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+arXiv:2301.00302v1  [math.CO]  31 Dec 2022
+On Harmonious coloring of hypergraphs
+Sebastian Czerwiński
+Institute of Mathematics, University of Zielona Góra, Poland
+January 3, 2023
+Abstract
+A harmonious coloring of a k-uniform hypergraph H is a vertex col-
+oring such that no two vertices in the same edge have the same color,
+and each k-element subset of colors appears on at most one edge. The
+harmonious number h(H) is the least number of colors needed for such a
+coloring.
+The paper contains a new proof of the upper bounded h(H) = O(
+k√
+k!m)
+on the harmonious number of k-hypergraphs of maximum degree ∆ with
+m edges. We use the local cut lemma of A. Bernshteyn.
+1
+Introducion
+Let H = (V, E) be a k-uniform hypergraph with the set of vertices V and the
+set of edges E. The set of edges is a family of k-elements sets of V , where k ≥ 2.
+A rainbow coloring h of a hypergraph H is a map c : V �→ {1, . . . , r} in
+which no two vertices in the same edge have the same color. If two vertices are
+in the same edge e with the same color, we say that the edge e is bad.
+A coloring c is called harmonious if c(e) ̸= c(f) for every pair of distinct
+edges e, f ∈ E and c is a rainbow coloring.
+We say that distinct edges e and f have the same pattern of colors if c(e\f) =
+c(f \ e) and there is no uncolored vertex in the set e \ f.
+Let h(H) be the least number of colors needed for a harmonious of H. In
+Bosek et al. (2016) proved that
+Theorem 1 (Bosek et al. (2016)). For every ε > 0 and every ∆ > 0 there
+exist integers k0 and m0 such that every k-uniform hypergraph H with m edges
+(where m ≥ m0 and k ≥ k0) and maximum degree ∆ satisfies
+h(H) ≤ (1 + ε)
+k
+k − 1
+k�
+∆(k − 1)k!m.
+Remark 1. The paper Bosek et al. (2016) contains an upper bound on the
+harmonious number
+h(H) ≤
+k
+k − 1
+k�
+∆(k − 1)k!m+1+∆2+(k−1)∆+
+k−1
+�
+i=2
+i
+i − 1
+i
+�
+(i − 1)i(k − 1)∆2
+k − i
+.
+1
+
+The proof of this theorem is based on the entropy compression method, see
+Grytczuk et al. (2013); Esperet and Parreau (2013).
+Because a number r of used colors must satisfy the inequality
+�r
+k
+�
+≤ m, we get
+lower bound Ω(
+k√
+k!m). By these observations, it is conjectured by Bosek et al.
+(2016) that
+Conjecture 1. For each k, ∆ ≥ 2 there exist a constant c = c(k, ∆) such that
+every k-uniform hypergraph H with m edges and maximum degree ∆ satisfies
+h(H) ≤
+k√
+k!m + c.
+This conjecture was posed by Edwards (1997b) for simple graphs. He prove
+there that
+h(G) ≤ (1 + o(1))
+√
+2m.
+There are many results about the harmonious number of particular classes of
+graphs, see Aflaki et al. (2012); Akbari et al. (2012); Edwards (1997a); Edwards and McDiarmid
+(1994a); Edwards (1996); Edwards and McDiarmid (1994b); Krasikov and Roditty
+(1994) or Aigner et al. (1992); Balister et al. (2002, 2003); Bazgan et al. (1999);
+Burris and Schelp (1997).
+The paper contains proof of the theorem of Bosek et al., we use a different
+method, the local cut lemma of Bernshteyn (2017, 2016). The proof is simpler
+and shorter than the original proof of Bosek et al.
+2
+A special version of the Local Cut Lemma
+Let A be a family of subsets of a powerset Pow(I), where I is a finite set. We
+say that it is downwards-closed if for each S ∈ A, implies Pow(S) ⊆ (A). A
+subset ∂A of I is called boundary of a downwards-closed family A if
+∂A := {i ∈ I : S ∈ A and S ∪ {i} ̸∈ A for some S ⊆ I \ {i}}.
+Let τ : T �→ [1; +∞) be a function, then for every X ⊆ I we denote by τ(X) a
+number
+τ(X) :=
+�
+x∈X
+τ(x).
+Let B a random event, X ⊆ I and i ∈ I. We introduce two quantities:
+σA
+τ (B, X) := max
+Z⊆I\X Pr(B and Z ∪ X ̸∈ A|Z ∈ A) · τ(X)
+and
+σA
+τ (B, i) := min
+i∈X⊆I σA
+τ (B, X).
+If Pr(Z ∈ A) = 0, then Pr(P|Z ∈ A) = 0 for all events P.
+2
+
+Theorem 2 (Bernshteyn (2017)). Let I be a finite set. Let Ω be a probabil-
+ity space and let A: Ω �→ Pow(Pow(I)) be a random variable such that with
+probability 1, A is a nonempty downwards-closed family of subsets of I. For
+each i ∈ I, Let B(i) be a finite collection of random events such that whenever
+i ∈ ∂A, at least one of the events in B(i) holds. Suppose that there is a function
+τ : I �→ [1, +∞) such that for all i ∈ I we have
+τ(i) ≥ 1 +
+�
+B∈B(i)
+σA
+τ (B, i).
+Then Pr(I ∈ A) ≥ 1/τ(I) > 0.
+3
+Proof of theorem
+We choose a coloring f : V �→ {1, . . ., t} uniformly at random. Let A be a subset
+of the power set of V given by
+A := {S ⊆ V : c is a harmonious coloring of H(V )}.
+It is a nonempty downwards-closed with probability 1 (the empty set is an
+element of A)
+By a set ∂A, we denote the set of all vertices v such that there is an element
+X of A such that the coloring c is not a harmonious coloring of X ∪ {v}. If
+the coloring c is not harmonious coloring there is a bad edge or there are two
+different edges with the same pattern of colors. So, we define for every v ∈ V a
+collection B(v) as union of sets:
+B1(v) := {Be : v ∈ e ∈ E(H) and e is not proper colored}
+and for every i ∈ {0, 1, . . ., k − 1}
+B2
+i (v) := {Be,f : v ∈ e, f ∈ E(H) and c(e) = c(f), |e \ f| = i}.
+That is B(v) = B1(v) ∪ �k−1
+i=1 B2
+i (v).
+We assume that the event Be happens if and only if the edge e is the bad
+edge and the event Be,f happens if and only if edges e and f have the same
+pattern of colors.
+We also assume that a function τ is a constant function, that is τ(v) = τ ∈
+[1, +∞). This implies that for any subset S of V , we have τ(S) = τ |S|.
+Now, we must find an upper bound on
+σA
+τ (B, v) =
+min
+X⊆V :v∈X max
+Z⊆V \X Pr(B ∧ Z ∪ X ̸∈ A|Z ∈ A)τ(X),
+where v ∈ V and B ∈ B(v).
+We will be use an estimation σA
+τ (B, v) ≤
+maxZ⊆V \X Pr(B|Z ∈ A)τ(X). Now, we consider two cases.
+3
+
+Case 1: B ∈ B1, i.e. B = Be
+We choose as X the set {e}. Because the colors of distinct vertices are indepen-
+dent, we get an upper bound σA
+τ (Be, v) ≤ Pr(Be)τ k (events Be and ”Z ∈ A”
+are independent). The probability Pr(Be) opposite to Pr(Be) full fields
+Pr(Be) = 1 − t
+t · t − 1
+t
+· . . . · t − k + 1
+t
+≥ 1 − (1 − k − 1
+t
+)k−1.
+Through Bernoulli’s inequality, we get
+Pr(Be) ≥ 1 − (1 − k − 1
+t
+· (k − 1)) = (k − 1)2
+t
+.
+So, Pr(Be) ≤ k2
+t .
+Case 2: B ∈ B2
+i , i.e. B = Be,f and |e \ f| = i
+Now, we set X = e \ f. The probability Pr(Be,f) is bonded above by i!
+ti . So, we
+get
+σA
+τ (Be,f, v) ≤ Pr(Be,f)τ i ≤ i!
+ti τ i.
+To end the proof we must find an upper bound on sizes of sets B1(v), B2
+0(v)
+and B2
+i (v), where i > 0. Because the degree of a vertex is bounded by above ∆
+and the number of edges is m we get that
+|B1(v)| ≤ ∆ and |B2
+0(v)| ≤ ∆m.
+The hardest part is an upper bound on B2
+i (v), i > 0. The number of edges
+f such that e \ f = i is bounded above by
+k∆
+k−i. There are at most k∆ edges
+with a nonempty intersection with the edge e and the edge f has exactly k − i
+common elements with e. So, we have |B2
+i (v)| ≤ ∆ k∆
+k−i. To apply theorem 2 we
+must find τ ∈ [1, +∞) and c ∈ N such that for all v ∈ V below inequality holds
+τ ≥ 1 + ∆k2
+t τ k + ∆mk!
+tk τ k +
+k−1
+�
+i=1
+∆ k∆
+k − i
+i!
+ti τ i.
+If we choose τ =
+k
+k−1 and t =
+k
+k−1
+k�
+∆(k − 1)k!m(1 + ε), it is easy to see that
+the inequality holds for sufficiently large hypergraph.
+Acknowledgments
+References
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+6
+

-NAyT4oBgHgl3EQfdfcz/content/tmp_files/load_file.txt

@@ -0,0 +1,341 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf,len=340
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='00302v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='CO]  31 Dec 2022  On Harmonious coloring of hypergraphs  Sebastian Czerwiński  Institute of Mathematics, University of Zielona Góra, Poland  January 3, 2023  Abstract  A harmonious coloring of a k-uniform hypergraph H is a vertex col-  oring such that no two vertices in the same edge have the same color,  and each k-element subset of colors appears on at most one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' The  harmonious number h(H) is the least number of colors needed for such a  coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  The paper contains a new proof of the upper bounded h(H) = O(  k√  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='m)  on the harmonious number of k-hypergraphs of maximum degree ∆ with  m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' We use the local cut lemma of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Bernshteyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  1  Introducion  Let H = (V, E) be a k-uniform hypergraph with the set of vertices V and the  set of edges E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' The set of edges is a family of k-elements sets of V , where k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  A rainbow coloring h of a hypergraph H is a map c : V �→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' , r} in  which no two vertices in the same edge have the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' If two vertices are  in the same edge e with the same color, we say that the edge e is bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  A coloring c is called harmonious if c(e) ̸= c(f) for every pair of distinct  edges e, f ∈ E and c is a rainbow coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  We say that distinct edges e and f have the same pattern of colors if c(e\\f) =  c(f \\ e) and there is no uncolored vertex in the set e \\ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Let h(H) be the least number of colors needed for a harmonious of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' In  Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' (2016) proved that  Theorem 1 (Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' For every ε > 0 and every ∆ > 0 there  exist integers k0 and m0 such that every k-uniform hypergraph H with m edges  (where m ≥ m0 and k ≥ k0) and maximum degree ∆ satisfies  h(H) ≤ (1 + ε)  k  k − 1  k�  ∆(k − 1)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' The paper Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' (2016) contains an upper bound on the  harmonious number  h(H) ≤  k  k − 1  k�  ∆(k − 1)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='m+1+∆2+(k−1)∆+  k−1  �  i=2  i  i − 1  i  �  (i − 1)i(k − 1)∆2  k − i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  1  The proof of this theorem is based on the entropy compression method, see  Grytczuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Esperet and Parreau (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Because a number r of used colors must satisfy the inequality  �r  k  �  ≤ m, we get  lower bound Ω(  k√  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' By these observations, it is conjectured by Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  (2016) that  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' For each k, ∆ ≥ 2 there exist a constant c = c(k, ∆) such that  every k-uniform hypergraph H with m edges and maximum degree ∆ satisfies  h(H) ≤  k√  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='m + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  This conjecture was posed by Edwards (1997b) for simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' He prove  there that  h(G) ≤ (1 + o(1))  √  2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  There are many results about the harmonious number of particular classes of  graphs, see Aflaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Akbari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Edwards (1997a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Edwards and McDiarmid  (1994a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Edwards (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Edwards and McDiarmid (1994b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Krasikov and Roditty  (1994) or Aigner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' (1992);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Balister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' (2002, 2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Bazgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Burris and Schelp (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  The paper contains proof of the theorem of Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=', we use a different  method, the local cut lemma of Bernshteyn (2017, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' The proof is simpler  and shorter than the original proof of Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  2  A special version of the Local Cut Lemma  Let A be a family of subsets of a powerset Pow(I), where I is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' We  say that it is downwards-closed if for each S ∈ A, implies Pow(S) ⊆ (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' A  subset ∂A of I is called boundary of a downwards-closed family A if  ∂A := {i ∈ I : S ∈ A and S ∪ {i} ̸∈ A for some S ⊆ I \\ {i}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Let τ : T �→ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' +∞) be a function, then for every X ⊆ I we denote by τ(X) a  number  τ(X) :=  �  x∈X  τ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Let B a random event, X ⊆ I and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' We introduce two quantities:  σA  τ (B, X) := max  Z⊆I\\X Pr(B and Z ∪ X ̸∈ A|Z ∈ A) · τ(X)  and  σA  τ (B, i) := min  i∈X⊆I σA  τ (B, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  If Pr(Z ∈ A) = 0, then Pr(P|Z ∈ A) = 0 for all events P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  2  Theorem 2 (Bernshteyn (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Let I be a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Let Ω be a probabil-  ity space and let A: Ω �→ Pow(Pow(I)) be a random variable such that with  probability 1, A is a nonempty downwards-closed family of subsets of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' For  each i ∈ I, Let B(i) be a finite collection of random events such that whenever  i ∈ ∂A, at least one of the events in B(i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Suppose that there is a function  τ : I �→ [1, +∞) such that for all i ∈ I we have  τ(i) ≥ 1 +  �  B∈B(i)  σA  τ (B, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Then Pr(I ∈ A) ≥ 1/τ(I) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  3  Proof of theorem  We choose a coloring f : V �→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=', t} uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Let A be a subset  of the power set of V given by  A := {S ⊆ V : c is a harmonious coloring of H(V )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  It is a nonempty downwards-closed with probability 1 (the empty set is an  element of A)  By a set ∂A, we denote the set of all vertices v such that there is an element  X of A such that the coloring c is not a harmonious coloring of X ∪ {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' If  the coloring c is not harmonious coloring there is a bad edge or there are two  different edges with the same pattern of colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' So, we define for every v ∈ V a  collection B(v) as union of sets:  B1(v) := {Be : v ∈ e ∈ E(H) and e is not proper colored}  and for every i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=', k − 1}  B2  i (v) := {Be,f : v ∈ e, f ∈ E(H) and c(e) = c(f), |e \\ f| = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  That is B(v) = B1(v) ∪ �k−1  i=1 B2  i (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  We assume that the event Be happens if and only if the edge e is the bad  edge and the event Be,f happens if and only if edges e and f have the same  pattern of colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  We also assume that a function τ is a constant function, that is τ(v) = τ ∈  [1, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' This implies that for any subset S of V , we have τ(S) = τ |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Now, we must find an upper bound on  σA  τ (B, v) =  min  X⊆V :v∈X max  Z⊆V \\X Pr(B ∧ Z ∪ X ̸∈ A|Z ∈ A)τ(X),  where v ∈ V and B ∈ B(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  We will be use an estimation σA  τ (B, v) ≤  maxZ⊆V \\X Pr(B|Z ∈ A)τ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Now, we consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  3  Case 1: B ∈ B1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' B = Be  We choose as X the set {e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Because the colors of distinct vertices are indepen-  dent, we get an upper bound σA  τ (Be, v) ≤ Pr(Be)τ k (events Be and ”Z ∈ A”  are independent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' The probability Pr(Be) opposite to Pr(Be) full fields  Pr(Be) = 1 − t  t · t − 1  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' · t − k + 1  t  ≥ 1 − (1 − k − 1  t  )k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Through Bernoulli’s inequality, we get  Pr(Be) ≥ 1 − (1 − k − 1  t  (k − 1)) = (k − 1)2  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  So, Pr(Be) ≤ k2  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Case 2: B ∈ B2  i , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' B = Be,f and |e \\ f| = i  Now, we set X = e \\ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' The probability Pr(Be,f) is bonded above by i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  ti .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' So, we  get  σA  τ (Be,f, v) ≤ Pr(Be,f)τ i ≤ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  ti τ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  To end the proof we must find an upper bound on sizes of sets B1(v), B2  0(v)  and B2  i (v), where i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Because the degree of a vertex is bounded by above ∆  and the number of edges is m we get that  |B1(v)| ≤ ∆ and |B2  0(v)| ≤ ∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  The hardest part is an upper bound on B2  i (v), i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' The number of edges  f such that e \\ f = i is bounded above by  k∆  k−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' There are at most k∆ edges  with a nonempty intersection with the edge e and the edge f has exactly k − i  common elements with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' So, we have |B2  i (v)| ≤ ∆ k∆  k−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' To apply theorem 2 we  must find τ ∈ [1, +∞) and c ∈ N such that for all v ∈ V below inequality holds  τ ≥ 1 + ∆k2  t τ k + ∆mk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  tk τ k +  k−1  �  i=1  ∆ k∆  k − i  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  ti τ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  If we choose τ =  k  k−1 and t =  k  k−1  k�  ∆(k − 1)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='m(1 + ε), it is easy to see that  the inequality holds for sufficiently large hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Acknowledgments  References  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Aflaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Akbari, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Edwards, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Eskandani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Jamaali, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Ra-  vanbod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' On harmonious colouring of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=', 19(1):Paper  3, 9, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='37236/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  4  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Aigner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Triesch, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Tuza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  Irregular assignments and vertex-  distinguishing edge-colorings of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' In Combinatorics ’90 (Gaeta, 1990),  volume 52 of Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content=', pages 1–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content='  URL  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content='1002/jgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='3190180305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content='  McDiarmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  New  upper  bounds  on  harmo-  nious  colorings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content='  URL  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='1002/jgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content=' Micek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  New approach to nonrepetitive  sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content='  URL  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='1002/rsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content='  Roditty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='1002/jgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='3190180212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}
+page_content='  6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'}

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@@ -0,0 +1,2040 @@
+EPR-Net: Constructing non-equilibrium potential landscape via a variational force
+projection formulation
+Yue Zhao,1 Wei Zhang,2, ∗ and Tiejun Li1, 3, 4, †
+1Center for Data Science, Peking University, Beijing 100871, China
+2Zuse Institute Berlin, D-14195 Berlin, Germany
+3LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
+4Center for Machine Learning Research, Peking University, Beijing 100871, China
+(Dated: January 6, 2023)
+We present a novel yet simple deep learning approach, dubbed EPR-Net, for constructing the
+potential landscape of high-dimensional non-equilibrium steady state (NESS) systems. The key idea
+of our approach is to utilize the fact that the negative potential gradient is the orthogonal projection
+of the driving force in a weighted Hilbert space with respect to the steady-state distribution. The
+constructed loss function also coincides with the entropy production rate (EPR) formula in NESS
+theory. This approach can be extended to dealing with dimensionality reduction and state-dependent
+diffusion coefficients in a unified fashion. The robustness and effectiveness of the proposed approach
+are demonstrated by numerical studies of several high-dimensional biophysical models with multi-
+stability, limit cycle, or strange attractor with non-vanishing noise.
+Since Waddington’s famous landscape metaphor on the
+development of cells in the 1950s [1], the construction of
+potential landscape for non-equilibrium biochemical reac-
+tion systems has been recognized as an important prob-
+lem in theoretical biology, as it provides insightful pic-
+tures for understanding complex dynamical mechanisms
+of biological processes. This problem has attracted con-
+siderable attention in recent decades in both biophysics
+and applied mathematics community. Until now, several
+approaches have been proposed to realize Waddington’s
+landscape metaphor in a rational way, see [2–10] and
+the references therein for details and [11–14] for reviews.
+Broadly speaking, these proposals can be classified into
+two types: (T1) the construction of potential landscape
+in the finite noise regime [3–5] and (T2) the construction
+of the quasi-potential in the zero noise limit [2, 6–9].
+For low-dimensional systems (i.e., dimension less than
+4), the potential landscape can be numerically computed
+either by solving a Fokker-Planck equation (FPE) using
+grid-based methods until the steady solution is reached
+approximately as in (T1) type proposals [3, 5], or by solv-
+ing a Hamilton-Jacobi-Bellman (HJB) equation using, for
+instance, the ordered upwind method [15] or minimum
+action type method [8] as in (T2) type proposals. How-
+ever, these approaches suffer from the curse of dimen-
+sionality when applied to high-dimensional systems. Al-
+though methods based on mean field approximations are
+able to provide a semi-quantitative description of the en-
+ergy landscape for typical systems [4, 16], direct and gen-
+eral approaches are still favored in applications. In this
+aspect, pioneering work has been done recently, which
+allows direct construction of high-dimensional potential
+landscape using deep neural networks (DNN), based on
+either the steady viscous HJB equation satisfied by the
+∗ [email protected]
+† [email protected]
+landscape function in (T1) case [17, 18], or the point-
+wise orthogonal decomposition of the force field in (T2)
+case [19]. These works have brought significant advances
+in the methodological developments in both cases. How-
+ever, these approaches, which are based on solving HJB
+equations alone, may encounter numerical difficulties due
+to the non-uniqueness of the weak solution to the non-
+viscous HJB equation in (T2) case [20], and challenges
+in solving the steady HJB equation with a small noise in
+(T1) case.
+Setup.
