able to take advantage of
+the parallelism available across the k-dimension.
+three strong-scaling GEMM scenarios where the number
+of output tiles is insufficient to produce a single full wave
+across the processor’s 108 SM cores.
+The first GEMM shape accumulates through a large-sized
+k-dimension to produce a short, wide output matrix. In this
+scenario, the reduction in MAC-loop time relative to the
+increasing costs of seam fixup is monotonically improving.
+Consequently, the optimal grid size coincides with maximal
+parallelism at g = 108 CTAs.
+The second shape accumulates through a medium-sized k-
+dimension to produce a square matrix with 64 output tiles. In
+this case, the fixup costs of b and d outweigh any reduction
+in MAC-loop iteration count, as seen by the global minima
+“dip” at g = 64 CTAs.
+The third shape produces a single output tile after accumu-
+lating through an enormous k-dimension, analogous to the
+execution schedule in Figure 9. Although the opportunity
+for strong scaling is quite large, the per-peer cost of serial
+reduction is entirely incurred by a single CTA. These accu-
+mulation costs begin to outweigh any further reductions in
+iteration count for grid sizes g > 8.
+13
+
+SMO
+0
+0
+DP CTA-0
+SK CTA-0
+B
+SM1
+0
+SK CTA-1
+SM2
+0
+SK CTA-2
+A
+SM3
+0
+SK CTA-3
+-
+tsk
+to
+(time)