+In this letter, we present a simple yet ef-
+fective DNN approach, EPR-Net, for constructing the
+potential landscape of high-dimensional non-equilibrium
+steady state (NESS) systems in (T1) type. Our key ob-
+servation is that the negative potential gradient is the or-
+thogonal projection of the driving force under a weighted
+inner product with respect to the steady-state distribu-
+tion. To be specific, let us consider the stochastic differ-
+ential equations (SDEs)
+dx(t)
+dt
+= F (x(t)) +
+√
+2D ˙w,
+x(0) = x0,
+(1)
+where x0 ∈ Rd, F : Rd → Rd is a smooth function,
+˙w = ( ˙w1, . . . , ˙wd)⊤ is the d-dimensional temporal Gaus-
+sian white noise with E ˙wi(t) = 0 and E[ ˙wi(t) ˙wj(s)] =
+δijδ(t − s) for i, j = 1, . . . , d, s, t > 0 and D > 0 is the
+noise strength, which is often related to the system’s tem-
+perature T by D = kBT, where kB is the Boltzmann con-
+stant. We assume that (1) is ergodic and denote by pss(x)
+its steady-state probability density function (PDF).
+We follow the (T1) type proposal in [3] to derive the po-
+tential landscape of (1) in the case of D > 0. That is, we
+define the potential U = −D ln pss and the steady proba-
+bility flux Jss = pssF −D∇pss in the domain Ω, which we
+assume for simplicity is either Rd or a d-dimensional hy-
+perrectangle. The steady-state PDF pss(x) satisfies the
+Fokker-Planck equation (FPE)
+∇ · (pssF ) − D∆pss = 0,
+for x ∈ Ω,
+(2)
+arXiv:2301.01946v1  [physics.bio-ph]  5 Jan 2023
+
+2
+and we assume the asymptotic boundary condition (BC)
+pss(x) → 0 as |x| → ∞ when Ω = Rd, or the re-
+flecting boundary condition Jss · n = 0 when Ω ⊂ Rd
+is a d-dimensional hyperrectangle, where n is the unit
+outer normal.
+In both cases, we have pss(x) ≥ 0 and
+�
+Ω pss(x) dx = 1.
+Learning approach. Aiming at an effective approach
+for high-dimensional applications, we employ DNNs to
+approximate U(x), and the key idea in this letter is to
+learn U by training DNN with the following loss function
+LEPR(V ) =
+�
+Ω
+|F (x) + ∇V (x; θ)|2 dπ(x),
+(3)
+where V := V (x; θ) is a neural network function with
+parameters θ [21], and dπ(x) = pss(x) dx.
+To justify
+(3), we note that U satisfies the important orthogonality
+relation: for any suitable function W : Rd → R,
+�
+Ω
+�
+F (x) + ∇U(x)
+�
+· ∇W(x) dπ(x) = 0.
+(4)
+Therefore, U(x) is the unique minimizer (up to a con-
+stant) of the loss LEPR and, moreover, the negative po-
+tential gradient −∇U is in fact the projection of the force
+field F in the π-weighted Hilbert space. See Sec. A and B
+in the Supplemental Material (SM) for derivations in de-
+tail.
+The minimum loss LEPR(U) has a clear physical inter-
+pretation. Indeed, we have (see SM Sec. B)
+LEPR(U) =
+�
+Ω
+|Jss|2 1
+pss
+dx = ess
+p ,
+(5)
+where ess
+p denotes the steady entropy production rate
+(EPR) of the NESS system (1) [3, 22, 23]. Therefore,
+minimizing (3) is equivalent to approximating the steady
+EPR. This explains the name EPR-Net of our approach.
+To utilize (3) in numerical computations, we replace
+the spatial integral in (3) with respect to the unknown π
+by its empirical average using data sampled from (1):
+�LEPR(θ) = 1
+N
+N
+�
+i=1
+��F (xi) + ∇V (xi; θ)
+��2,
+(6)
+where (xi)1≤i≤N could be either the final states (at time
+T) of N trajectories starting from different initializations
+or equally spaced time series along a single long trajec-
+tory up to time T, where T ≫ 1.
+In both cases, the
+ergodicity of SDE (1) guarantees that (6) is a good ap-
+proximation of (3) as long as T is large [24]. We adopt
+the former approach in the numerical experiments in this
+work, where the gradients of both V (with respect to x)
+and the loss itself (with respect to θ) in (6) are calculated
+by auto-differentiation through PyTorch [25]. The stabil-
+ity analysis of this approximation is presented in detail
+in SM Sec. C.
+We apply our method to a toy model first in order to
+check its applicability and accuracy. We take
+F (x) = −(I + A) · ∇U0(x),
+(7)
+where A ∈ Rd×d is a constant skew-symmetric matrix,
+i.e., A⊤ = −A, and U0 is some known function. With this
+choice of F , one can check that the true potential land-
+scape is U(x) = U0(x). In particular, the system is re-
+versible when A = 0. Based on the proposed method, we
+construct a double-well model with known potential U0
+for verification. We take D = 0.1. As shown in Fig. 1(A),
+the learned potential agrees well with the simulated sam-
+ples.
+Also, the decomposition of the force field shows
+that the negative gradient part −∇V (x; θ) around the
+wells points towards the attractor and is nearly orthog-
+onal to the non-gradient part. The overall non-gradient
+field shows a counter-clockwise rotation.
+The relative
+root mean square error (rRMSE) of the potential V (x; θ)
+learned by EPR loss is 0.0987 (averaged over 3 runs),
+which supports the effectiveness of our approach.
+See
+SM Sec. F F.1 for details of the problem setting.
+The correct interpretation of the computational results
+based on the EPR loss (3) is that the accuracy of V (x)
+is guaranteed only when π(x) is evidently above zero
+for any specific x.
+In the “visible” domain of π (i.e.,
+the places where there are sample points of {xi}), the
+trained potential V gives reliable approximation; while
+in the weakly visible or invisible domain, especially in
+local transition regions between meta-stable states and
+boundaries of the visible domain, we must resort to the
+original FPE (2) which holds pointwise in space.
+Learning strategy for small D. Substituting the rela-
+tion pss(x) = exp(−U(x)/D) into (2), we get the viscous
+HJB equation
+NHJB(U) := F · ∇U + |∇U|2 − D∆U − D∇ · F = 0 (8)
+with the asymptotic BC U → ∞ as |x| → ∞ in the case
+of Ω = Rd, or the reflecting BC (F + ∇U) · n = 0 on ∂Ω
+when Ω is a d-dimensional hyperrectangle, respectively.
+As in the framework of physics-informed neural networks
+(PINNs) [26], (8) motivates the HJB loss
+LHJB(V ) =
+�
+Ω
+��NHJB(V (x; θ))
+��2 dµ(x),
+(9)
+where µ is any desirable distribution.
+By choosing µ
+properly, this loss allows the use of sample data that
+better cover the domain Ω and, when combined with the
+loss in (3), leads to significant improvement of the train-
+ing results in our numerical experiments when D is small.
+Specifically, for small D, we propose the enhanced loss in
+training which has the form
+�Lenh(θ) = �LEPR(θ) + λ �LHJB(θ),
+(10)
+where
+�LEPR(θ)
+is
+defined
+in
+(6),
+�LHJB(θ)
+=
+1
+N ′
+�N ′
+i=1 |NHJB(V (x′
+i; θ))|2 is an approximation of (9) us-
+ing an independent data set (x′
+i)1≤i≤N ′ obtained by sam-
+pling the trajectories of (1) with a larger D′ > D, and
+λ > 0 is a weight parameter balancing the contribution
+of the two terms in (10). Note that the proposed strategy
+is both general and easily adaptable. For instance, one
+
+3
+FIG. 1. Filled contour plots of the learned potential V (x; θ) for (A) toy model learned by EPR loss (3) with D = 0.1, and
+(B)-(C) a biochemical oscillation network model [3] and a tri-stable cell development model [5] learned by enhanced loss (10).
+The force field F (x) is decomposed into the gradient part −∇V (x; θ) (white arrows) and the non-gradient part (gray arrows).
+The length of an arrow denotes the scale of the vector. The solid dots are samples from the simulated invariant distribution.
+can alternatively use data (x′
+i)1≤i≤N ′ that contains more
+samples in the transition region, or employ a modification
+of the loss (9) in (10) [17].
+We apply our enhanced loss (10) to construct the land-
+scape for a 2D biological system with a limit cycle [3]
+and a 2D multistable system [5]. The potential V (x; θ)
+learned by the enhanced loss (10), the force decomposi-
+tion, and sample points from the simulated invariant dis-
+tribution are shown in Fig. 1(B) and (C). As in the toy
+model case, the gradient part (white arrows) points di-
+rectly towards the attractors, while the non-gradient part
+(gray arrows) shows a counter-clockwise rotation for the
+limit cycle, and a splitting-and-back flow from the mid-
+dle attractor to the other two attractors for the tri-stable
+dynamical model. To further verify the accuracy of the
+method, we numerically solve the FPE (2) as reference
+solutions by a fine grid discretization. Comparisons be-
+tween the proposed method and the method based on
+the naive HJB loss on these two problems are demon-
+strated in SM. Averaged over 3 runs, the rRMSE of the
+potential V learned by our enhanced loss is 0.0524 and
+0.0402, respectively, which shows an evident advantage
+over the naive HJB loss. See SM Sec. F for details of the
+comparisons.
+Dimensionality reduction.
+When applying the ap-
+proach above to high-dimensional problems, dimensional-
+ity reduction is necessary in order to visualize the results
+and gain physical insights. A straightforward approach is
+to first learn the high-dimensional potential U and then
+find its low-dimensional representation, i.e., the reduced
+potential or the free energy function, using dimension-
+ality reduction techniques (see SM Sec. D D.1). In the
+following, we present an alternative approach that allows
+to directly learn the low-dimensional reduced potential.
+For simplicity, we consider the linear case and, with a
+slight abuse of notation, denote by x = (y, z)⊤, where
+z = (xi, xj) ∈ R2 contains the coordinates of two vari-
+ables of interest, and y ∈ Rd−2 corresponds to the
+other d − 2 variables.
+The domain Ω (either Rd or
+a d-dimensional hyperrectangle) has the decomposition
+Ω = Σ × �Ω, where Σ ⊆ Rd−2 and �Ω ⊆ R2 are the do-
+mains of y and z, respectively. As can be seen in the
+numerical examples, this setting is applicable to many
+interesting biochemical systems. Extensions to nonlinear
+low-dimensional reduced variables with general domains
+are possible, e.g., by applying the approach developed
+in [27]. In the current setting, the reduced potential is
+�U(z) = −D ln �pss(z) = −D ln
+�
+Σ
+pss(y, z) dy,
+(11)
+and one can show that �U minimizes the following loss
+function:
+LP-EPR(�V ) =
+�
+Ω
+��Fz(y, z)+∇z �V (z; θ)
+��2 dπ(y, z), (12)
+where Fz(y, z) ∈ R2 is the z-component of the force
+field F = (Fy, Fz)⊤. Similar to (6), the empirical form
+of (12) can be used in learning the reduced potential �U.
+Moreover, one can derive an enhanced loss as in (10) that
+could be used for systems with small D. To this end, we
+note that �U satisfies the projected HJB equation
+NP-HJB(�U) := �F · ∇z �U + |∇z �U|2
+− D∆z �U − D∇z · �F = 0 ,
+(13)
+with asymptotic BC �U
+→ ∞ as |z| → ∞, or the
+reflecting BC ( �F + ∇z �U) · �n
+=
+0 on ∂�Ω, where
+�F (z)
+:=
+�
+Σ Fz(y, z)dπ(y|z) is the projected force
+defined using the conditional distribution dπ(y|z) =
+pss(y, z)/�pss(z) dy, and �n denotes the unit outer normal
+on ∂�Ω. Based on (13), we can formulate the projected
+HJB loss
+LP-HJB(�V ) =
+�
+�Ω
+��NP-HJB(�V (z; θ))
+��2 dµ(z),
+(14)
+
+A
+B
+3.0
+2.00
+2.00
+0.20
+8 
+2.5
+7
+2.5
+1.60
+1.60
+0.16
+6
+2.0
+2.0
+5
+0.12
+1.20
+1.20
+1.5
+>1.5
+4 
+0.04
+0.80
+-0.80
+1.0
+3 
+1.0-
+2 
+-0.40 0.5
+0.40
+0.04
+0.5
+0.00
+0.0
+0.00
+-0.00 0.0
+0
+0.5
+2.5
+6
+0.5
+2.0
+1.5
+2.0
+2
+8
+1.0
+1.5
+2.5
+0.0
+1.0
+3.0
+0
+4
+0.0
++
+X
+X4
+where µ is any suitable distribution over �Ω, and �F in (13)
+is learned beforehand by training a DNN with the loss
+LP-For( �
+G) =
+�
+Ω
+��Fz(y, z) − �
+G(z; θ)
+��2 dπ(y, z).
+(15)
+The overall enhanced loss used in numerical computa-
+tions comprises two terms, which are empirical estimates
+of (12) and (14) based on two different sets of sample
+data. See SM Sec. D for derivation details.
+We then apply our dimensionality reduction approach
+to construct the landscape for an 8D cell cycle model con-
+taining both a limit cycle and a stable equilibrium point
+for the chosen parameters, and take CycB and Cdc20 as
+the reduced variables following [4]. As shown in Fig. 2, we
+can find that the depth of the reduced potential and force
+strength agree well with the density of projected samples.
+Moreover, we can also get some important insights from
+Fig. 2 on the projection of the high-dimensional dynam-
+ics with a limit cycle to two dimensions. One particular
+feature is that the limit cycle induced by the projected
+force �
+G (outer red circle) has minor differences with the
+limit cycle directly projected from high dimensions (yel-
+low circle), and the difference is slight or moderate de-
+pending on whether the density of samples is high or
+low. This is natural in the reduction since the distribu-
+tion π(y|z) in the projection is not of Dirac type when
+D > 0, and this difference will disappear as D → 0.
+Another feature is that we unexpectedly get an addi-
+tional stable limit cycle (inner red circle) and a stable
+point (red dot in the center) emerging inside the limit
+cycle.
+Though virtual in high dimensions and biologi-
+cally irrelevant, the existence of such two limit sets is
+reminiscent of the Poincar´e-Bendixson theorem in pla-
+nar dynamics theory [28, Chapter 10.6], which depicts
+a common phenomenon when performing dimensionality
+reduction with limit cycles to 2D plane. The emergence
+of these two limit sets, though being not a general sit-
+uation, is specific in the considered model due to the
+relatively flat landscape of the potential in the centering
+region. In addition, close to the saddle point (0.13, 0.55)
+of �V (green star), there is a barrier domain along the
+limit cycle direction, while a local well domain along the
+Cdc20 direction, which characterizes the region that bi-
+ological cycle paths mainly go through.
+Last but not
+the least, a zoom-in view of the local well domain out-
+side of the limit cycle shows its detailed spiral structure
+(Fig. 2C), which has not been revealed before by mak-
+ing a Gaussian approximation. Some other applications
+of our approach to Ferrell’s three-ODE model [29], 52D
+stem cell network model [16] and 3D Lorenz model are
+demonstrated in SM Sec. G and H.
+Extension to variable diffusion coefficient case.
+The
+EPR-Net formulation can be extended to the case of
+state-dependent diffusion coefficients without any diffi-
+culty. Consider the Ito SDEs
+dx(t)
+dt
+= F (x(t)) +
+√
+2Dσ(x(t)) ˙w,
+x(0) = x0,
+(16)
+FIG. 2. Dimensionality reduction of an 8D cell cycle model
+with two reduced variables. (A) Reduced potential landscape
+�V with projected contour lines. (B) Projected sample points,
+streamlines of the projected force field �
+G and the filled con-
+tour plot of �V . The red circles and dots are stable limit sets
+of the projected force field. The yellow circle is the projection
+of the original high-dimensional limit cycle. (C) The detailed
+spiral structure of the streamlines of �
+G around the stable
+point by zooming in the square domain in (B).
+with diffusion matrix σ(x) ∈ Rd×m and
+˙w is an m-
+dimensional temporal Gaussian white noise. We assume
+that m ≥ d and the matrix a(x) := (σσ⊤)(x) satisfies
+u⊤a(x)u ≥ c0|u|2 for all x, u ∈ Rd, where c0 > 0 is a
+positive constant. Using a similar derivation as before,
+we can again show that the high-dimensional landscape
+function U of (16) minimizes the EPR loss
+LV-EPR(V ) =
+�
+Ω
+|F v(x) + a(x)∇V (x)|2
+a−1(x) dπ(x),
+(17)
+where F v(x) = F (x) − D∇ · a(x) and |u|2
+a−1(x) :=
+u⊤a−1(x)u for u ∈ Rd. We provide derivation details
+of (17) in SM Sec. E. However, we will not pursue a nu-
+merical study of (16)–(17) in this paper.
+Discussions and Conclusion. Below we make some fi-
+nal remarks. First, concerning the use of the steady-state
+distribution π(x) in (3) and its approximation by a long
+time series of the SDE (1) in EPR-Net, we emphasize that
+it is the sampling approximation of π that naturally cap-
+tures the important parts of the potential function, and
+the landscape beyond the sampled regions is not that
+essential in practice.
+Second, as is exemplified in SM
+Sec. F F.4, we found that a direct application of density
+estimation methods (DEM), e.g., normalizing flows [30],
+to the sampled time series data does not give potential
+
+A
+B
+1.0
+0.08
+0.08
+0.07
+0.06
+0.04
+0.06
+0.02
+0.05
+0.00
+-0.02
+0.8
+0.04
+-0.04
+0.03
+1.0
+0.8
+0.02
+0.60
+0.01
+0.4
+0.2
+0.0
+0.1
+0.2
+0.6
+0.3
+0.4
+0.5
+CycB
+Cdc20
+C
+3.0
+0.16
+0.4
+2.5
+0.14
+2.0
+0.12
+1.5
+0.10
+0.2
+1.0
+0.08
+0.5
+0.06
+0.18
+0.20
+0.22
+0.24
+0.26
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+CycB5
+landscape with satisfactory accuracy. We speculate that
+such deficiency of DEM is due to its over-generality and
+the fact that it does not take advantage of the force field
+information explicitly compared to (3).
+Overall, we have presented the EPR-Net, a simple
+yet effective DNN approach, for constructing the non-
+equilibrium potential landscape of NESS systems. This
+approach is both elegant and robust due to its variational
+structure and its flexibility to be combined with other
+types of loss functions. Further extension of dimensional-
+ity reduction to nonlinear reduced variables and numeri-
+cal investigations in the case of state-dependents diffusion
+coefficients will be explored in future work.
+Acknowledgement.
+We thank Professors Chunhe Li,
+Xiaoliang Wan and Dr. Yufei Ma for helpful discus-
+sions. TL and YZ acknowledge the support from NSFC
+and MSTC under Grant No.s 11825102, 12288101 and
+2021YFA1003300.
+WZ is supported by the DFG un-
+der Germany’s Excellence Strategy-MATH+: The Berlin
+Mathematics Research Centre (EXC-2046/1)-project ID:
+390685689.
+The numerical computations of this work
+were conducted on the High-performance Computing
+Platform of Peking University.
+
+6
+Supplemental Material for:
+EPR-Net: Constructing non-equilibrium potential landscape via
+a variational force projection formulation
+CONTENTS
+Part 1: Theory
+6
+A. Validation of the EPR loss
+6
+B. EPR loss and entropy production rate
+7
+C. Stability of the EPR minimizer
+7
+D. Dimensionality reduction
+8
+D.1. Gradient projection loss
+8
+D.2. Projected EPR loss
+8
+D.3. Force projection loss
+9
+D.4. HJB equation for the reduced potential
+9
+E. State-dependent diffusion coefficients
+9
+Part 2: Computation
+10
+F. 2D models and comparisons
+10
+F.1. Toy model and enhanced EPR
+10
+F.2. 2D limit cycle model
+11
+F.3. 2D multi-stable model
+12
+F.4. Numerical comparisons
+12
+G. 3D models
+13
+G.1. 3D Lorenz system
+14
+G.2. Ferrell’s three-ODE model
+14
+H. High dimensional models
+15
+H.1. 8D complex system
+15
+H.2. 52D multi-stable system
+16
+References
+17
+In this supplemental material (SM), we will present
+further theoretical derivations and computational details
+of the contents in the main text (MT). This SM consists
+of two parts: Theory and computation.
+PART 1: THEORY
+We will first provide details of theoretical derivations
+omitted in the MT.
+A.
+VALIDATION OF THE EPR LOSS
+In this section, we show that, up to an additive con-
+stant, the potential function U(x) := −D ln pss(x) is the
+unique minimizer of the EPR loss (3) defined in the MT.
+First, we show that the orthogonality relation
+�
+Ω
+(F + ∇U) · ∇W dπ = 0
+(18)
+holds for any suitable function W(x) : Rd → R under
+both choices of the boundary conditions (BC) considered
+in the MT, where dπ(x) := pss(x)dx. To see this, we
+note that
+�
+Ω
+(F + ∇U) · ∇W dπ
+=
+�
+Ω
+(F pss − D∇pss) · ∇W dx
+=
+�
+∂Ω
+W(F pss − D∇pss) · n dx
+−
+�
+Ω
+W∇ · (F pss − D∇pss) dx
+:=P1 − P2
+where we have used integration by parts and the relation
+pss(x) = exp(−U(x)/D).
+The term P1 is zero due to
+the fact that pss(x) tends to 0 exponentially as |x| → ∞
+when Ω = Rd, and the reflecting BC Jss · n = 0 which
+holds on ∂Ω when Ω is bounded. The term P2 is zero
+due to the steady state Fokker-Planck equation (FPE)
+satisfied by pss.
+Now consider the EPR loss, we have
+LEPR(V ) =
+�
+Ω
+|F + ∇V |2 dπ
+=
+�
+Ω
+|F + ∇U + ∇V − ∇U|2 dπ
+=
+�
+Ω
+�
+|F + ∇U|2 + |∇V − ∇U|2�
+dπ
++ 2
+�
+Ω
+(F + ∇U) · ∇(V − U) dπ
+=
+�
+Ω
+|F + ∇U|2 + |∇V − ∇U|2 dπ,
+where we have used the orthogonality relation (18) to
+arrive at the last equality, from which we conclude that
+
+7
+U(x) is the unique minimizer of the EPR loss up to an
+additive constant.
+In fact, define the π-weighted inner product for any
+square integrable functions f, g on Ω:
+(f, g)π :=
+�
+Ω
+f(x)g(x) dπ(x)
+(19)
+and the corresponding L2
+π-norm ∥·∥π by ∥f∥2
+π := (f, f)π,
+we get a Hilbert space L2
+π (see, e.g., [31, Chapter II.1]).
+Choosing W = U in (18), we observe that the minimiza-
+tion of EPR loss finds the orthogonal projection of F
+under the π-weighted inner product, i.e.,
+F (x) = −∇U(x) + l(x), such that (∇U, l)π = 0. (20)
+However, we remark that this orthogonality holds only
+in the L2
+π-inner product sense instead of the pointwise
+sense. Furthermore, the two orthogonality relations (18)
+and (20) can be understood as follows. Using (20), the
+relation (18) is equivalent to
+�
+Ω l · ∇Wdπ = 0 for any
+W. Integration by parts gives ∇ · (l e−U/D) = 0, which
+is equivalent to ∇U · l + D∇ · l = 0. When D → 0, we
+recover the pointwise orthogonality, which is adopted in
+computing quasi-potentials in [19].
+B.
+EPR LOSS AND ENTROPY PRODUCTION
+RATE
+In this section, we show that the minimum EPR loss
+coincides with the steady entropy production rate in non-
+equilibrium steady state (NESS) theory.
+Following [22, 23], we have the important identity con-
+cerning the entropy production for the SDE (1) defined
+in the MT:
+DdS(t)
+dt
+= ep(t) − hd(t),
+(21)
+where S(t) := −
+�
+Ω p(x, t) ln p(x, t) dx is the entropy of
+the probability density function p(x, t) at time t, ep is
+the entropy production rate (EPR)
+ep(t) =
+�
+Ω
+|F (x) − D∇ ln p(x, t)|2 p(x, t) dx,
+(22)
+and hd is the heat dissipation rate
+hd(t) =
+�
+Ω
+F (x) · J(x, t) dx,
+(23)
+with the probability flux J(x, t)
+:=
+p(x, t)(F (x) −
+D∇ ln p(x, t)) at time t. When D = kBT, the above for-
+mulas have clear physical meaning in statistical physics.
+At the steady state, we get the steady EPR
+ess
+p =
+�
+Ω
+|F − D∇ ln pss|2 pss dx
+=
+�
+Ω
+|F + ∇U|2 pss dx
+=
+�
+Ω
+|Jss|2 1
+pss
+dx = LEPR(U),
+where Jss(x) = pss(x)(F (x)+∇U(x)) is the steady prob-
+ability flux.
+This shows the relation between the pro-
+posed EPR loss function and the entropy production rate
+in the NESS theory.
+C.
+STABILITY OF THE EPR MINIMIZER
+In this section, we formally show that small perturba-
+tions of the invariant distribution π will not introduce
+a disastrous change to the minimizer of the correspond-
+ing EPR loss. We only consider the bounded domain,
+i.e., Ω is a hyperrectangle. The argument for unbounded
+domains is similar.
+Suppose dπ(x) = p(x)dx, dµ(x) = q(x)dx, and the
+functions U(x) and ¯U(x) are the unique minimizers (up
+to a constant) of the following two EPR losses
+U = arg min
+V
+�
+Ω
+|F + ∇V |2 dπ,
+¯U = arg min
+V
+�
+Ω
+|F + ∇V |2 dµ,
+respectively.
+It is not difficult to find that the Euler-
+Lagrange equations of U, ¯U are given by the following
+partial differential equation (PDE) with suitable BCs:
+∇ · ((F + ∇U)p) = 0 in Ω, (F + ∇U) · n = 0 on ∂Ω,
+∇ · ((F + ∇ ¯U)q) = 0 in Ω, (F + ∇ ¯U) · n = 0 on ∂Ω.
+The PDEs above defined inside the domain Ω can be
+converted to
+∆Up + ∇U · ∇p = −∇ · (pF ),
+∆ ¯Uq + ∇ ¯U · ∇q = −∇ · (qF ).
+Define U0(x) = −D ln p(x) and ¯U0(x) = −D ln q(x). We
+then obtain
+−∇U · ∇U0 + D∆U = F · ∇U0 − D∇ · F ,
+(24)
+−∇ ¯U · ∇ ¯U0 + D∆ ¯U = F · ∇ ¯U0 − D∇ · F .
+(25)
+Assuming that δU0 := U0 − ¯U0 = O(ε), where 0 < ϵ ≪ 1
+denotes a small constant, we have the PDE for U − ¯U by
+subtracting (25) from (24):
+−∇(U− ¯U) · ∇U0 + D∆(U − ¯U)
+= F · ∇(δU0) + ∇ ¯U · ∇(δU0)
+with BC ∇(U − ¯U) · n = 0. Since U0, ¯U, F ∼ O(1), we
+can obtain that
+U(x) − ¯U(x) = O(ε)
+by the regularity theory of elliptic PDE [32, Section 6.3]
+when D ∼ O(1), or by the matched asymptotic expan-
+sion when D ≪ 1 [33, Chapter 2]. In fact, the closeness
+between U(x) and ¯U(x) can be ensured as long as U0 and
+¯U0 are close enough in the region where p(x) and q(x) are
+bounded away from zero by the method of characteristics
+analysis [32, Section 2.1] and matched asymptotics.
+
+8
+D.
+DIMENSIONALITY REDUCTION
+In this section, we study dimensionality reduction for
+high-dimensional problems in order to learn the projected
+potential.
+Denote by x = (y, z)⊤ ∈ Ω. As in the MT, we assume
+the domain
+Ω = �Ω × Σ,
+where �Ω ⊆ R2 and Σ ⊆ Rd−2 are the domain of y and z,
+respectively. The reduced potential �U(z) is defined as
+�U(z) = −D ln �pss(z) = −D ln
+�
+Σ
+pss(y, z) dy.
+(26)
+One natural approach for constructing �U(z) is directly
+integrating pss(y, z) based on the learned U(y, z) with
+the EPR loss, i.e.,
+�U(z) = −D ln
+�
+Σ
+exp(−U(y, z)/D) dy.
+(27)
+However, performing this integration is not a straightfor-
+ward numerical task (see, e.g., [34, Chapter 7]).
+D.1.
+Gradient projection loss
+In this subsection, we study a simple approach to
+approximate �U(z) based on sample points, which ap-
+proximately obey the invariant distribution π(x), and
+the learned high dimensional potential function U(x) by
+EPR loss. This approach is not investigated numerically
+in this work, but it will be useful for the derivations in
+the next subsection. The idea is to utilize the gradient
+projection (GP) loss on the z components of ∇U:
+LGP(�V ) =
+�
+Ω
+��∇zU(y, z) − ∇z �V (z)
+��2 dπ(y, z).
+(28)
+To justify (28), we note that
+LGP(�V ) =
+�
+Ω
+��∇zU − ∇z �V
+��2 dπ(x)
+=
+�
+Ω
+��∇zU − ∇z �U + ∇z �U − ∇z �V
+��2 dπ(x)
+=
+�
+Ω
+���∇zU − ∇z �U
+��2 +
+��∇z �U − ∇z �V
+��2�
+dπ(x)
++ 2
+�
+Ω
+(∇zU − ∇z �U) · ∇z(�U − �V ) dπ(x)
+=:P1 + P2,
+where P1 and P2 denote the terms in the third and the
+fourth line above, respectively. The term P2 = 0 since
+�
+Ω
+∇zU · ∇z(�U − �V ) dπ(x)
+=
+�
+�Ω
+��
+Σ
+∇zUe− U
+D dy
+�
+· ∇z(�U − �V ) dz
+= − D
+�
+�Ω
+∇z
+��
+Σ
+e− U
+D dy
+�
+· ∇z(�U − �V ) dz
+= − D
+�
+�Ω
+∇z�pss · ∇z(�U − �V ) dz
+=
+�
+�Ω
+∇z �U · ∇z(�U − �V ) �pss dz
+and
+�
+Ω
+∇z �U · ∇z(�U − �V ) dπ(x)
+=
+�
+�Ω
+∇z �U · ∇z(�U − �V ) �pss dz,
+which cancel with each other in P2.
+Therefore, the minimization of GP loss is equivalent to
+minimizing
+�
+�Ω
+��∇z �U − ∇z �V
+��2 �pss dz,
+which clearly implies that �U(z) is the unique minimizer
+(up to a constant) of the proposed GP loss.
+D.2.
+Projected EPR loss
+In this subsection, we study the projected EPR (P-
+EPR) loss, which has the form
+LP-EPR(�V ) =
+�
+Ω
+��Fz(y, z) + ∇z �V (z)
+��2 dπ(y, z),
+(29)
+where Fz(y, z) ∈ R2 is the z-component of the force field
+F = (Fy, Fz)⊤.
+Define
+�LP-EPR(�V ) =
+�
+Ω
+��F (y, z) + ∇�V (z)
+��2 dπ(y, z),
+(30)
+where ∇ is the full gradient with respect to x. To justify
+(29), we first note the following equivalence
+min LP-EPR(�V )
+⇐⇒
+min �LP-EPR(�V ),
+(31)
+since ∇y �V (z) = 0 and the y-components of F +∇�V only
+introduce an irrelevant constant in (30). Furthermore, we
+have
+�LP-EPR(�V ) =
+�
+Ω
+��F + ∇�V
+��2 dπ(x)
+=
+�
+Ω
+��F + ∇U + ∇�V − ∇U
+��2 dπ(x)
+=
+�
+Ω
+��F + ∇U
+��2 +
+��∇�V − ∇U
+��2 dπ(x),
+
+9
+where the last equality is due to the orthogonality rela-
+tion (18). Using a similar argument for deriving (31), the
+equivalence (31) itself, as well as the GP loss in (28), we
+get
+min LP-EPR(�V )
+⇐⇒
+min LGP(�V ).
+(32)
+Since �U minimizes the GP loss as is shown in the previous
+subsection, we conclude that �U minimizes the loss in (29).
+D.3.
+Force projection loss
+In this subsection, we study the force projection (P-
+For) loss for approximating the projection of Fz onto the
+z-space.
+Denote by
+�F (z) :=
+�
+Σ
+Fz(y, z) dπ(y|z)
+(33)
+the projected force defined using the conditional distri-
+bution
+dπ(y|z) = pss(y, z)/�pss(z) dy.
+(34)
+We can learn �F (z) via the following force projection loss
+LP-For( �
+G) =
+�
+Ω
+��Fz(y, z) − �
+G(z)
+��2 dπ(y, z).
+(35)
+To justify (35), we note that
+�
+Ω
+��Fz(y, z) − �
+G(z)
+��2 dπ(y, z)
+=
+�
+Ω
+�
+|Fz(y, z)|2 + | �
+G(z)|2�
+dπ(y, z)
+− 2
+�
+Ω
+Fz(y, z) · �
+G(z) dπ(y, z)
+=:P1 − 2P2.
+The term P2 can be simplified as
+P2 =
+�
+�Ω
+��
+Σ
+Fz(y, z)dπ(y|z)
+�
+· �
+G(z) �pss(z) dz
+=
+�
+�Ω
+�F (z) · �
+G(z) �pss(z) dz.
+Therefore, we have the equivalence
+min LP-For( �
+G)
+⇐⇒
+min �LP-For( �
+G),
+(36)
+where
+�LP-For( �
+G) :=
+�
+�Ω
+�� �F (z) − �
+G(z)
+��2�pss(z) dz.
+From the analysis above we can conclude that �F(z) min-
+imizes the loss in (35).
+D.4.
+HJB equation for the reduced potential
+In this subsection, we show that the reduced potential
+�U satisfies the projected HJB equation
+�F · ∇z �U + |∇z �U|2 − D∆z �U − D∇z · �F = 0 ,
+(37)
+with asymptotic BC �U → ∞ as |z| → ∞, or the reflecting
+BC ( �F + ∇z �U) · �n = 0 on ∂�Ω, where �n denotes the unit
+outer normal on ∂�Ω. We will only consider the rectangu-
+lar domain case here. The argument for the unbounded
+case is similar.
+Recall that pss(x) satisfies the FPE
+∇ · (pssF ) − D∆pss = 0.
+(38)
+Integrating both sides of (38) on Σ with respect to y
+and utilizing the boundary condition Jss · n = 0, where
+Jss = pssF − D∇pss, we get
+∇z ·
+� �
+Σ
+Fzpss dy
+�
+− D∆z�pss = 0.
+(39)
+Taking (33) and (34) into account, we obtain
+∇z ·
+�
+�pss �F
+�
+− D∆z�pss = ∇z · �
+J = 0,
+(40)
+i.e., a FPE for �pss(z) with the reduced force field �F ,
+where �
+J := �pss �F −D∇z�pss. The corresponding boundary
+condition can be also derived by integrating the original
+BC Jss · n = 0 on Σ with respect to y for z ∈ ∂�Ω, which
+gives
+�
+J · �n =
+�
+�pss �F − D∇z�pss
+�
+· �n = 0.
+(41)
+Substituting the relation �pss(z) = exp
+�
+−�U(z)/D
+�
+into
+(40) and (41), we get (37) and the corresponding reflect-
+ing BC after some algebraic manipulations.
+E.
+STATE-DEPENDENT DIFFUSION
+COEFFICIENTS
+In this section, we study the EPR loss for NESS sys-
+tems with a state-dependent diffusion coefficient.
+Consider the Ito SDEs
+dx(t)
+dt
+= F (x(t)) +
+√
+2Dσ(x(t)) ˙w
+(42)
+with the state-dependent diffusion matrix σ(x). Under
+the same assumptions as in the MT, we have the FPE
+∇ · (pssF ) − D∇2 : (pssa) = 0.
+(43)
+We show that the high dimensional landscape function
+U of (42) minimizes the EPR loss
+LV-EPR(V ) =
+�
+Ω
+|F v(x) + a(x)∇V (x)|2
+a−1(x) dπ(x),
+(44)
+
+10
+where F v(x) := F (x) − D∇ · a(x) and |u|2
+a−1(x) :=
+u⊤a−1(x)u for u ∈ Rd.
+To justify (44), we first note that (43) can be rewritten
+as
+∇ · (pssF v − Da∇pss) = 0 ,
+(45)
+which, together with the BC, implies the orthogonality
+relation
+�
+Ω
+�
+F v + a∇U
+�
+· ∇W dπ = 0
+(46)
+for a suitable test function W(x). Following the same
+reasoning used in establishing (18) and utilizing (46), we
+have
+�
+Ω
+|F v + a∇V |2
+a−1 dπ
+=
+�
+Ω
+��F v + a∇U + a∇(V − U)
+��2
+a−1 dπ
+=
+�
+Ω
+|F v + a∇U
+��2
+a−1 dπ +
+�
+Ω
+��a∇(V − U)
+��2
+a−1 dπ.
+The last expression implies that U(x) is the unique min-
+imizer of LV-EPR(V ) up to a constant.
+The above derivation for the state-dependent diffusion
+case will permit us to construct the landscape for the
+chemical Langevin dynamics, which will be studied in
+future work.
+PART 2: COMPUTATION
+Now we present the computational details and results
+omitted in the MT in the computation part.
+F.
+2D MODELS AND COMPARISONS
+In this section, we will describe the computational
+setup and results for some 2D models which we utilize
+for the test of different formulations, including the toy
+model with known potential in the MT, a 2D biologi-
+cal system with a limit cycle [3] and a 2D multi-stable
+system [5]. We will also demonstrate the motivation for
+enhanced EPR and its advantage over other methods.
+F.1.
+Toy model and enhanced EPR
+In the toy model, we set the force field as
+F (x) = −(I + A) · ∇U0(x),
+(47)
+and choose the potential
+U0 = ((x − 1.5)2 − 1.0))2 + 0.5(y − 1.5)2,
+(48)
+where x = (x, y)⊤. We take the anti-symmetric matrix
+A =
+�
+0
+0.5
+−0.5
+0
+�
+,
+(49)
+which introduces a counter-clockwise rotation for a fo-
+cusing central force field.
+This sets up a simple non-
+equilibrium system. In this model, we have
+F (x) = −∇U0(x) + l(x), l(x) = −A · ∇U0(x)
+and
+l(x) · ∇U0(x) = 0
+holds in the pointwise sense. So, we have constructed
+a double-well non-reversible system with analytically
+known potential which can be used to verify the accu-
+racy of the learned potential. We focus on the domain
+Ω = [0, 3] × [0, 3].
+Primarily, the single EPR loss works well for the toy
+model with a relatively large diffusion coefficient D = 0.1,
+as shown in Fig. 1(A) in the MT. A slice plot of the poten-
+tial at y = 1.5 (Fig. 3(A)) shows the EPR solution coin-
+cides well with the analytical solution. The relative root
+mean square error (rRMSE) and the relative mean abso-
+lute error (rMAE), which will be defined in Section F F.4,
+have mean and standard deviation of 0.099 ± 0.010 and
+0.081 ± 0.013 over 3 runs, respectively.
+However, when decreasing D to 0.05, the samples from
+simulated invariant distribution mainly stay in the dou-
+ble wells and away from the transition region (orange
+
+11
+FIG. 3. An illustration for the motivation of enhanced EPR. (A) and (B) show the comparisons of the learned potentials and
+true solution on the line y = 1.5 in the toy model with D = 0.1 and D = 0.05, respectively. (C) shows the filled contour plot
+of the potential learned by only the EPR loss. The orange points are samples from the simulated invariant distribution with
+D = 0.05, While green points are enhanced samples simulated from a more diffusive distribution with D′ = 0.1, which are used
+in the enhanced EPR.
+points in Fig. 3(C)). In this case, the double well do-
+main can still be learned well, yet the transition region,
+without enough samples, has not been effectively trained.
+Thus, as shown in Fig. 3(B), the single EPR result cap-
+tures the double wells, but cannot accurately connect
+them in the transition domain, which makes the left well
+a bit higher than the right one. The pointwise HJB loss
+with enhanced samples that better cover the transition
+domain thus helps the EPR loss with samples for small
+D, which mainly focuses on the local well domain. Us-
+ing these enhanced samples for D′ = 0.1 (green points
+in Fig. 3(A)), the enhanced EPR method performs much
+better in the transition domain between the two wells
+and thus agrees well with the true solution.
+The above strategy is general, and we apply it to com-
+pute the landscape for all of the continued 2D problems
+and compare it with other methods in Section F F.4.
+F.2.
+2D limit cycle model
+We apply our approach to the limit cycle dynamics
+with a Mexican-hat shape landscape [3].
+Before proceeding to the concrete dynamical model,
+we have the following observation. For any SDEs like
+dx
+dt = F (x) +
+√
+2D ˙w,
+(50)
+the corresponding steady FPE is
+∇ · (F pss) − D∆pss = 0.
+If we make the transformation
+F → κF , D → κD
+in (50), then the steady state PDF
+pss(x) ∝ exp
+�
+−U(x)
+D
+�
+= exp
+�
+−κU(x)
+κD
+�
+is not changed.
+The transformation only changes the
+timescale of the dynamics (50) from t0 to κt0. However,
+this transformation changes the learned potential from U
+to κU if we utilize the drift κF (x) and noise strength κD
+in the system (50), which is helpful to set the scale of U
+to be O(1) by adjusting κ suitably for a specific problem.
+An alternative approach to accomplish this task is by
+choosing F to be κF in the EPR loss.
+We take D = 0.1 and consider the limit cycle dynamics
+dx
+dt = κ
+�α2 + x2
+1 + x2
+1
+1 + y − ax
+�
+,
+(51)
+dy
+dt = κ
+τ0
+�
+b −
+y
+1 + cx2
+�
+,
+(52)
+where the parameters are κ = 100, α = a = b = 0.1, c =
+100, and τ0 = 5. Here the choice of κ = 100 is to make
+U ∼ O(1) following [17]. We focus on the domain Ω =
+[0, 8] × [0, 8] and compute the potential landscape and
+force decomposition which is presented in the MT. As
+explained in the above paragraph, this corresponds to
+the case D = 0.1/κ = 0.001 for the force field considered
+in [3].
+
+B
+A
+C
+1.4
+1.4
+3.0
+EPR
+EPR
+True
+1.2
+1.2 -
+True
+一
+2.5
+Enhanced EPR
+1.0 
+1.0
+2.0
+0.8 -
+0.8
+>
+y1.5
+>
+0.6
+0.6
+1.0
+0.4 
+0.4
+0.5
+0.2
+0.2 
+0.0
+0.0
+0.0 -
+2.5
+0.5
+1.5
+0.5
+1.0
+1.5
+2.0
+1.0
+1.5
+2.0
+2.5
+2.5
+0.5
+1.0
+2.0
+3.0
+0.0
+3.0
+0.0
+3.0
+0.0
+X
++
++12
+FIG. 4. Filled contour plots of the potential V (x; θ) for the toy model with D = 0.05 learned by (A) Enhanced EPR, (B) Naive
+HJB, and (C) Normalizing Flow. The force field F (x) is decomposed into the gradient part −∇V (x; θ) (white arrows) and the
+non-gradient part (gray arrows). The length of an arrow denotes the scale of the vector. The solid dots are samples from the
+simulated invariant distribution.
+F.3.
+2D multi-stable model
+We also apply the enhanced approach to study the
+dynamics of a multi-stable system [5]
+dx
+dt =
+axn
+Sn + xn +
+bSn
+Sn + yn − k1x,
+(53)
+dy
+dt =
+ayn
+Sn + yn +
+bSn
+Sn + xn − k2y,
+(54)
+where the parameters are a = b = k1 = k2 = 1, S = 0.5,
+and n = 4. We focus on the domain Ω = [0, 3] × [0, 3]
+and present the results for D = 0.01 in the MT.
+F.4.
+Numerical comparisons
+In this subsection, we conduct a comparison study on
+the previous 2D problems to show the priority of our
+enhanced EPR approach over other methods.
+For the
+toy model, we have the analytical solution; while for the
+other two 2D examples, we take the reference solution as
+the numerical solution of the steady FPE by a piecewise
+bilinear finite element method with fine rectangular grids
+and the least squares solver for the obtained sparse lin-
+ear system (a normalization condition
+�
+Ω pss(x)dx = 1 is
+added to fix the extra shifting degree of freedom).
+We use a fully connected neural network with 3 lay-
+ers and 20 hidden states as the potential V (x; θ). We
+train the network with a batch size of 2048 and a learn-
+ing rate of 0.001 by the Adam [35] optimizer for 3000
+epochs. We simulate the SDEs by the Euler-Maruyama
+scheme with reflecting boundaries on the boundary of
+the domain and obtain a dataset of size 10000 to approx-
+imate the invariant distribution. We update the dataset
+by one time step at each training iteration to make it
+closer to the invariant distribution.
+In the toy model,
+we try different scales to enhance samples and report the
+best performance (when D′ = 2D) for naive HJB. For
+fairness, we use the same enhanced samples in enhanced
+EPR as naive HJB does. In SM, we denote the enhanced
+loss as λ1 LEPR +λ2 LHJB and use λ1 = 0.1, λ2 = 1.0 in
+the three models. We can also use Gaussian disturbances
+of the SDE data to obtain enhanced data, as we do in
+the limit cycle problem. We use D′ = 5D in the multi-
+stable problem for a better covering of the transition do-
+main.
+For the comparison with normalizing flows, we
+train a neural spline flow [36] using the implementation
+from [37]. We repeat 4 blocks of the rational quadratic
+spline with 3 layers of 64 hidden units and a followed LU
+linear permutation. We train the flow model by Adam of
+the learning rate 0.0001 for 20000 epochs, based on the
+same sample dataset as enhanced EPR.
+We shift the potential to the origin by its minimum
+and focus on the domain
+D = {x ∈ Ω|V (x; θ) ≤ 20D}.
+We then define the modified potential
+U m
+0 (x) := min(U0(x), 20D),
+V m(x; θ) := min(V (x; θ), 20D)
+for the shifted potential U0(x) and V (x; θ) since only the
+potential values in the domain D is of practical interest.
+We use the relative root mean square error (rRMSE) and
+the relative mean absolute error (rMAE) to describe the
+accuracy.
+rRMSE =
+��
+Ω |V m(x; θ) − U m
+0 (x)|2 dx
+�
+Ω |U m
+0 (x)|2dx
+,
+(55)
+rMAE =
+�
+Ω |V m(x; θ) − U m
+0 (x)| dx
+�
+Ω |U m
+0 (x)|dx
+.
+(56)
+We summarize the comparison of numerical errors for
+the 2D problems in Table I. The advantages of enhanced
+EPR over both naive HJB and normalizing flow can be
+identified from the following points.
+
+A
+B
+C
+3.0
+3.0
+3.0
+1.4
+1.4
+1.4
+2.5
+2.5
+2.5
+1.2
+1.2
+1.2
+2.0
+2.0
+2.0
+1.0
+1.0
+1.0
+0.8
+0.8
+0.8
+1.5
+1.5
+>1.5
+0.6
+0.6
+0.6
+1.0
+1.0
+1.0
+0.4
+0.4
+0.4
+0.5
+0.5
+0.5
+0.2
+0.2
+0.2
+0.0
+0.0
+0.0
+0.0
+0.0
+0.0
+1.5
+0.5
+1.0
+2.5
+0.5
+1.0
+1.5
+2.0
+2.5
+0.5
+1.5
+2.0
+3.0
+0.0
+0.0
+2.0
+2.5
+0.0
+3.0
+1.0
+3.0
+X
+X
+X13
+FIG. 5. Slices of the learned 3D potential V (x; θ) in the Lorenz system. The solid dots are samples from the simulated invariant
+distribution.
+TABLE I. Comparisons on Numerical Methods. We report
+the mean and the standard deviation over 3 random seeds.
+Problem
+Method
+rRMSE
+rMAE
+Toy, D=0.1
+Enhanced EPR
+0.027±0.012 0.023±0.011
+Naive HJB
+0.195±0.007 0.094±0.020
+Normalizing Flow 0.260±0.007 0.222±0.010
+Toy, D=0.05
+Enhanced EPR
+0.048±0.021 0.030±0.012
+Naive HJB
+0.237±0.020 0.142±0.042
+Normalizing Flow 0.284±0.028 0.231±0.030
+Limit Cycle
+Enhanced EPR
+0.052±0.039 0.029±0.016
+Naive HJB
+0.107±0.043 0.048±0.019
+Normalizing Flow 0.255±0.007 0.210±0.015
+Multi-stable
+Enhanced EPR
+0.040±0.008 0.022±0.005
+Naive HJB
+0.103±0.014 0.053±0.006
+Normalizing Flow 0.199±0.059 0.123±0.055
+• Without the guidance of EPR loss, naive HJB can
+not effectively optimize to the true solution with
+the heuristically chosen distribution. As shown in
+Table I, the enhanced EPR significantly achieves
+much better performances than naive HJB. Also,
+in the toy model with D = 0.05, naively training
+by HJB leads to an unreliable solution in Fig. 4(B)
+with relative errors larger than 0.1. Our computa-
+tional experiences show that the enhanced EPR is
+more robust than naive HJB and less sensitive to
+the enhanced data distribution and parameters.
+• The enhanced EPR converges faster than the naive
+HJB. For instance, in the toy model with D = 0.1,
+the enhanced EPR has achieved rRMSE of 0.087 ±
+0.069 and rMAE of 0.066 ± 0.013 in 2000 epochs,
+while the naive HJB can not attain the same level
+even after 3000 epochs.
+• Without information from the dynamics, the nor-
+malizing flow performs the worst only based on
+the simulated invariant distribution dataset. The
+learned potential tends to be rough and non-
+smooth at the edge of samples as shown in Fig. 4.
+Thus the enhanced EPR explicitly utilizing the in-
+formation of the force field does help in more accu-
+rate training of the potential.
+We further compare the potential landscape computed
+by different methods in Fig. 4. We remark that we omit
+the space {x|V (x) ≥ 30D} in both Fig. 3 and Fig. 4 since
+these domains are not of practical interest (their proba-
+bility is less than 10−9 according to the Gibbs form of
+the invariant distribution). The enhanced EPR presents
+the landscape more consistent with the simulated sam-
+ples and the true/reference solution than other methods.
+The decomposition of the force also shows better match-
+ing for the toy model. The normalizing flow captures the
+high probability domain but lacks information on the dy-
+namics, thus making its error larger than enhanced EPR
+and naive HJB.
+G.
+3D MODELS
+In this section, we describe the computational setup
+for the Lorenz system in three dimensions and Ferrell’s
+three-ODE model. We demonstrate the slices of the 3D
+
+20.0
+40
+17.5
+35
+15.0
+30
+12.5
+25
+Z
+10.0
+20
+7.5
+15
+5.0
+2.5
+10
+0.0
+-15 -10 -5
+12 1416
+0
+0
+2
+5
+10
+4 
+6
+X
+1514
+FIG. 6. Streamlines of the projected force ˜
+G(z) and filled contour plot of the reduced potential ˜V (z; θ) for Ferrell’s three-ODE
+model learned by enhanced EPR.
+potential for the former and conduct the proposed di-
+mensionality reduction on the latter.
+G.1.
+3D Lorenz system
+In this subsection, we apply our landscape construction
+approach to the 3D Lorenz system [38] with isotropic
+temporal Gaussian white noise.
+The Lorenz system has the form
+dx
+dt = β1(y − x),
+(57)
+dy
+dt = x (β2 − z) − y,
+(58)
+dz
+dt = xy − β3z,
+(59)
+where β1 = 10, β2 = 28 and β3 = 8
+3. We add the noise
+with strength D = 1. This model was also considered
+in [18] with D = 20.
+We obtain the enhanced data by adding Gaussian
+noises with standard deviation σ = 5 to the SDEs-
+simulation data.
+We directly train the 3D potential
+V (x; θ) by enhanced EPR with λ1 = 10.0, λ2 = 1.0
+and present a slice view of the landscape in Fig. 5. The
+learned 3D potential agrees well with the simulated sam-
+ples and shows a butterfly-like shape as the original sys-
+tem does.
+G.2.
+Ferrell’s three-ODE model
+In this subsection, we consider Ferrell’s three-ODE
+model for a simplified cell cycle dynamics [29] denoted
+by
+x = [CDK1], y = [Plk1], z = [APC]
+
+0.7
+0.200
+0.175
+0.6
+0.150
+0.5
+0.125
+0.4
+Plk1
+0.100
+0.3
+0.075
+0.2
+0.050
+0.1
+0.025
+0.0
+0.000
+0.0
+0.3
+0.5
+0.1
+0.2
+0.6
+0.4
+0.7
+CDK115
+for the concentration of CDK1, Plk1, and APC. We have
+the ODEs
+dx
+dt = α1 − β1x
+zn1
+Kn1
+1
++ zn1 ,
+(60)
+dy
+dt = α2 (1 − y)
+xn2
+Kn2
+2
++ xn2 − β2y,
+(61)
+dz
+dt = α3 (1 − z)
+yn3
+Kn3
+3
++ yn3 − β3z,
+(62)
+where α1 = 0.1, α2 = α3 = β1 = 3, β2 = β3 = 1, K1 =
+K2 = K3 = 0.5, n1 = n2 = 8, and n3 = 8. We add the
+noise scale D = 0.01 with isotropic temporal Gaussian
+white noise.
+By taking the reduced variables z = (x, y)⊤, we can
+apply our force projection loss and enhanced loss to learn
+the projected force ˜G(x) and potential ˜V (x; θ), and the
+results are shown in Fig. 6.
+We use three-layer net-
+works with 80 hidden states in this problem and en-
+hanced samples simulated from a more diffusive distribu-
+tion with D′ = 5D. We train the projected force ˜G(x)
+for 1000 epochs and then conduct enhanced EPR with
+λ1 = 0.1, λ2 = 1.0 for 4000 epochs to compute the pro-
+jected potential. The obtained reduced potential shows
+a plateau in the centering region and a local-well tube
+domain along the reduced limit cycle.
+H.
+HIGH DIMENSIONAL MODELS
+In this section, we apply our approach to 8D limit cy-
+cle dynamics [4] and 52D multistable dynamics [39]. We
+directly train the reduced force field ˜G(z) and poten-
+tial ˜V (z; θ) according to the selected reduction variables
+suggested in the corresponding literature. We use three-
+layer networks with 80 hidden states for both force and
+potential. The training strategies are similar to previous
+examples.
+H.1.
+8D complex system
+We consider an 8D system in which the dynamics and
+parameters are the same as the supporting information
+of [4], and take CycB and Cyc20 as the reduction variable
+z. We set the mass in this problem as m = 0.8.
+In [4], the noise strength D = 0.0005 is not suitable for
+direct neural network training since the scale of the po-
+tential is O(10−5). Borrowing the idea in Section F F.2,
+we amplify the original force field F considered in [4]
+by κ = 1000 times, and take D = 0.01 for the trans-
+formed force field. This amounts to set D = 10−5 for the
+original force field, which is even smaller than the case
+considered in [4]. We simulate the SDEs without bound-
+aries first and then fix the dataset without updating. We
+obtain the enhanced samples by adding Gaussian pertur-
+bations to the obtained dataset. Only the data within the
+FIG. 7. Streamlines and limit sets of the projected force field
+of the 8D cell cycle model by two reduced variables CycB and
+Cdc20. The outer red circle is the stable limit cycle of the
+reduced force field corresponding to the yellow circle as the
+projection of the original high dimensional limit cycle. The
+inner red circle, red dot and two green circles are stable and
+unstable limit sets of the reduced dynamics, which are virtual
+in high dimensions.
+biologically meaningful domain of [0, 1.5]8 is utilized for
+computation.
+We train the projected force ˜G(z; θ) for 5000 epochs
+and conduct the enhanced EPR with λ1 = 0.1, λ2 = 1.0
+for 10000 epochs. Some essential features of the reduced
+potential and dynamics on the plane have been presented
+in MT.
+In the SM Fig. 7, we present a more thorough picture
+of the reduced dynamics for the 8D model than the MT
+Fig. 2. To be more specific, we further show two unstable
+
+0.8
+0.6
+Cdc20
+0.4
+0.2
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+CycB16
+FIG. 8. Projected force ˜
+G(x) and potential ˜V (x; θ) of the 52D double-well model learned by enhanced EPR.
+limit cycles of the projected force field, two green circles
+obtained by reverse time integration, in SM Fig. 7. They
+fall between the outer and inner stable limit cycles (inner
+and outer red circles), and the inner stable limit cycle and
+inner stable node (red dot in the center), which play the
+role of separatrices between the neighboring stable limit
+sets. This picture occurs as the result that the landscape
+of the considered system in the centering region is very
+flat. These inner limit sets are virtual in high dimensions,
+but they naturally appear in the reduced dynamics on the
+plane. Similar features might also occur in other reduced
+dynamics in two dimensions.
+H.2.
+52D multi-stable system
+We also apply our approach to a biological system
+with 52 ODEs constructed by [39] and take GATA6 and
+NANOG as the reduction variable z. We define Ai as
+the set of indices for activating xi and Ri as the set of
+indices for repressing xi, the corresponding relationships
+are defined as the 52D node network shown in [39]. For
+i = 1, ..., 52,
+dxi
+dt = −kxi +
+�
+j∈Aj
+axn
+j
+Sn + xn
+j
++
+�
+j∈Rj
+bSn
+Sn + xn
+j
+,
+(63)
+where a = 0.37, b = 0.5, k = 1, S = 0.5, and n = 3. We
+choose the noise strength D = 0.01.
+We train the force ˜G(z; θ) for 500 epochs and conduct
+enhanced EPR with λ1 = 100.0, λ2 = 1.0 for 500 epochs.
+We use enhanced samples simulated from a more diffusive
+distribution with D′ = 5D.
+As shown in Fig. 8, the
+projected force demonstrate the reduced dynamics and
+the depth of the constructed potential agrees well with
+the density of the sample points.
+
+2.00
+0.200
+1.75
+0.175
+1.50
+0.150
+1.25
+0.125
+IANOG
+1.00
+0.100
+0.75
+0.075
+0.050
+0.50
+0.025
+0.25
+0.000
+0.00
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+GATA617
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+EZInterviewer: To Improve Job Interview Performance with
+Mock Interview Generator
+Mingzhe Li∗†
+Peking University
+[email protected]
+Xiuying Chen*
+CBRC, KAUST
+CEMSE, KAUST
+[email protected]
+Weiheng Liao
+Made by DATA
+[email protected]
+Yang Song
+BOSS Zhipin NLP Center
+[email protected]
+Tao Zhang
+BOSS Zhipin
+[email protected]
+Dongyan Zhao
+Peking University
+[email protected]
+Rui Yan‡
+Gaoling School of AI
+Renmin University of China
+[email protected]
+ABSTRACT
+Interview has been regarded as one of the most crucial step for
+recruitment. To fully prepare for the interview with the recruiters,
+job seekers usually practice with mock interviews between each
+other. However, such a mock interview with peers is generally far
+away from the real interview experience: the mock interviewers are
+not guaranteed to be professional and are not likely to behave like
+a real interviewer. Due to the rapid growth of online recruitment in
+recent years, recruiters tend to have online interviews, which makes
+it possible to collect real interview data from real interviewers. In
+this paper, we propose a novel application named EZInterviewer,
+which aims to learn from the online interview data and provides
+mock interview services to the job seekers. The task is challenging
+in two ways: (1) the interview data are now available but still of
+low-resource; (2) to generate meaningful and relevant interview
+dialogs requires thorough understanding of both resumes and job
+descriptions. To address the low-resource challenge, EZInterviewer
+is trained on a very small set of interview dialogs. The key idea is
+to reduce the number of parameters that rely on interview dialogs
+by disentangling the knowledge selector and dialog generator so
+that most parameters can be trained with ungrounded dialogs as
+well as the resume data that are not low-resource. Specifically, to
+keep the dialog on track for professional interviews, we pre-train
+a knowledge selector module to extract information from resume
+in the job-resume matching. A dialog generator is also pre-trained
+with ungrounded dialogs, learning to generate fluent responses.
+* Both authors contributed equally to this research.
+† Work done during an internship at BOSS Zhipin.
+‡ Corresponding author: Rui Yan ([email protected]).
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than the
+author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or
+republish, to post on servers or to redistribute to lists, requires prior specific permission
+and/or a fee. Request permissions from [email protected].
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+© 2023 Copyright held by the owner/author(s). Publication rights licensed to ACM.
+ACM ISBN 978-1-4503-9407-9/23/02...$15.00
+https://doi.org/10.1145/3539597.3570476
+Then, a decoding manager is finetuned to combine information
+from the two pre-trained modules to generate the interview ques-
+tion. Evaluation results on a real-world job interview dialog dataset
+indicate that we achieve promising results to generate mock in-
+terviews. With the help of EZInterviewer, we hope to make mock
+interview practice become easier for job seekers.
+CCS CONCEPTS
+• Computing methodologies → Natural language generation.
+KEYWORDS
+EZInterviewer, mock interview generation, knowledge-grounded
+dialogs, online recruitment, low-resource deep learning
+ACM Reference Format:
+Mingzhe Li, Xiuying Chen, Weiheng Liao, Yang Song, Tao Zhang, Dongyan
+Zhao, Rui Yan. 2023. EZInterviewer: To Improve Job Interview Performance
+with Mock Interview Generator. In Proceedings of the Sixteenth ACM Inter-
+national Conference on Web Search and Data Mining (WSDM ’23), February
+27-March 3, 2023, Singapore, Singapore. ACM, New York, NY, USA, 9 pages.
+https://doi.org/10.1145/3539597.3570476
+1
+INTRODUCTION
+To make better preparations, job seekers practice mock interviews,
+which aims to anticipate interview questions and prepare them
+for what they might get asked in their real turn. However, the
+outcome of such an approach is unsatisfactory, since those “mock
+interviewers” do not have interview experience themselves, and
+do not know what the real recruiters would be interested in. Mock
+Interview Generation (MIG) represents a plausible solution to this
+problem. Not only makes interviews more cost-effective, but mock
+interview generators also appear to be feasible, since much can be
+learned about the job seekers from their resumes, as can the job
+itself from the job description (JD). An illustration of MIG task is
+shown in Figure 1.
+There are two main challenges in this task. One is that the
+knowledge-grounded interviews are extremely time-consuming
+and costly to collect. Without a sufficient amount of training data,
+arXiv:2301.00972v1  [cs.CL]  3 Jan 2023
+
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+Mingzhe Li et al.
+Figure 1: An example of the Mock Interview Generation task.
+Based on the candidate’s work experience and the current di-
+alog on the experience of web page development, the system
+generates an interview question “If a product needs a three-
+level classification selection, which component would you
+use and how to achieve it?”.
+the performance of such dialog generation models drops dramati-
+cally [37]. The second challenge is to make the knowledge-grounded
+dialog relevant to the candidate resume, job description, and previ-
+ous dialog utterances. This makes MIG a complex task involving
+text understanding, knowledge selection, and dialog generation.
+In this paper, we propose EZInterviewer, a novel mock interview
+generator, with the aim of making interviews easier to prepare. The
+key idea is to train EZInterviewer in a low-resource setting: the
+model is first pre-trained on large-scale ungrounded dialogs and
+resume data, and then fine-tuned on a very small set of resume-
+grounded interview dialogs. Specifically, the knowledge selector
+consists of a resume encoder to encode the resume, and a key-value
+memory network with mask self-attention mechanism, responsible
+for selecting relevant information in the resume to focus on to help
+generate the next interview utterance. The dialog generator also
+has two components, a context encoder which encodes the current
+dialog context, and a response decoder, responsible for generating
+the next dialog utterance without knowledge from the resumes.
+This knowledge-insensitive dialog generator is coordinated with
+the knowledge selector by a decoding manager that dynamically
+determines which component is activated for utterance generation.
+It is noted that the number of parameters in the decoding man-
+ager can be small, therefore it only requires a small number of
+resume-grounded interview dialogs. Extensive experiments on real-
+world interview dataset demonstrate the effectiveness of our model.
+To summarize, our contributions are three-fold:
+• We introduce a novel Mock Interview Generation task, which
+is a pilot study of intelligent online recruitment with potential
+commercial values.
+• To address the low-resource challenge, we propose to reduce
+the number of parameters that rely on interview dialogs by dis-
+entangling knowledge selector and dialog generator so that the
+majority of parameters can be trained with large-scale ungrounded
+dialog and resume data.
+• We propose a novel model to jointly process dialog contexts,
+candidate resumes, and job descriptions and generate highly rele-
+vant, knowledge-aware interview dialogs.
+2
+RELATED WORK
+Multi-turn response generation aims to generate a response that is
+natural and relevant to the entire context, based on utterances in its
+previous turns. [36] concatenated multiple utterances into one sen-
+tence and utilized RNN encoder or Transformer to encode the long
+sequence, simplifying multi-turn dialog into a single-turn dialog.
+To better model the relationship between multi-turn utterances,
+[4, 10] introduced interaction between utterances after encoding
+each utterance.
+As human conversations are almost always grounded with exter-
+nal knowledge, the absence of knowledge grounding has become
+one of the major gaps between current open-domain dialog systems
+and real human conversations [8, 24, 35]. A series of work [20, 29]
+focused on generating a response based on the interaction between
+context and unstructured document knowledge, while a few oth-
+ers [22, 33] introduced knowledge graphs into conversations. These
+models, however, usually under-perform in a low-resource setting.
+To address the low resource problem, [16] proposed to enhance
+the context-dependent cross-lingual mapping upon the pre-trained
+monolingual BERT representations. [28] extended the meta-learning
+algorithm, which utilized knowledge learned from high-resource
+domains to boost the performance of low-resource unsupervised
+neural machine translation. Different from the above methods, [37]
+proposed a disentangled response decoder in order to isolate pa-
+rameters that depend on knowledge-grounded dialogs from the
+entire generation model. Our model takes a step further, taking
+into account the changes in attention on knowledge in multi-turn
+dialog scenarios.
+3
+MODEL
+3.1
+Problem Formulation
+For an input multi-turn dialog context 𝑈 = {𝑢1,𝑢2, . . . ,𝑢𝑚} be-
+tween a job candidate and an interviewer, where 𝑢𝑖 represents the
+𝑖-th utterance, we assume there is a ground truth textual interview
+question 𝑌 = {𝑦1,𝑦2, . . . ,𝑦𝑛}. 𝑚 is the utterance number in the
+dialog context and 𝑛 is the total number of words in question 𝑌.
+In the 𝑖-th utterance, 𝑢𝑖 = {𝑥𝑖
+1,𝑥𝑖
+2, . . . ,𝑥𝑖
+𝑇 𝑖𝑢 }. Meanwhile, there is a
+candidate resume 𝑅 = {(𝑘1, 𝑣1), (𝑘2, 𝑣2), . . . , (𝑘𝑇𝑟 , 𝑣𝑇𝑟 )} correspond-
+ing to the candidate in the interview, which has 𝑇𝑟 key-value pairs,
+and each of which represents an attribute in the resume. For the
+job-resume matching pre-training task, there is an external job
+description 𝐽 = {𝑗1, 𝑗2, . . . , 𝑗𝑇𝑗 }, which has 𝑇𝑗 words. The goal is to
+generate an interview question 𝑌
+′ that is not only coherent with the
+dialog context 𝑈 but also pertinent to the job candidate’s resume 𝑅.
+3.2
+System Overview
+In this section, we propose our Low-resource Mock Interview Gen-
+erator (EZInterviewer) model, which is divided into three parts as
+shown in Figure 2:
+
+O
+100%10:38
+100%10:47
+99%11:09
+MyOnlineResume
+Preview
+<
+<
+Mock Interview
+Web Front-end
+Expected Position
+Hello, I'm an undergraduate, and I
+Development Engineer
+am confident that I am qualified for
+Python 8-9K
+ Fresh graduate
+ Undergraduate
+the intern position of web front-end
+San Francisco
+engineer.I hope you can see my
+information.
+HR
+Work Experience
+Do you have experience in
+Job Description
+Company
+Emini programs or pc website
+2019.01-now>
+Web front-end development
+Web front-end development
+Webpack
+development?
+Content.
+GIT
+Gulp
+JavaScript
+Vue
+:Iusedtodevelopfront-endwebpagesbasedor
+Adjust the style of the system
+:HTML and CSS.
+I'm sorry that I have not done any
+back-stage on the PC front-end and call
+mini program work, but I used to
+Web front-end
+Vue
+Mini program
+develop Web page in the past.
+the interface to decelop a small part of
+thefunction
+Work closely with the back-end devel-
+Project Experience
+:Let's do a test. If a product needs a
+opment team to ensure limited code
+:three-level .classification. selection...
+which component would you use
+:docking,optimize front-end perfor-
+and how to achieve it?
+Education Experience
+④
+:mance, and participate in mobile inter-
+:face development and architecture
+university
+2019-2022
+:design;
+Message
+?
+Undergraduate
+Computer ScienceEZInterviewer: To Improve Job Interview Performance with Mock Interview Generator
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+Job Desc
+Job Encoder
+Resume
+Resume Encoder
+Multi-turn 
+Interview History
+Self Attention
+ Add & Norm
+Position-Wise FeedForward
+[CLS]
+[CLS]
+[CLS]
+...
+...
+...
+Utterance 
+States
+Masked Self Attention
+Cross Attention Manager
+Add & Norm
+Position-Wise FeedForward
+Decoder Input
+xN
+Linear
+Linear
+Linear
+Decoder 
+State
+Utterance 
+States
+Cross
+Attention
+Cross
+Attention
+xN
+updated output
+Fusion gate
+Pretrained 
+Soft Target 
+Updated 
+Distribution
+One-hot 
+Target 
+Masked 
+Self-attention
+Visible Matrix
+Cross
+Attention
+Transfer
+Memory
+Job-resume 
+Matching
+Knowledge Selector (Pretraining)
+Dialog Generator (Pretraining)
+Decoding Manager
+Self Attention
+Key-Value 
+Memory Network
+Figure 2: Overview of EZInterviewer, which consists of three parts: (1) Knowledge Selector selects salient knowledge infor-
+mation from the candidate resume; (2) Dialog Generator predicts the next word without knowledge of resumes; (3) Decoding
+Manager coordinates the output from knowledge selector and dialog generator to produce the interview question.
+• Dialog Generator predicts the next word of a response based on
+the prior sub-sequence. In our model, we pre-train it by large-scale
+ungrounded dialogs.
+• Knowledge Selector selects salient knowledge information from
+the candidate resume for interview question generation. In our
+model, we augment the ability of the knowledge selector by em-
+ploying it to perform job-resume matching.
+• Decoding Manager coordinates the output from knowledge
+selector and dialog generator to predict the interview question.
+It is important to note that to train an EZInterviewer model,
+two pre-train techniques are employed. Firstly, we pre-train the
+knowledge selector in a job-matching task. This is because while
+it is hard to attend to appropriate content in a resume just on its
+own, the salient information in a resume can be identified in a
+job-resume matching task [13, 34]. Secondly, the context encoder
+and response decoder of the dialog generator are pre-trained with
+a large scale of ungrounded dialogs, so as to predict the next word
+of response based on the prior sub-sequence. Finally, the decoding
+manager, which relies on a few parameters, coordinates the two
+components to generate knowledge grounded interview utterance.
+3.3
+Dialog Generator
+Context Encoder. Instead of processing the dialog context as a
+flat sequence, we employ a hierarchical encoder [3] to capture intra-
+and inter-utterance relations, which is composed of a local sentence
+encoder and a global context encoder. For the sentence encoder, to
+model the semantic meaning of the dialog context, we learn the
+representation of each utterance 𝑢𝑖 by a self-attention mechanism
+(SAM) initialized by BERT [5]:
+ℎ𝑖
+𝑗 = SAMu(𝑒(𝑥𝑖
+𝑗),ℎ𝑖
+∗).
+(1)
+We extract the state at “[cls]” position to denote the utterance state,
+abbreviated as ℎ𝑖. Apart from the local information exchange in
+each utterance, we let information flow across multi-turn context:
+ℎ𝑐
+𝑡 = SAMc(ℎ𝑡,ℎ𝑐
+∗),
+(2)
+where ℎ𝑐
+𝑡 denotes the hidden state of the 𝑡-th utterance in SAMc.
+Response Decoder. Response decoder is responsible for under-
+standing the previous dialog context and generates the response
+without the knowledge of resume information [19]. Our decoder
+also follows the style of Transformer.
+Concretely, we first apply the self-attention on the masked de-
+coder input, obtaining𝑑𝑡. Based on𝑑𝑡 we compute the cross-attention
+scores over previous utterances:
+𝛼𝑐
+𝑡 = ReLU([𝑑𝑡𝑊𝑑 (ℎ𝑐
+𝑖𝑊ℎ)𝑇 ]).
+(3)
+The attention weights 𝛼𝑐
+𝑡 is then used to obtain the context vectors
+as𝑐𝑡 = �𝑚
+𝑖=1 𝛼𝑐
+𝑡 ℎ𝑐
+𝑖 . The context vectors𝑐𝑡, treated as salient contents
+of various sources, are concatenated with the decoder hidden state
+𝑑𝑡 to produce the distribution over the target vocabulary:
+𝑃𝑤
+𝑣 = Softmax (𝑊𝑜 [𝑑𝑡;𝑐𝑡]) .
+(4)
+Pre-training process. While interview dialogs are hard to come
+by, online conversation is abundant on the internet, and can be
+easily collected. Hence, we pre-train the dialog generator on un-
+grounded conversations. Concretely, during pre-training process,
+we employ the context encoder to first encode the multi-turn pre-
+vious dialog context. Then, at the 𝑡-th decoding step, we use the
+response decoder to predict the 𝑡-th word in the response. We set
+the loss as the negative log likelihood of the target word 𝑦𝑡:
+𝐿𝑜𝑠𝑠𝑔 = − 1
+𝑛
+�𝑛
+𝑡=1 log 𝑃𝑤
+𝑣 (𝑦𝑡).
+(5)
+3.4
+Knowledge Selector
+Resume Encoder. As shown in Figure 2, a resume contains several
+key-value pairs (𝑘𝑖, 𝑣𝑖). Most of key and value fields include a single
+word or a phrase such as “skills” or “gender”, and we can obtain the
+feature representation through an embedding matrix. Concretely,
+for each key or value field with a single word or a phrase, we estab-
+lish a corresponding resume embedding matrix 𝑒𝑖𝑟 that is different
+from the previous one. Then we use the resume embedding matrix
+to map each field word 𝑘𝑖 or 𝑣𝑖 into to a high-dimensional vector
+space, denoted as 𝑒𝑖𝑟 (𝑘𝑖) or 𝑒𝑖𝑟 (𝑣𝑖). For fields with more than one
+word such as “work experience” or “I used to...”, we denote them as
+𝑣𝑖 = (𝑣1
+𝑖 , ...𝑣𝑙𝑖
+𝑖 ), where 𝑙𝑖 denotes the word number of the current
+
+Birthday
+19980501
+Gender
+MaleWSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+Mingzhe Li et al.
+field. We first process them through the previous word embedding
+matrix 𝑒, then there is an SAMR, similar with SAMu in Section 3.3,
+to model the temporal interactions between words:
+ℎ𝑟𝑖
+𝑡 = SAMR(𝑒(𝑣 𝑗
+𝑖 ),ℎ𝑟𝑖
+𝑡−1).
+(6)
+We use the last hidden state of the SAMR, i.e., ℎ𝑟𝑖
+𝑙𝑖 to denote the
+overall representation for field 𝑣𝑖.
+For brevity, in the following sections, we use ℎ𝑘
+𝑖 and ℎ𝑣
+𝑖 to denote
+the encoded key-value pair (𝑘𝑖, 𝑣𝑖) in the resume.
+Masked Self-attention. Traditional self-attention can be used
+to update representation of each resume item due to its flexibility in
+relating two elements in a distance-agnostic manner [17]. However,
+as shown in [21], too much knowledge incorporation may divert the
+representation from its correct meaning, which is called knowledge
+noise (KN) issue. In our scenario, the information in the resume
+is divided into several parts, i.e., basic personal information, work
+experiences and extended work, each of which contains variable
+number of items. The items within each part are closely connected,
+while different parts can be considered as different domains, and
+the interaction may introduce a certain amount of noise. To over-
+come this problem, we introduce a visible matrix, in which items
+belonging to the same part are visible to each other, while the visi-
+bility degree between items is determined by the cosine similarity
+of semantic representations, i.e., 𝐶𝑖,𝑗 = cos_sim(ℎ𝑣
+𝑖 ,ℎ𝑣
+𝑗 ). Then, the
+scaled dot-product masked self-attention is defined as:
+𝛼𝑖,𝑗 =
+exp
+�
+(ℎ𝑘
+𝑖 𝑊𝑞)𝐶𝑖,𝑗 (ℎ𝑘
+𝑗𝑊𝑘)𝑇 �
+�𝑇𝑟
+𝑛=1 exp
+�
+(ℎ𝑘
+𝑖 𝑊𝑞)𝐶𝑖,𝑛(ℎ𝑘
+𝑗𝑊𝑘)𝑇
+� ,
+(7)
+ˆℎ𝑣
+𝑖 =
+∑︁𝑇𝑟
+𝑗=1
+𝛼𝑖,𝑗ℎ𝑣
+𝑗
+√
+𝑑
+,
+(8)
+where 𝑑 stands for hidden dimension and 𝐶 is the visible matrix.
+ˆℎ𝑣
+𝑖 is then utilized as the updated resume value representation.
+Key-Value Memory Network. The goal of key matching is to
+calculate the relevance between each attribute of the resume and
+the previous dialog context. Given dialog context ℎ𝑖, for the 𝑗-th
+attribute pair (𝑘𝑗, 𝑣𝑗), we calculate the probability of ℎ𝑖 over 𝑘𝑗,
+i.e., 𝑃(𝑘𝑗 |ℎ𝑖), as the matching score 𝛽𝑖,𝑗. To this end, we exploit the
+context representation ℎ𝑖 to calculate the matching score:
+𝛽𝑖,𝑗 =
+exp
+�
+ℎ𝑖𝑊𝑎ℎ𝑘
+𝑗
+�
+�𝑇𝑟
+𝑛=1 exp
+�
+ℎ𝑖𝑊𝑎ℎ𝑘𝑛
+� .
+(9)
+Since context representation ℎ𝑖 and resume key representation ℎ𝑘
+𝑗
+are not in the same semantic space, we use a trainable key matching
+parameter𝑊𝑎 to transform these representations into a same space.
+As the relevance between context ℎ𝑖 and each pair in the resume
+table (𝑘𝑗, 𝑣𝑗), the matching score 𝛽𝑖,𝑗 can help to capture the most
+relevant pair for generating a correct question. Therefore, as shown
+in Equation 10, the knowledge selector reads the information 𝑀𝑖
+from KVMN via summing over the stored values, and guides the
+follow-up response generation, so we have:
+𝑀𝑖 =
+∑︁𝑇𝑟
+𝑗=1 𝛽𝑖,𝑗 ˆℎ𝑣
+𝑗,
+(10)
+where ˆℎ𝑣
+𝑗 is the representation of value 𝑣𝑗, and 𝛽𝑖,𝑗 is the matching
+score between dialog context ℎ𝑖 and key 𝑘𝑗.
+Pre-training Process. In practice, the resume knowledge con-
+tains a variety of professional and advanced scientific concepts such
+as “Web front-end”, “HTML”, and “CSS”. These technical terms are
+difficult to understand for people not familiar with the specific
+domain, not to mention for the model that is not able to access
+a large-scale resume-grounded dialog dataset. Hence, it would be
+difficult for the knowledge selector to understand the resume con-
+tent and previous context about the resume, so as to select the next
+resume pair to focus on.
+On the other hand, we notice that in job-resume matching task,
+it is crucial to capture the decisive information in the resume to
+perform a good matching. For example, recruiters may tend to hire
+the candidate with particular experiences among several candidates
+with similar backgrounds [34]. Intuitively, the key-value pair that is
+important for job-resume matching is also the key factor to consider
+in a job interview. Hence, if we can let the model learn the salient
+information in the resume by performing the job-resume matching
+task on large-scale job-resume data, then it would also bring benefits
+for selecting salient information in interview question generation.
+Concretely, we use the job description to attend to the resume to
+perform a job-resume matching task, as a pre-training process for
+knowledge selector module. As shown in Figure 2, the Job Encoder
+encodes the job description by a SAMjd:
+ℎ𝑗𝑑
+𝑖
+= SAMjd(𝑒(𝑗𝑖),ℎ𝑗𝑑
+𝑖−1),
+(11)
+where 𝑗𝑖 denotes the 𝑖-th word in the job description, and 𝑒(𝑗𝑖)
+is mapped by the previous embedding matrix 𝑒. We use the final
+hidden state of the SAMjd, i.e., ℎ𝑗𝑑
+𝑇𝑗 as the overall representation
+for the description, abbreviated as ℎ𝑗𝑑. ℎ𝑗𝑑 plays a similar part as
+the context representation ℎ𝑖, which first attends to the keys in the
+resume, and then is used to “weightedly” read the values in the
+resume. We use 𝑚𝑗𝑑 to denote the weighted read result.
+In the training process, we first pre-train the knowledge selector
+by job-resume matching task, which can be formulated as a classi-
+fication problem [26]. The objective is to maximize the scores of
+positive samples while minimizing that of the negative samples.
+Specifically, we concatenateℎ𝑗𝑑 and𝑚𝑗𝑑 since vector concatenation
+for matching is known to be effective [27]. Then the concatenated
+vector is fed to a multi-layer, fully-connected, feed-forward neural
+network, and the job-resume matching score 𝑠𝑗𝑟 is obtained as:
+𝑠𝑗𝑟 = 𝜎
+�
+𝐹𝑠 ([ℎ𝑗𝑑;𝑚𝑗𝑑]])
+�
+,
+(12)
+where [; ] denotes concatenation operation, and the outputs are the
+probabilities of successfully matching. We use the job-resume pairs
+in interviews as positive samples, and then use the job-resume pairs
+without interviews as negative instances.
+After pre-training, the job description is replaced by the context
+representations, while the key matching and value combination
+processes remain the same. We use a knowledge memory 𝑀 to store
+the selection result, where each slot stores the value combination
+result 𝑀𝑖 in Equation 10.
+
+EZInterviewer: To Improve Job Interview Performance with Mock Interview Generator
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+3.5
+Decoding Manager
+The decoding manager is supposed to generate the proper word
+based on the knowledge memory and the response decoder. Our
+idea is inspired by an observation on the nature of interview dialogs:
+despite the fact that a dialog is based on the resume, words and utter-
+ances in the dialog are not always related to resume. Therefore, we
+postulate that formation of a response can be decomposed into two
+uncorrelated actions: (1) selecting a word according to the context
+to make the dialog coherent (corresponding to the dialog generator);
+(2) selecting a word according to the extra knowledge memory to
+ground the dialog (corresponding to the knowledge selector). The
+two actions can be independently performed, which becomes the
+key reason why the large resume-job matching and ungrounded
+dialog datasets, although seemingly unrelated to interview dialogs,
+can be very useful in an MIG task.
+Note that in Section §3.4, we store the selected knowledge 𝑀𝑖 in
+a knowledge memory 𝑀. To select a word based on it, similar to
+the response decoder, we use 𝑑𝑡 to attend to each slot of knowledge
+memory, and we can obtain the knowledge context vector 𝑔𝑘
+𝑡 and
+the output decoder state 𝑑𝑘𝑜
+𝑡 .
+The response decoder and knowledge selector are controlled by
+the decoding manager with a “fusion gate” to decide how much
+information from each side should be focused on at each step of
+interview question prediction.
+𝛾𝑓 = 𝜎 (𝐹𝑚(𝑑𝑡)) ,
+(13)
+where 𝑑𝑡 is the 𝑡-th decoder hidden state. Then, the probability to
+predict word 𝑦𝑡 can be formulated as:
+𝑑𝑜
+𝑡 = 𝛾𝑓 𝑑𝑤𝑜
+𝑡
++ (1 − 𝛾𝑓 )𝑑𝑘𝑜
+𝑡 ,
+(14)
+𝑃𝑣 = softmax �𝑊𝑣𝑑𝑜
+𝑡 + 𝑏𝑣
+� .
+(15)
+As for the optimization goal, generation models that use one-
+hot distribution optimization target always suffer from the over-
+confidence issue, which leads to poor generation diversity [32].
+Hence, aside from the ground truth one-hot label 𝑃, we also propose
+a soft target label 𝑃𝑤
+𝑣 (see Equation 4), which is borrowed from the
+pre-trained Dialog Generator in Section 3.3. Forcing the decoding
+manager to simulate the pre-trained decoder can help it learn the
+context of the interview dialog. We combine the one-hot label with
+the soft label by an editing gate 𝜆, as shown in Figure 2. Concretely,
+a smooth target distribution 𝑃 ′ is proposed to replace the hard
+target distribution 𝑃 as:
+𝑃 ′ = 𝜆𝑃 + (1 − 𝜆)𝑃𝑤
+𝑣 .
+(16)
+where 𝜆 ∈ [0, 1] is an adaption factor, 𝑃𝑤
+𝑣 is obtained from Equa-
+tion 4, and 𝑃 is the hard target as one-hot distribution which assigns
+a probability of 1 for the target word 𝑦𝑡 and 0 otherwise.
+4
+EXPERIMENTAL SETUP
+4.1
+Dataset
+In this paper, we conduct experiments on a real-world dataset pro-
+vided by “Boss Zhipin” 1, the largest online recruiting platform
+in China. To protect the privacy of candidates, user records are
+anonymized with all personal identity information removed. The
+1https://www.zhipin.com
+Table 1: Statistics of the datasets used in the experiments.
+Statistics
+Values
+Interview Dialog Dataset
+Total number of resumes
+12,666
+Total number of dialog utterances
+49,214
+Avg turns # per dialog context
+4.47
+Avg words # per utterance
+13.18
+Job-Resume Dataset
+Key-value pairs # per resume
+22
+Avg words # per work experience in resume
+72.80
+Avg words # per self description in resume
+51.13
+Avg words # per job description
+74.26
+Ungrounded Dialog Dataset
+Total number of context-response pairs
+2,995,000
+Avg turns # per dialog context
+4
+Avg words # per utterance
+15.15
+dataset includes 12,666 resumes, 8,032 job descriptions, and 49,214
+interview dialog utterances. The statistics of the dataset is summa-
+rized in Table 1. We then tokenize each sentence into words with
+the benchmark Chinese tokenizer toolkit “JieBa” 2.
+To pre-train the knowledge selector module, we use a job-resume
+matching dataset [34], again from “Boss Zhipin”. The training
+set and the validation set include 355,000 and 1,006 job-resume
+pairs, respectively. To pre-train dialog generator, we choose Weibo
+dataset [2], which includes a massive number of multi-turn con-
+versations collected from “Weibo”3. The data includes 2,990,000
+context-response pairs for training and 5,000 pairs for validation.
+The details are also summarized in Table 1.
+4.2
+Comparisons
+We compare our proposed model against traditional knowledge-
+insensitive dialog generation baselines, and knowledge-aware dia-
+log generation baselines.
+• Knowledge-insensitive dialog generation baselines:
+Transformer [30]: is based solely on attention mechanisms.
+BERT [5]: initializes Transformer with BERT as the encoder. Di-
+aloGPT [36]: proposes a large, tunable neural conversational re-
+sponse generation model trained on more conversation-like ex-
+changes. T5-CLAPS [14]: generates samples for contrastive learn-
+ing by adding small and large perturbations, respectively.
+• Knowledge-aware dialog generation baselines:
+TMN [6]: is built upon a transformer architecture with an ex-
+ternal memory hosting the knowledge. ITDD [20]: incrementally
+encodes multi-turn dialogs and knowledge and decodes responses
+with a deliberation technique. DiffKS [38]: utilizes the differential
+information between selected knowledge in multi-turn conversa-
+tion for knowledge selection. DRD [37]: tackles the low-resource
+challenge with pre-training techniques using ungrounded dialogs
+and documents. DDMN [31]: dynamically keeps track of dialog
+context for multi-turn interactions and incorporates KB knowledge
+2https://github.com/fxsjy/jieba
+3https://www.weibo.com
+
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+Mingzhe Li et al.
+Table 2: Comparing model performance on full dataset: automatic evaluation metrics.
+BLEU-1
+BLEU-2
+BLEU-3
+BLEU-4
+Extrema
+Average
+Greedy
+Dist-1
+Dist-2
+Entity F1
+Cor
+Knowledge-insensitive dialog generation
+Transformer [30]
+0.5339
+0.3811
+0.2836
+0.2530
+0.4859
+0.7673
+0.6803
+0.0928
+0.3157
+0.3606
+0.2711
+BERT [5]
+0.5671
+0.3864
+0.2735
+0.2583
+0.4861
+0.7669
+0.6792
+0.0947
+0.3558
+0.3711
+0.2894
+DialoGPT [36]
+0.5722
+0.4015
+0.3004
+0.2697
+0.4858
+0.7670
+0.6814
+0.1001
+0.3620
+0.3843
+0.3002
+T5-CLAPS [14]
+0.5846
+0.4126
+0.3020
+0.2783
+0.4837
+0.7851
+0.6674
+0.0970
+0.3702
+0.3549
+0.2870
+Knowledge-aware dialog generation
+TMN [6]
+0.5437
+0.3891
+0.2963
+0.2630
+0.4841
+0.7655
+0.6811
+0.0996
+0.3299
+0.3830
+0.2652
+ITDD [20]
+0.5484
+0.4009
+0.2929
+0.2656
+0.4833
+0.7650
+0.6859
+0.1055
+0.3703
+0.3661
+0.2715
+DiffKS [38]
+0.5617
+0.3898
+0.2776
+0.2441
+0.4826
+0.7830
+0.6752
+0.0937
+0.3612
+0.3672
+0.2750
+DRD [37]
+0.5711
+0.4001
+0.2914
+0.2548
+0.4824
+0.7813
+0.6783
+0.0867
+0.3661
+0.3825
+0.2883
+DDMN [31]
+0.5693
+0.4065
+0.2968
+0.2694
+0.4831
+0.7655
+0.6811
+0.0944
+0.3640
+0.3754
+0.2869
+Persona [9]
+0.5532
+0.3829
+0.2715
+0.2377
+0.4823
+0.7822
+0.6783
+0.0911
+0.3598
+0.3833
+0.2928
+EZInterviewer
+0.6106
+0.4320
+0.3284
+0.2917
+0.4893
+0.7884
+0.6886
+0.1071
+0.3747
+0.3927
+0.3145
+No Pre-train
+0.5738
+0.4029
+0.2929
+0.2599
+0.4846
+0.7833
+0.6831
+0.0981
+0.3673
+0.3819
+0.3007
+w/o KM
+0.5795
+0.4127
+0.3069
+0.2754
+0.4847
+0.7841
+0.6762
+0.0979
+0.3685
+0.3803
+0.3010
+w/o KS
+0.5775
+0.4122
+0.3067
+0.2746
+0.4781
+0.7668
+0.6787
+0.1003
+0.3691
+0.3848
+0.2994
+w/o LS
+0.6007
+0.4232
+0.3176
+0.2821
+0.4869
+0.7863
+0.6832
+0.0969
+0.3664
+0.3902
+0.3127
+into generation. Persona [9]: introduces personal memory into
+knowledge selection to address the personalization issue.
+4.3
+Implementation Details
+We implement our experiments in TensorFlow [1] on an NVIDIA
+GTX 1080 Ti GPU. For our model and all baselines, we follow the
+same setting as described below. We truncate input dialog to 100
+words with 20 words in each utterance, as we did not find significant
+improvement when increasing input length from 100 to 200 tokens.
+The minimum decoding step is 10, and the maximum step is 20.
+The word embedding dimension is set to 128 and the number of
+hidden units is 256. Experiments are performed with a batch size
+of 256, and the vocabulary is comprised of the most frequent 50k
+words. We use Adam optimizer [12] as our optimizing algorithm.
+We selected the 5 best checkpoints based on performance on the
+validation set and report averaged results on the test set. Note that
+for better performance, our model is built based on BERT, and the
+decoding process is the same as Transformer [30]. Finally, due to
+the limitation of time and memory, small settings are used in the
+pre-trained baselines.
+4.4
+Evaluation Metrics
+To evaluate the performance of EZInterviewer against baselines,
+we adopt the following metrics widely used in existing studies.
+Overlap-based Metric. Following [18], we utilize BLEU score
+[25] to measure n-grams overlaps between ground-truth and gener-
+ated response. In addition, we apply Correlation (Cor) to calculate
+the words overlap between generated question and job description,
+which measures how well the generated questions line up with the
+recruitment intention.
+Embedding Metrics. We compute the similarity between the
+bag-of-words (BOW) embeddings of generated results and reference
+to capture their semantic matching degrees [11]. In particular we
+adopt three metrics: 1) Greedy, i.e., greedily matching words in two
+Table 3: Human evaluation results on: Readability (Read),
+Informativeness (Info), Meaningfulness (Mean), Usefulness
+(Use), Relevance (Rel), and Coherence (Coh).
+Model
+Dialog-level
+Interview-level
+Read
+Info
+Mean
+Use
+Rel
+Coh
+DiffKS
+1.79
+2.01
+1.87
+2.03
+1.99
+2.10
+DDMN
+1.97
+1.83
+1.63
+2.12
+2.14
+1.91
+DRD
+2.05
+2.11
+2.09
+2.08
+2.17
+2.02
+EZInterviewer
+2.42▲
+2.51▲
+2.39▲
+2.46▲
+2.57▲
+2.38▲
+utterances based on cosine similarities; 2) Average, cosine similarity
+between the averaged word embeddings in two utterances [23];
+3) Extrema, cosine similarity between the largest extreme values
+among the word embeddings in the two utterances [7].
+Distinctness. The distinctness score [15] measures word-level
+diversity by calculating the ratio of distinct uni-gram and bi-grams
+in generated responses.
+Entity F1. Entity F1 is computed by micro-averaging precision
+and recall over knowledge-based entities in the entire set of sys-
+tem responses, and evaluates the ability of a model to generate
+relevant entities to achieve specific tasks from the provided knowl-
+edge base [31]. The entities we use are extracted from an entity
+vocabulary provided by “Boss Zhipin”.
+Human Evaluation Metrics. We further employ human eval-
+uations aside from automatic evaluations. Three well-educated
+annotators from different majors are hired to evaluate the quality
+of generated responses, where the evaluation is conducted in a
+double-blind fashion. In total 100 randomly sampled responses gen-
+erated by each model are rated by each annotator on both dialog
+level and interview level. We adopt the Readability (is the response
+grammatically correct?) and Informativeness (does the response
+include informative words?) to judge the quality of the generated
+
+EZInterviewer: To Improve Job Interview Performance with Mock Interview Generator
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+responses on the dialog level. On the interview level, we adopt
+Meaningfulness (is the generated question meaningful?), Usefulness
+(is the question worth the job candidate preparing in advance?),
+Relevance (is the question relevant to the resume?) and Coherence (is
+the generated text coherent with the context?) to assess the overall
+performance of a model and the quality of user experience. Each
+metric is given a score between 1 and 3 (1 = bad, 2 = average, 3 =
+good).
+5
+EXPERIMENTAL RESULT
+5.1
+Overall Performance
+Automatic evaluation. The comparison between EZInterviewer
+and state-of-the-art generative baselines is listed in Table 2.
+We take note that the knowledge-aware dialog generation mod-
+els outperform traditional dialog models, suggesting that utilizing
+external knowledge introduces advantages in generating relevant
+response. We also notice the pre-train based model DRD outper-
+forms other baselines, showing that initializing parameters by pre-
+training on large-scale data can lead to a substantial improvement
+in performance. It is worth noting some models achieve better En-
+tity F1 but a lower BLEU score; this suggests that those models tend
+to copy necessary entity words from the knowledge but are not
+able to use them properly.
+EZInterviewer outperforms baselines on all automatic metrics.
+Firstly, our model improves BLEU-1 by 6.92% over DRD. On the Dis-
+tinctness metric Dist-1, our model outperforms DialoGPT by 6.99%,
+suggesting that the generated interview questions are diversified
+and personalized with different candidates’ resumes. Moreover our
+model attains a good score of 0.3927 on entity F1, which evaluates
+the degree to which the generated question is grounded on the
+knowledge base. Finally, Cor score of 0.3145 suggests the ques-
+tions generated by EZInterviewer is in line with the job description,
+hence reflect the intention of the recruiters. Overall the metrics
+demonstrate that our model successfully learns an interviewer’s
+points of interest in a resume, and incorporates this knowledge into
+interview questions properly.
+Human evaluation. The results of human evaluations on all
+models are listed in Table 3. EZInterviewer is the top performer on
+all the metrics. Specifically, our model outperforms DiffKS by 35.20%
+on Readability, suggesting that EZInterviewer manages to reduce
+the grammatical errors and improve the readability of the generated
+response. As for the Informativeness metric, our model scores 0.68
+higher than DDMN. This indicates that EZInterviewer captures
+salient information in the resume. On the interview level, EZInter-
+viewer’s Usefulness score is 18.27% better than DRD, demonstrating
+its capabilities to help job seekers to pick the right questions to
+prepare. On Relevance metric, our model outperforms all baselines
+by a considerable margin, suggesting that the generated questions
+are closely related to the interview process. Our model also per-
+forms better than other baselines in Meaningfulness and Coherence
+metrics, suggesting the overall higher quality of our model.
+The above results demonstrate the competence of EZInterviewer
+in producing meaningful and useful interview questions whilst
+keeping the interview dialog flowing smoothly, just like a human
+recruiter. Note that the average kappa statistics of human evaluation
+are 0.51 and 0.48 on dialog level and interview level, respectively,
+Figure 3: Visualization of key matching between dialog con-
+text and selected resume keys, i.e., work experiment (Exp),
+self description (Desc), skills (Ski), work years (Year), ex-
+pected position (Pos), school (Sch), and major (Maj). 𝑈𝑖 de-
+notes the 𝑖-th utterance.
+which indicates moderate agreement between annotators. To prove
+the significance of these results, we also conduct the two-tailed
+paired student t-test between our model and DRD (row with shaded
+background). The statistical significance of observed differences is
+denoted using ▲(or ▼) for strong (or weak) significance for 𝛼 = 0.01.
+Moreover, we obtain an average p-value of 5 × 10−6 and 3 × 10−4
+for both levels, respectively.
+5.2
+Ablation Study
+We conduct an ablation study to assess the contribution of individ-
+ual components in the model. The results are shown in Table 2.
+To verify the effectiveness of knowledge memory, we omit the
+knowledge selection of dialog context history and directly use the
+last utterance representation to select knowledge. The results (see
+row w/o KM) confirm that employing each turn of historical dialog
+to select knowledge and saving it in memory contribute to gener-
+ating better responses. To confirm whether selecting knowledge
+helps with the response generation process, we remove it from the
+model, then simply add the representation of each utterance with
+all resume values, and store it into the memory. This results in a
+drop of 5.42% in BLEU-1 (see row w/o KS), suggesting that selecting
+resume knowledge is beneficial in response generation.
+5.3
+Analysis of Knowledge Selector
+In Section § 3.4, we introduce the selecting mechanism of knowl-
+edge selector, where the final attention (matching) score is obtained
+in Equation 9. To study what specific information is attended by
+the knowledge selector, and whether the selected information is
+suitable for the next interview question, we conduct a case study to
+visualize the matching score produced by the knowledge selector,
+as shown in Table 4 and Figure 3. The first utterance in the history
+is “Have you been engaged in front-end development work before?”,
+and the knowledge selector learns that this utterance focuses on the
+work experience in the resume. Accordingly, the fourth utterance “I
+have more than 10 years of work experience.” pays more attention
+to work years and work experience than other items in the resume.
+This demonstrates that the knowledge selector learns which item
+in the resume to focus on when generating each utterance. Hence,
+when we want to ask the candidate to “introduce a React related
+project”, the knowledge selector focuses on the work experience in
+the resume and generates the mock interview question.
+
+U1
+U2
+U3
+U4
+Ski
+Sch
+Exp
+Year
+Desc
+Pos
+MajWSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+Mingzhe Li et al.
+Table 4: Translated interview questions generated by baselines and EZInterviewer: an example.
+denotes information
+extracted and words generated by knowledge selector, whereas
+denotes words generated from dialog generator.
+Resume
+Interview
+Gender
+Male
+Job Description:
+The main content of this work includes design and development based on the React
+front-end framework. It requires the ability to efficiently complete front-end
+development work and serve customers well.
+Age
+28
+Education
+Undergraduate
+Major
+Computer Science
+Work Years
+10
+Context:
+U1: Have you been engaged in front-end development work before?
+U2: Yes, I am good at Vue, Node.js and some other skills.
+U3: Okay, so do you have any React related experience?
+U4: Yes, I have more than 10 years of work experience.
+Expected Position
+Front-end Engineer
+Low Salary
+5
+High Salary
+6
+Skills
+Vue, Node.js, Java
+Experience
+I was engaged in front-end design and was re-
+sponsible for the project development based
+on the React front-end framework and par-
+ticipated in the system architecture process.
+Ground Truth: So can you introduce a React related project you have done?
+DDMN: What other front-end frameworks would you use?
+DRD: Hello, can you tell us about your previous work?
+EZInterviewer: Well, can you introduce the experience based on React framework?
+Figure 4: Automatic evaluation metrics of DDMN, DRD and EZInterviewer on training data of different scales.
+5.4
+Impact of Training Data Scales
+To understand how our model and baseline models perform in a low-
+resource scenario, we first evaluate them on the full training dataset,
+then on smaller portions of the training dataset. Figure 4 presents
+the performance of the models, DDMN, DRD, and EZInterviewer,
+on the full, 1/2, 1/4, 1/8 and 1/10 of the training dataset (data scale),
+respectively. It is observed that as the size of training dataset re-
+duces, DDMN suffers a massive drop across all metrics, whereas the
+scores of pre-training based models, i.e., DRD and EZInterviewer,
+stay relatively stable. This demonstrates pre-training as an effective
+strategy to tackle the low-resource challenge. Moreover, our model
+outperforms DRD on all data scales, demonstrating the superiority
+of our model. Figure 4 shows EZInterviewer eventually achieves the
+best performance on all metrics and outperforms (albeit slightly),
+with only 1/10 training data against all state-of-the-art baselines
+trained with the full training dataset.
+5.5
+Case Study
+Table 4 presents a translated example of EZInterviewer and baseline
+models. We observe that the question from EZInterviewer not only
+catches the context, but also expands the conversation with proper
+knowledge. This is highlighted in color codes: pink-colored words,
+i.e., “experience” and “React framework”, are what knowledge selec-
+tor extracts from resume knowledge, whereas blue-colored words,
+i.e., “Well, can you introduce the...” and “based on”, which closely
+connect to the context, are generated by dialog generator. In con-
+trast, the questions from the baselines respond to the dialog but fail
+to make connection with the resume knowledge.
+6
+CONCLUSION
+In this paper, we conduct a pilot study for the novel application of
+intelligent online recruitment, namely EZInterviewer, which aims
+to serve as mock interviewers for job-seekers. The mock interview
+is generated with thorough understanding of the candidate’s re-
+sume, the job requirements, the previous utterances in the context,
+as well as the selected knowledge for grounded interviews. To ad-
+dress the low-resource challenge, EZInterviewer is trained on a very
+small set of interview dialogs. The key idea is to reduce the number
+of parameters that rely on interview dialogs by disentangling the
+knowledge selector and dialog generator so that most parameters
+can be trained with ungrounded dialogs as well as the resume data
+that are not low-resource. We conduct extensive experiments to
+demonstrate the effectiveness of the proposed solution EZInter-
+viewer. Our model achieves the best results using full training data
+as well as small subsets of the training data in terms of various
+metrics such as BLEU, embedding based similarity and diversity,
+as well as human judgments. In particular, the human evaluation
+indicates that our solution EZInterviewer can provide satisfactory
+mock interviews to help the job-seekers prepare the real interview,
+making the interview preparation process easier.
+ACKNOWLEDGMENTS
+We would like to thank the anonymous reviewers for their con-
+structive comments. This work was supported by National Natural
+Science Foundation of China (NSFC Grant No. 62122089). Rui Yan
+is supported by Beijing Academy of Artificial Intelligence (BAAI).
+
+0.600
+0.575
+Score
+0.550
+0.525
+BLEU
+0.500
+DDMN
+0.475
+DRD
+0.450
+EZInterviewer
+1
+1/2
+1/4
+1/8
+1/10
+Data Scale0.105
+: Score
+0.100
+Distinct 
+0.095
+0.090
+DDMN
+DRD
+0.085
+EZinterviewer
+i
+1/2
+1/4
+1/8
+1/10
+Data ScaleEmbedding Score
+0.78
+0.76
+DDMN
+0.74
+DRD
+EZInterviewer
+0.72
+0.70
+i
+1/2
+1/4
+1/8
+1/10
+Data Scale0.39
+0.38
+score
+0.37
+S
+0.36
+Entity
+0.35
+0.34
+DDMN
+DRD
+0.33
+EZInterviewer
+0.32
+i
+1/2
+1/4
+1/8
+1/10
+Data ScaleScore
+0.31
+0.30
+Correlation s
+0.29
+0.28
+DDMN
+DRD
+0.27
+EZlnterviewer
+i
+1/2
+1/4
+1/8
+1/10
+Data ScaleEZInterviewer: To Improve Job Interview Performance with Mock Interview Generator
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+REFERENCES
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+

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+arXiv:2301.05057v1  [q-bio.QM]  19 Dec 2022
+AN OVERVIEW OF OPEN SOURCE DEEP LEARNING-BASED
+LIBRARIES FOR NEUROSCIENCE
+Louis Fabrice Tshimanga
+Department of Neuroscience (DNS)
+University of Padova
+[email protected]
+Manfredo Atzori
+Department of Neuroscience (DNS),
+Padova Neuroscience Center (PNC)
+University of Padova
+Information Systems Institute
+University of Applied Sciences Western Switzerland (HES-SO Valais)
+[email protected]
+Federico Del Pup
+Department of Neuroscience (DNS),
+Department of Information Engineering (DEI)
+University of Padova
+[email protected]
+Maurizio Corbetta
+Department of Neuroscience (DNS),
+Padova Neuroscience Center (PNC)
+University of Padova
+Department of Neurology
+Washington University School of Medicine
+[email protected]
+ABSTRACT
+In recent years, deep learning revolutionized machine learning and its applications, producing re-
+sults comparable to human experts in several domains, including neuroscience. Each year, hundreds
+of scientific publications present applications of deep neural networks for biomedical data analysis.
+Due to the fast growth of the domain, it could be a complicated and extremely time-consuming task
+for worldwide researchers to have a clear perspective of the most recent and advanced software
+libraries. This work contributes to clarify the current situation in the domain, outlining the most
+useful libraries that implement and facilitate deep learning application to neuroscience, allowing
+scientists to identify the most suitable options for their research or clinical projects. This paper
+summarizes the main developments in Deep Learning and their relevance to Neuroscience; it then
+reviews neuroinformatic toolboxes and libraries, collected from the literature and from specific hubs
+of software projects oriented to neuroscience research. The selected tools are presented in tables
+detailing key features grouped by domain of application (e.g. data type, neuroscience area, task),
+model engineering (e.g. programming language, model customization) and technological aspect
+(e.g. interface, code source). The results show that, among a high number of available software
+tools, several libraries are standing out in terms of functionalities for neuroscience applications. The
+aggregation and discussion of this information can help the neuroscience community to devolop
+their research projects more efficiently and quickly, both by means of readily available tools, and by
+knowing which modules may be improved, connected or added.
+Keywords Deep Learning · Neuroscience · Neuroinformatics · Open source
+1
+Introduction
+In the last decade, Deep Learning (DL) has taken over most classic approaches in Machine Learning (ML), Computer
+Vision, Natural Language Processing, showing an unprecedented versatility, and matching or surpassing the perfor-
+mances of human experts in narrow tasks.
+The recent growth of DL applications to several domains, including Neuroscience, consequently offers numerous open-
+
+source software opportunities for researchers.
+Mapping available resources can allow a faster and more precise exploitation.
+Neuroscience is a diversified field on its own, as much for the objects and scales it focuses on, as for the types of data
+it relies on.
+The discipline is also historically tied to developments in electrical, electronic, and information technology. Modern
+Neuroscience relies on computerization in many aspects of data generation, acquisition, and analysis. Statistical and
+Machine Learning techniques already empower many software packages, that have become de facto standards in sev-
+eral subfields of Neuroscience, such as Principal and Independent Component Analysis in Electroencephalography
+and Neuroimaging, to name a few.
+Meanwhile, the rich and rapidly evolving taxonomy of Deep Neural Networks (DNNs) is becoming both an opportu-
+nity and hindrance. On the one hand, currently open-source DL libraries allow an increasing number of applications
+and studies in Neuroscience. On the other hand, the adoption of available methods is slowed down by a lack of stan-
+dards, reference frameworks and established workflows. Scientific communities whose primary focus or background
+is not in machine learning engineering may be left partially aside from the ongoing Artificial Intelligence (AI) gold
+rush.
+For such reasons it is fundamental to overview open-source libraries and toolkits. Framing a panorama could help
+researchers in selecting ready-made tools and solutions when convenient, as well as in pointing out and filling in the
+blanks with new applications. This work would contribute to advancing the community’s possibilities, reducing the
+workload for researchers to exploit DL, allowing Neuroscience to benefit of its most recent advancements.
+2
+Background
+2.1
+Deep Learning
+Deep Learning (DL) has contributed many best solutions to problems in its parent field, Machine Learning, thanks to
+theoretical and technological achievements that unlocked its intrinsic versatility.
+Machine Learning is the study of computer algorithms that tackle problems without complete access to predefined
+rules or analytical, closed-form solutions.
+The algorithms often require a training phase to adjust parameters and satisfy internal or external constraints (e.g. of
+exactness, approximation or generality) on dedicated data for which solutions might be already known.
+Machine Learning comprises a wide array of statistical and mathematical methods, including Artificial Neural Net-
+works (ANNs), biologically inspired systems that connect inputs and outputs through simple computing units (neu-
+rons), which act as function approximators.
+Each unit implements a nonlinear function of the weighted sum of its inputs, thus the output of the whole ANN is a
+composite function, as formally intended in mathematics. The networks of neurons are most often layered and "feed-
+forward", meaning that units from any layer only output results to units in subsequent layers. The width of a layer
+refers to its neuron count, while the depth of a network refers to its layer count. The typical architecture instantiating
+the above characteristics is the MultiLayer Perceptron [1] (MLP).
+Universal approximation theorems [2] [3] ensure that, whenever a nonlinear network as the MLP is either bound in
+width and unbound in depth or viceversa, its weights can then be set to represent virtually any function (i.e. a wide
+variety of functions families).
+The training problem thus consists in building networks with sets of weights so to instantiate or approximate the func-
+tion that would solve the assigned task, or that represents the input-output relation. This search is not trivial: it can
+be framed as the optimization problem for a functional over the ANN weights. Such functional, typically called "loss
+function", associates the "errors" made on the training data to the neural net parameters (its weights), acting as a total
+performance score. Approaching local minima of the loss function and improving the network performance on the
+training data is the prerequisite to generalize on real world and unseen data.
+DL is concerned with the use of deep ANNs, namely characterized by depth, stacking several intermediate, (hidden)
+layers between input and output units.
+As mentioned above, other dimensions being equal, depth increases the representational power of ANNs and, more
+specifically, aims at modeling complicated functions as meaningful compositions of simpler ones.
+As with their biological counterparts [4], depth is supposed to manage hierarchies of features from larger input por-
+tions, capturing characteristics often inherent to real world objects and effective in modeling actual data.
+Overall, depth is one of the key features that allowed to overcome historical limits [5] of simpler ANNs such as the
+Perceptron. At the same time, depth comes with numerical and methodological hardships in models training.
+Part of the difficulties arise as the search space for the optimal set of parameters grows considerably with the number
+of layers (and their width as well).
+Other issues are strictly numerical, since the training algorithms include long computation chains that may affect the
+stability of training and learning.
+2
+
+Hence, new or rediscovered ideas in training protocols and mathematical optimization (e.g. applying the "backpropa-
+gation of errors" algorithm to neural nets [6]) played an important role through times when the scientific interest and
+hopes in ANNs faded (so called "AI winters"), paving the way for later advancement.
+The main drivers for the latest success of deep neural networks are of varied nature, and can be schematised as techni-
+cal and human related factors.
+On a technical side DL has profited from [7]:
+• the datafication of the world, i.e. the growing availability of (Big) data
+• the diffusion of Graphical Processing Units (GPUs) as hardware tools.
+To outperform classic machine learning models, deep neural networks often require larger quantities of data samples.
+Such data hunger and high parameters count contribute to the high requirements of deep models in terms of memory,
+number of operations and computation time. Training models with highly parallelized and smartly scheduled compu-
+tations gained momentum thanks to GPUs.
+In 2012 a milestone exemplified both the above technical aspects, when AlexNet [8], a deep Convolutional Neural
+Network (CNN) based on ideas from Fukushima [4] and LeCun [9] - [10], won the ImageNet Large Scale Visual
+Recognition Challenge after being trained using two GPUs [11]. Since then, Deep Learning has brought new out-
+standing results in various tasks and domains, processing different data types. Deep networks can nowadays work on
+image, video, audio, text, and speech data, time series and sequences, graphs, and more; the main tasks consist in
+classification, prediction, or estimating the probability density of data distributions, with the possibility of modifying,
+completing the input or even generating new instances.
+On a more sociological side, the drivers of Deep Learning success can be related to the synergy of big tech companies,
+advanced research centers, and developer communities [12]. Investments of economical and scientific resources in
+relatively independent, collective projects, such as open-source libraries, frameworks, and APIs (Application Program-
+ming Interfaces), have offered varied tools adapted to multiple specific situations and objectives, exploiting horizontal
+organization [13] and mixing top-down and bottom-up approaches. It is difficult to imagine a rapid rise of successful
+endeavors, without both active communities and the technical means to incorporate and manage lower-level aspects.
+In fact, applying Deep Learning to a relevant problem in any research field requires, in addition to specific domain
+knowledge, a vast background of statistical, mathematical, and programming notions and skills. The tools that support
+scientists and engineers in focusing on their main tasks encompass the languages to express numerical operations on
+GPUs, such as CUDA [14] and cuDNN [15] by NVIDIA, as well as the frameworks to design models, like Tensor-
+Flow [16] and Keras [17] by Google, and PyTorch by Meta [18], or the supporting strategies to build data pipelines.
+Many Deep Learning achievements are relevant to biomedical and clinical research, and the above presented tools
+have enabled explorations of the capabilities of deep neural networks with neuroscience and biomedical data.
+A fuller exploitation and routinely employment of modern algorithms are yet to come, both in research and clinical
+practice. This process would accelerate by popularizing, democratizing, and jointly developing models, improving
+their usability, and expanding their environments, i.e. by wrapping solutions into libraries and shared frameworks.
+2.2
+Neuroscience
+As per the Nature journal, «Neuroscience is a multidisciplinary science that is concerned with the study of the structure
+and function of the nervous system. It encompasses the evolution, development, cellular and molecular biology,
+physiology, anatomy and pharmacology of the nervous system, as well as computational, behavioural and cognitive
+neuroscience» [19].
+Expanding, neuroscience investigates:
+• the evolutionary and individual development of the nervous system;
+• the cellular and molecular biology that characterizes neurons and glial cells;
+• the physiology of living organisms and the role of the nervous system in the homeostatic function;
+• the anatomy, i.e. the identification and description of the system’s structures;
+• pharmacology, i.e. the effect of chemicals of external origin on the nervous system, their interactions with
+endogenous molecules;
+• the computational features of the brain and nerves, how information is processed, which mathematical and
+physical models best predict and approximate the behaviour of neurons;
+• cognition, the mental processes at the intersection of psychology and computational neuroscience;
+• behaviour as a phenomenon rooted in genetics, development, mental states, and so forth.
+3
+
+The techniques to access tissues and structures of the nervous system are often shared by disciplines focused on other
+physiological systems, and some of these processes have been computer aided for long.
+Moreover, nerve cells have distinctive electromagnetic properties and their activity directly and indirectly generates
+detectable signals, adding physical and technical specificity to Neuroscience.
+Overall, neuroscience research is profoundly multi-modal. Data are managed and processed inside a model depending
+on their type and format. The most prominent categories of data involved in neuroscience research comprise 2,3-D
+images or video on the one side, and sequences or signals on the other. Still it is important to acknowledge the differ-
+ent phenomena, autonomous or provoked by the measurement apparatus, underlying data generation and acquisition.
+Bioimages may be produced from:
+• Magnetic Resonance Imaging (MRI)
+• X-rays
+• Tomography with different penetrating waves
+• Histopathology microscopy
+• Fundus photography (retinal images)
+and more.
+Neuroscience sequences may come from:
+• Electromiography (EMG)
+• Electroencephalography (EEG)
+• Natural language, text records
+• Genetic sequencing
+• Eye-tracking
+and more.
+Adding to the above, other data types are common in neuroscience, e.g. tabular data, text that may come from
+medical records written by physicians for diagnostic purposes, test scores, inspections of cognitive and sensorimotor
+functions, as the National Institute of Health (NIH) Stroke Scale test scores [20], and more broadly clinical reports
+from anamneses or surveys.
+2.3
+Neuroinformatics
+Neuroscience is evolving into a data-centric discipline. Modern research heavily depends on human researchers as
+well as machine agents to store, manage and process computerized data from the experimental apparatus to the end
+stage.
+Before delving in the specifics of artificial neural networks applied to the study of biological neural systems, it is
+useful to outline the broader concepts of Neuroinformatics, regarding data and coding, especially in the light of open
+culture.
+According to the International Neuroinformatics Coordinating Facility (INCF), «Neuroinformatics is a research field
+devoted to the development of neuroscience data and knowledge bases together with computational models and ana-
+lytical tools for sharing, integration, and analysis of experimental data and advancement of theories about the nervous
+system function. In the INCF context, neuroinformatics refers to scientific information about primary experimental
+data, ontology, metadata, analytical tools, and computational models of the nervous system. The primary data includes
+experiments and experimental conditions concerning the genomic, molecular, structural, cellular, networks, systems
+and behavioural level, in all species and preparations in both the normal and disordered states» [21]. Given the rele-
+vance of Neuroinformatics to Neuroscience, supporting open and reproducible science implies and requires attention
+to standards and best practices regarding open data and code.
+The INCF itself is an independent organization devoted to validate and promote such standards and practices, inter-
+acting with the research communities [22] and aiming at the "FAIR principles for scientific data management and
+stewardship" [23].
+FAIR principles consist in:
+• being Findable, registered and indexed, searchable, richly described in metadata;
+• being Accessible, through open, free, universally implementable protocols;
+• being Interoperable, with appropriate standards for metadata in the context of knowledge representation;
+4
+
+• being Reusable, clearly licensed, well described, relevant to a domain and meeting community standards.
+Among free and open resources, several software and organized packages integrating pre-processing and data analysis
+workflows for neuroimaging and signal processing became the reference for worldwide researchers in Neuroscience.
+Such tools allow to perform scientific research in neuroscience easily in solid and repeatable ways. It can be useful to
+mention, for neuroimaging, Freesurfer1 [24] and FSL2 [25] that are standalone softwares, and the MATLAB-connected
+SPM3 [26]. In the domain of signal processing, examples are EEGLAB4 [27], Brainstorm5 [28], PaWFE6 [29], all
+MATLAB related yet free and open, and MNE7 [30], that runs on Python. Regarding applications for neurorobotics
+and Brain Computer Interfaces (BCIs), a recent opensource platform can be found in ROS-neuro8 [31].
+The interested readers can find lists of open resources for computational neuroscience (including code, data, mod-
+els, repositories, textbooks, analysis, simulation and management software) at Open Computational Neuroscience
+Resource 9 (by Austin Soplata), and at Open Neuroscience 10. Additional software resources oriented to Neuroinfor-
+matics in general, but not necessarily open, can also be found as indexed at "COMPUTATIONAL NEUROSCIENCE
+on the Web" 11 (by Jim Perlewitz).
+2.4
+Bringing Deep Learning to the Neurosciences
+The Deep Learning community is accustomed to open science, as many datasets, models, programming frameworks
+and scientific outcomes are publicly released by both academia and companies continuously. However, while Deep
+Learning can openly provide state-of-the-art models to old and new problems in Neuroscience, theoretical understand-
+ing, formalization and standardisation are often yet to be achieved, which may prevent adoption in other research
+endeavors. From a technical standpoint, deep networks are a viable tool for many tasks involving data from the brain
+sciences. Image classification has arguably been the task in which deep neural networks have had the highest mo-
+mentum, in terms of pushing the state of the art forward. This translates now in a rich taxonomy of architectures
+and pre-trained models that consistently maintain interesting performances in pattern recognition, across a number of
+image domains.
+Pattern recognition is indeed central for diagnostic purposes, in the form of classification of images with pathological
+features (e.g. types of brain tumors or meningiomas), segmentation of structures (such as the brain, brain tumors
+or stroke lesions), classification of signals (e.g. classification of electromyography or electro encephalography data),
+as well as for action recognition in Human-Computer Interfaces (HCIs). The initiatives BRain Tumor Segmentation
+(BRATS) Challenge12 [32], Ischemic Stroke LEsion Segmentation (ISLES) Challenge13 [33]- [34], and Ninapro14 [35]
+are examples of data releases for which above-mentioned tools proved effective.
+There are models learning image-to-image functions, capable of enhancing data, preprocessing it, correcting artifacts
+and aberrations, allowing smart compression as well as super-resolution, and even expressing cross-modal transforma-
+tions between different acquisition apparatus.
+In the related tasks of object tracking, action recognition and pose estimation, research results from the automotive
+sector or crowd analysis have inspired solutions for behavioural neuroscience, especially in animal behavioral studies.
+When dealing with sequences, deep networks success in Computer Vision has inspired CNN-based approaches to EEG
+and EMG studies [36] - [37], either with or without relying on 2D data, given that mathematical convolution has a 1D
+version, and 1D signals have 2D spectra. Other architectures more directly instantiate temporal and sequential aspects,
+e.g. Recurrent Neural Networks (RNNs) such as the Long Short Term Memory (LSTM) [38] and Gated Recurrent
+Units (GRUs) [39], and they too can be applied to sequence problems and sub-tasks in neuroscience, such as decoding
+time-dependent brain signals.
+Although deep neural network do not explicitly model the nervous system, they are inspired by biological knowledge
+and mimic some aspects of biological computation and dynamical systems. This has inspired new comparative studies,
+1https://surfer.nmr.mgh.harvard.edu/
+2https://fsl.fmrib.ox.ac.uk/fsl/fslwiki
+3https://www.fil.ion.ucl.ac.uk/spm/
+4https://sccn.ucsd.edu/eeglab/index.php
+5https://neuroimage.usc.edu/brainstorm/Introduction
+6http://ninapro.hevs.ch/node/229
+7https://mne.tools/stable/index.html
+8https://github.com/rosneuro
+9https://github.com/asoplata/open-computational-neuroscience-resources
+10https://open-neuroscience.com/
+11https://compneuroweb.com/sftwr.html
+12https://www.med.upenn.edu/cbica/brats/
+13https://www.isles-challenge.org/
+14http://ninaweb.hevs.ch/node/7
+5
+
+and analogy approaches to learning and perception, in a unique way among machine learning algorithms [40].
+Many neuroinformatic studies demonstrate how novel deep learning concepts and methods apply to neurological
+data [12]. However, they often showcase new further achievements in performance metrics that do not translate di-
+rectly to new accepted neuroscience discoveries or clinical best practices.
+Such results are very often published together with open code repositories, allowing reproducibility, yet they may not
+be explicitly organized for widespread routinely adoption in domains different from machine learning. Algorithms are
+usually written in open programming languages like Python [41], R [42], Julia [43], and deep learning design frame-
+works such as TensorFlow, PyTorch or Flux [44]. Still, they are more inspiring to the experienced machine learning
+researcher, rather than practically helpful to end-users such as neuroscientists.
+In fact, to successfully build a deep learning application from scratch, a vast knowledge is needed in the data science
+aspect of the task and in coding , as much as in the theoretical and experimental foundations and frontiers of the
+application domain, here being Neuroscience.
+For the above reasons, the open source and open science domains are promising frames for common development
+and testing of relevant solutions for Neuroscience, as they provide an active flow of ideas and robust diversification,
+avoiding "reinvention of the wheel", harmful redundancies or starting from completely blank states.
+As a contribution in clarifying the current situation and reducing the workload for researchers, this work collects and
+analyzes several open libraries that implement and facilitate Deep Learning application in Neuroscience, with the aim
+of allowing worldwide scientists to identify the most suitable options for their inquiries and clinical tasks.
+3
+Methods
+The large corpus of available open code makes useful to specify what qualifies as a coding library or a framework,
+rather than as a model accompanied by utilities, for the present scope.
+In programming, a library is a collection of pre-coded functions and object definitions, often relying on one another,
+and written to optimize programming for custom tasks. The functions are considered useful and unmodified across
+multiple unrelated programs and tasks. The main program at hand calls the library, in the control flow specified by the
+end-users.
+A framework is a higher level concept, akin to the library, but typically with a pre-designed control flows in which
+custom code from the end-users is inserted.
+For instance, a repository that simply collects the functions that define and instantiate a deep model would not be
+considered a library. On the other hand, collections of notebooks that allow to train, retrain and test models with
+several architectures, while possibly taking care also of data pre-processing and preparation, would be considered
+libraries (and frameworks) for the present scopes.
+The explicit definition of the authors, their aims and their
+maintainance of the library is relevant as well, in determining if a repository would be considered a library, toolkit,
+toolbox, or other.
+For the sake of the review, several resources were queried or scanned. Google Scholar was queried with:
+• "deep learning library" OR "deep learning toolbox" OR "deep learning package" -"MATLAB deep learning
+toolbox" -"deep learning toolbox MATLAB"
+preserving the top 100 search results, ordered for relevance by the engine algorithm. On PubMed the queries were:
+• opensource (deep learning) AND (toolbox OR toolkit OR library);
+• (EEG OR EMG OR MRI OR (brain (X-ray OR CT OR PT))) (deep learning) AND (toolbox OR toolkit OR
+library).
+Moreover, the site https://open-neuroscience.com/ was scanned specifically for "deep learning" mentions, and
+relevant papers cited or automatically suggested throughout the query process were considered for evaluation, as well
+as the platform of the Journal of Open Source Software at https://joss.theoj.org/.
+The collected libraries were organized according to the principal aim, in the form of data type processed, or the
+supporting function in the workflow, thus dividing:
+1. libraries for sequence data (e.g. EMG, EEG)
+2. libraries for image data (including scalar volumes, 4-dimensional data as in fMRI, video)
+3. libraries and frameworks to support model building, evaluation, data ingestion
+In each category, a set of three tables present separately the results related to the following libraries characteristics:
+6
+
+1. domain of application
+2. model engineering
+3. technology and sources
+The domain of application comprises the Neuroscience area, the Data types handled, the provision of Datasets, and
+the machine learning Task to which the library is dedicated.
+The model engineering tables include informations on the architecture of DL Models manageable in the library, the
+DL framework and Programming language main dependencies, and the possibility of Customization for the model
+structure or training parameters.
+Technology and sources refer to the type of Interface available for a library, whether it works Online//Offline, specif-
+ically with real-time data or logged data. Maintenance refers to the ongoing activity of releasing features, solving
+issues and bugs or offering support through channels, Source specifies where code files and instructions are made
+available.
+4
+Results: Deep Learning Libraries
+The analysis of the literature allowed to select a total of 48 publications describing libraries that implement or em-
+power deep learning applications for neuroscience. Despite open source and effectiveness, several publications did not
+provide an ecosystem of reusable functions. Proofs of concept and single-shot experiments were discarded.
+4.1
+Libraries for sequence data
+Libraries and frameworks for sequence data are shown in Tables 1 (domains of application), 2 (models characteris-
+tics), 3 (technologies and sources). The majority of process EEG sygnals, which are among the most common types
+of sequential data in Neuroscience research. A common objective is deducing the activity or state of the subject,
+based on temporal or spectral (2D) patterns. Deep Learning is capable of bypassing some of the preprocessing steps
+often required by other common statistical and engineering techniques, and it comprises both 1D and 2D approaches,
+through MLPs, CNNs or RNNs architectures. BioPyC is an example of such scenario. It offers the possibility to
+train a pre-set CNN architecture as well as loading and training a custom model. Moreover, It can process different
+types of sequence data, making it very versatile and applicable/ suitable/usable in/for different neuroscience area.
+Another example of sequence-oriented library is gumpy, whose intended area of application is that of Brain Computer
+Interfaces (BCIs), where decoding a signal is the first step towards communication and interaction with a computer
+or robotic system. Given the setting, gumpy allows working with EEG or EMG data and suits them with specific
+defaults, e.g. 1-D CNNs, or LSTMs.
+Notable mentions in the sequence category are the library Traja and the VARDNN toolbox, as they depart from
+the common scenarios of previous examples. Traja stands out as an example of less usual sequential data, namely
+trajectory data (sequences of coordinates in 2 or 3 dimensions, through time). Moreover, in Traja sequences are
+modeled and analyzed employing the advanced architectures of Variational AutoEncoders (VAEs) and Generative
+Adversarial Networks (GANs), usually encountered in image tasks. With different theoretical backgrounds, both
+architectures allow simulation and characterization of data through their statistical properties. The VARDNN toolbox
+allows analyses on BOLD signals, in the established domain of functional Magnetic Resonance Imaging (fMRI), but
+uses a unique approach to autoregressive processes mixed with deep neural networks, allowing to perform causal
+analysis and to study functional connections between brain regions through their patterns of activity in time.
+7
+
+Name
+Neuroscience
+area
+Data type
+Datasets
+Task
+BioPyC [45]
+General
+Sequences (EEG, miscellaneous)
+No
+Classification
+braindecode [46]
+General
+Sequences (EEG, MEG)
+External
+Classification
+DeLINEATE [47]
+General
+Images, sequences
+External
+Classification
+EEG-DL [48]
+BCI
+Sequences (EEG)
+No
+Classification
+gumpy [49]
+BCI
+Sequences (EEG, EMG)
+No
+Classification
+DeepEEG
+Electrophysiology
+Sequences (EEG)
+No
+Classification
+ExBrainable [50]
+Electrophysiology
+Sequences (EEG)
+External
+Classification, XAI
+Traja [51]
+Behavioural
+neuro-
+science
+Sequences (Trajectory coordinates over time)
+No
+Prediction, Classification, Synthesis
+VARDNN toolbox [52] toolbox
+Connectomics
+(Functional
+Connectiv-
+ity)
+Sequences (BOLD signal)
+No
+Time series causal analysis
+Table 1: Domains of applications for the libraries and frameworks processing sequence data
+8
+
+Name
+Models
+DL framework
+Customization
+Programming language
+BioPyC
+1-D CNN
+Lasagne
+Yes (weights, model)
+Python
+braindecode
+1-D CNN
+PyTorch
+Yes (weights, model)
+Python
+DeLINEATE
+CNN
+Keras, TensorFlow
+Yes (weights, model)
+Python
+EEG-DL
+Miscellaneous
+TensorFlow
+Yes (weights, model)
+Python, MATLAB
+gumpy
+CNN, LSTM
+Keras, Theano
+Yes (weights, model)
+Python
+DeepEEG
+MLP, 1,2,3-D CNN, LSTM
+Keras, TensorFlow
+Yes (weights)
+Python
+ExBrainable
+CNN
+PyTorch
+Yes (weights)
+Python
+Traja
+LSTM, VAE, GAN
+PyTorch
+Yes (weights, model)
+Python
+VARDNN toolbox
+Vector Auto-Regressive DNN
+Deep Learning Toolbox (MATLAB)
+Yes (weights)
+MATLAB
+Table 2: Model engineering specifications for the libraries and frameworks processing sequence data
+9
+
+Name
+Interface
+Online/Offline
+Maintenance
+Source
+BioPyC
+Jupyter Notebooks
+Offline
+Active
+gitlab.inria.fr/biopyc/BioPyC/
+braindecode
+None
+Offline
+Active
+github.com/braindecode/braindecode
+DeLINEATE
+GUI, Colab Notebooks
+Offline
+Active
+bitbucket.org/delineate/delineate
+EEG-DL
+None
+Offline
+Active
+github.com/SuperBruceJia/EEG-DL
+gumpy
+None
+Online, Offline
+Inactive
+github.com/gumpy-bci
+DeepEEG
+Colab Notebooks
+Offline
+Inactive
+github.com/kylemath/DeepEEG
+ExBrainable
+GUI
+Offline
+Active
+github.com/CECNL/ExBrainable
+Traja
+None
+Offline
+Active
+github.com/traja-team/traja
+VARDNN toolbox
+None
+Offline
+Active
+github.com/takuto-okuno-riken/vardnn
+Table 3: Technological aspects and code sources for the libraries and frameworks processing sequence data
+10
+
+4.2
+Libraries for image data
+Libraries and frameworks for image data are shown in Tables 4 (domains of application), 5 (models characteristics),6
+(technologies and sources). Computer vision and 2D image processing are arguably the fields in which DL has
+achieved the most impressive and state-of-art defining results, often inspiring and translating breakthroughs in other
+domanis. Classification and segmentation (i.e. the separation of parts of the image based on their classes) are the
+most common tasks addressed by the image processing libraries. Magnetic resonance is the primary source of data;
+however, various deep learning libraries are built microscopic and eye-tracking data as well. Most of the libraries
+collected in our analysis take advantage of classical CNN architectures for classification, Convolutional AutoEncoders
+(CAEs) for segmentation, and GANs for synthesis. It is common to employ transfer learning to lessen the compu-
+tational and memory burden during the training phase, and take advantage of pre-trained models. Transfer learning
+consists in initializing models with parameters learnt on usually larger data sets, possibly from different domains and
+tasks, with varying amounts of further training in the target domain. The best such examples are pose-estimation
+libraries extending the DeepLabCut system, arguably the most relevant project on the topic. DeepLabCut is an
+interactive framework for labelling, training, testing and refining models, that originally exploits the weights learned
+from ResNets (or newer architectures) on the ImageNet data. The results match human annotation using quite few
+training samples, holding for many (human and non-human) animals, and settings. The documentation and demon-
+strative notebooks and tools offered by the Mathis Lab allow different levels of understanding and customization of
+the process, with high levels of robustness. Among the considered libraries, two set apart from the majority given
+the type of tasks they perform: GaNDLF addresses eXplainable AI (XAI), i.e. Artificial Intelligence whose deci-
+sions and outputs can be understood by humans through more transparent mental models; ANTsX performs both the
+co-registration step and super-resolution as a quality enhancing step for neuroimages, with the former being usually
+performed by traditional algorithms. GaNDLF sets its goal as the provision of deep learning resources in different
+layers of abstraction, allowing medical researchers with virtually no ML knowledge to perform robust experiments
+with models trained on carefully split data, with augmentations and preprocessing, under standardized protocols that
+can easily integrate interpretability tools such as Grad-CAM [53] and attention maps, which highlight the parts of
+an image according to how they influenced a model outcome. The ANTsX ecosystem is of similar wide scope, and
+is intended to build workflows on quantitative biology and medical imaging data, both in Python and R languages.
+Packages from the same ecosystem perform registration of brain structures (by classical methods) as well as brain
+extraction by deep networks.
+11
+
+Name
+Neuroscience
+area
+Data type
+Datasets
+Task
+AxonDeepSeg [54]
+Microbiology,
+Histology
+Img (SEM, TEM)
+External
+Segm.
+DeepCINAC [55]
+Electrophys.
+Vid (2-photon calcium)
+No
+Class.
+DeepLabCut [56]
+Behavioral
+neuroscience
+Vid
+No
+Pose est.
+DeepNeuro [57]
+Neuroimaging
+Img (fMRI, misc.)
+No
+Class., Segm., Synthesis
+DeepVOG [58]
+Oculography
+Img, Vid
+Demo
+Segm.
+DeLINEATE [47]
+General
+Img, sequences
+External
+Class.
+DNNBrain [59]
+Brain
+map-
+ping
+Img
+No
+Class.
+ivadomed [60]
+Neuroimaging
+Img (2D, 3D)
+No
+Class., Segm.
+MEYE [61]
+Oculography
+Img, Vid
+Yes
+Segm.
+Allen Cell Structure Segmenter [62]
+Microbiology,
+Histology
+Img (3D-fluor. microscopy)
+No
+Segm.
+VesicleSeg [63]
+Microbiology,
+Histology
+Img (EM)
+No
+Segm.
+CDeep3M2 [64]
+Microbiology,
+Histology
+Img (misc. microscopy)
+Yes
+Segm.
+CASCADE [65]
+Electrophys.
+Vid (2-photon calcium), Seq
+Yes
+Event detection
+ScLimibic [66]
+Neuroimaging
+Img (MRI)
+External
+Segm.
+ALMA [67]
+Behavioral
+neuroscience
+Vid
+External
+Pose est., Class.
+fetal-code [68]
+Neuroimaging
+Img (rs-fMRI)
+External
+Segm.
+ClinicaDL [69]
+Neuroimaging
+Img (MRI, PET)
+External
+Class., Segm.
+DeepNeuron [70]
+Microbiology,
+Histology
+Img (confocal microscopy)
+No
+Obj. detect., Segm.
+GaNDLF [71]
+Medical
+Imaging
+Img (2D, 3D)
+External
+Segm., Regression, XAI
+MesoNet [72]
+Neuroimaging
+Img (fluoresc. microscopy)
+External
+Segm., Registration
+MARS, BENTO [73]
+Behavioral
+neuroscience
+Vid
+Yes
+Pose est., Class., Action rec., Tag
+NiftyNet [74]
+Medical
+Imaging
+Img (MRI, CT)
+No
+Class., Segm., Synth.
+ANTsX [75] (ANTsPyNet, ANTsRNet)
+Neuroimaging
+Img (MRI)
+No
+Classificastion, Segm., Registr., Super-res.
+MARS, BENTO [73]
+Behavioral
+neuroscience
+Vid
+Yes
+Pose est., Class., Action rec., Tag
+Visual Fields Analysis [76]
+Eye tracking,
+Behavioral
+neuroscience
+Vid
+No
+Pose est., Class.
+Table 4: Domains of applications for the libraries and frameworks processing image data
+12
+
+Name
+Models
+DL framework
+Customization
+Programming language
+AxonDeepSeg
+CAE
+TensorFlow
+Yes (weights)
+Python
+DeepCINAC
+DeepCINAC
+(CNN+LSTM)
+Keras, TensorFlow
+Yes (weights)
+Python
+DeepLabCut
+CNN
+TensorFlow
+Yes (weights)
+Python
+DeepNeuro
+CNN, CAE, GAN
+Keras, TensorFlow
+Yes (weights, model)
+Python
+DeepVOG
+CAE
+TensorFlow
+No
+Python
+DeLINEATE
+CNN
+Keras, TensorFlow
+Yes (weights, model)
+Python
+DNNBrain
+CNN
+PyTorch
+Yes (model)
+Python
+ivadomed
+2,3-D CNN, CAE
+PyTorch
+Yes (weights, model)
+Python
+MEYE
+CAE, CNN
+TensorFlow
+Yes (model)
+Python
+Allen Cell Structure Segmenter
+CAE
+PyTorch
+No
+Python
+VesicleSeg
+CNN
+PyTorch
+No
+Python
+CDeep3M2
+CAE
+TensorFlow
+Yes (weights)
+Python
+CASCADE
+1-D CNN
+TensorFlow
+Yes (weights)
+Python
+ScLimibic
+3-D CAE
+neurite, TensorFlow
+No
+Python
+ALMA
+CNN
+Unspecified
+No
+Python
+fetal-code
+2-D CNN
+TensorFlow
+No
+Python
+ClinicaDL
+CNN, CAE
+PyTorch
+Yes
+Python
+DeepNeuron
+CNN
+Unspecified
+No
+C++
+GaNDLF
+CNN, CAE
+PyTorch
+Yes
+Python
+MesoNet
+CNN, CAE
+Keras, TensorFlow
+No
+Python
+NiftyNet
+CNN
+TensorFlow
+Yes
+Python
+ANTsX (ANTsPyNet, ANTsRNet)
+CNN, CAE, GAN
+Keras, TensorFlow
+Yes
+Python, R, C++
+MARS, BENTO
+CNN
+TensorFlow
+Yes (weights)
+Python
+Visual Fields Analysis
+DeepLabCut
+TensorFlow, DeepLabCut
+Yes (weights)
+Python
+Table 5: Model engineering specifications for the libraries and frameworks processing image data
+13
+
+Name
+Interface
+Online/Offline
+Maintenance
+Source
+AxonDeepSeg
+Jupyter
+Note-
+books
+Offline
+Active
+github.com/axondeepseg/axondeepseg
+DeepCINAC
+GUI,
+Colab
+Notebooks
+Offline
+Active
+gitlab.com/cossartlab/deepcinac
+DeepLabCut
+GUI,
+Colab
+Notebooks
+Offline
+Active
+github.com/DeepLabCut/DeepLabCut
+DeepNeuro
+None
+Offline
+Active
+github.com/QTIM-Lab/DeepNeuro
+DeepVOG
+None
+Offline
+Inactive
+github.com/pydsgz/DeepVOG
+DeLINEATE
+GUI,
+Colab
+Notebooks
+Offline
+Active
+bitbucket.org/delineate/delineate
+DNNBrain
+None
+Offline
+Active
+github.com/BNUCNL/dnnbrain
+ivadomed
+None
+Offline
+Active
+github.com/ivadomed/ivadomed
+MEYE
+Web app
+Online, Offline
+Active
+pupillometry.it
+Allen Cell Structure Segmenter
+GUI,
+Jupyter
+Notebooks
+Offline
+Active
+github.com/AllenCell/aics-ml-segmentation
+VesicleSeg
+GUI
+Offline
+Active
+github.com/Imbrosci/synaptic-vesicles-detection
+CDeep3M2
+GUI,
+Colab
+Notebooks
+Offline
+Active
+github.com/CRBS/cdeep3m2
+CASCADE
+GUI,
+Colab
+Notebooks
+Offline
+Active
+github.com/HelmchenLabSoftware/Cascade
+ScLimibic
+Unspecified
+Offline
+Active
+surfer.nmr.mgh.harvard.edu/fswiki/ScLimbic
+ALMA
+GUI
+Offline
+Active
+github.com/sollan/alma
+fetal-code
+GUI,
+Colab
+Notebooks
+Offline
+Active
+github.com/saigerutherford/fetal-code
+ClinicaDL
+GUI,
+Colab
+Notebooks
+Offline
+Active
+github.com/aramis-lab/clinicadl
+DeepNeuron
+GUI
+Online, Offline
+Inactive
+github.com/Vaa3D/Vaa3D_Data/releases/tag/1.0
+GaNDLF
+GUI
+Offline
+Active
+github.com/CBICA/GaNDLF
+MesoNet
+GUI,
+Colab
+Notebooks
+Offline
+Active
+osf.io/svztu
+NiftyNet
+None
+Offline
+Inactive
+github.com/NifTK/NiftyNet
+ANTsX (ANTsPyNet, ANTsRNet)
+None
+Offline
+Active
+github.com/ANTsX
+MARS, BENTO
+GUI,
+MATLAB
+GUI,
+Jupyter
+Notebooks
+Offline
+Active
+github.com/neuroethology
+Visual Fields Analysis
+GUI
+Offline
+Active
+github.com/mathjoss/VisualFieldsAnalysis
+Table 6: Technological aspects and code sources for the libraries and frameworks processing image data
+14
+
+4.3
+Libraries targeting data types different from sequences or images and general applications
+Libraries and frameworks for sequence data are shown in Tables 7 (domains of application), 8 (models characteris-
+tics), 9 (technologies and sources). In this category fall libraries and projects with either varying input data type, or
+other than sequence and image data analysis; other libraries target computational platforms, higher hierarchy frame-
+works, or supporting functions for deep learning like specific preprocessing and augmentations. NeuroCAAS is an
+ambitious project that both standardizes experimental schedules, analyses and offers computational resources on the
+cloud. The platform lifts the burden of configuring and deploying data analysis tool, guaranteeing also replicability
+and readily available usage of pre-made pipelines, with high efficiency. MONAI is a project that brings deep learning
+tools to many health and biology problems, and is a commonly used framework for the 3D variations of UNet [77]
+lately dominating the yearly BraTS challenge [32] (see at http://braintumorsegmentation.org/). The paradigm
+builds on PyTorch and aims at unifying healthcare AI practices throughout both academia and enterprise research, not
+only in the model development but also in the creation of shared annotated datasets. Lastly, it focuses on deployment
+and work in real world clinical production, settling as a strong candidate for being the standard solution in the do-
+main. Predify and THINGvision are two libraries that bridge deep learning research and computational neuroscience.
+The former allows to include an implementation of a «predictive coding mechanism» (as hypothesized in [78]) into
+virtually any pre-built architectures, evaluating its impact on performance. The latter offers a single environment for
+Representational Similarity Analysis, i.e. the study of the encodings of biological and artificial neural networks that
+process visual data.
+15
+
+Name
+Neuroscience area
+Data type
+Datasets
+Task
+NeuroCAAS [79]
+Virtually all
+Virtually all
+External availability
+Virtually all
+MONAI [80]
+Virtually all
+Virtually all
+External availability
+Virtually all
+Predify [81]
+Computational
+Neuro-
+science
+Images, Virtually all
+No
+Classification, Adversarial attacks, virtually all
+THINGvision [82]
+Computational
+Neuro-
+science
+Images, Text
+External availability
+Classification
+TorchIO [83]
+Imaging
+All images
+No
+Augmentation
+Table 7: Domains of applications for the libraries and frameworks for special applications
+16
+
+Name
+Models
+DL framework
+Customization
+Programming language
+NeuroCAAS
+CNN
+TensorFlow
+Yes
+Python
+MONAI
+Virtually All
+PyTorch
+Yes
+Python
+Predify
+CNN, Virtually all
+PyTorch
+Yes
+Python
+THINGvision
+CNN, RNN, Transformers
+PyTorch, TensorFlow
+No
+Python
+TorchIO
+CNN
+PyTorch
+Yes
+Python
+Table 8: Model engineering specifications for the libraries and frameworks for special applications
+17
+
+Name
+Interface
+Online/Offline
+Maintenance
+Source
+NeuroCAAS
+GUI, Jupyter Notebooks
+Offline
+Active
+github.com/cunningham-lab/neurocaas
+MONAI
+GUI, Colab Notebooks
+Offline
+Active
+github.com/Project-MONAI/MONAI
+Predify
+Text UI (TOML)
+Offline
+Active
+github.com/miladmozafari/predify
+THINGvision
+None
+Offline
+Active
+github.com/ViCCo-Group/THINGSvision
+TorchIO
+GUI, Command line
+Offline
+Active
+torchio.rtfd.io
+Table 9: Technological aspects and code sources for the libraries and frameworks for special applications
+18
+
+5
+Discussion
+The panorama of open-source libraries dedicated to deep learning applications in neuroscience is quite rich and diver-
+sified. There is a corpus of organized packages that integrate preprocessing, training, testing and performance analyses
+of deep neural networks for neurological research. Most of these projects are tuned to specific data modalities and
+formats, but some libraries are quite versatile and customizabile, and there are projects that encompass quantitative
+biology and medical analysis as a whole. There is a common tendency to develop GUIs, enhancing user-friendliness of
+toolkits for non-programmers and researchers unacquainted with the command line interfaces, for example. Moreover,
+for the many libraries developed in Python, the (Jupyter) Notebook format appears as a widespread tool both for tutori-
+als, documentation and as an interface to cloud computational resources (e.g. Google Colab [84]). Apart from specific
+papers and documentation, and outside of deep learning per se, it is important to make researchers and developers
+aware of the main topics and initiatives in open culture and Neuroinformatics. For this reason, the interested reader
+is invited to rely on competent institutions (e.g. INCF) and databases of open resources (e.g. open-neuroscience)
+dedicated to Neuroscience. Among the possibly missing technologies, the queries employed did not retrieve results
+in Natural Language Processing libraries dedicated to neuroscience, nor toolkits specifically employing Graph Neural
+Networks (GNNs), although available in EEG-DL. NLP is actually fundamental in healthcare, since medical reports
+often come in non standardized forms. Large language models, Named Entity Recognition (NER) systems and text
+mining approaches in biomedical research exist [85], [86]. GNNs comprise recent architectures that are extremely
+promising in a variety of fields [87], including biomedical research and particularly neuroscience [88], [89]. Even if
+promising, their application is still less mature than that of computer vision models or time series analysis.
+Considering the available software for imaging and signal processing in the domain of neuroscience, at this moment a
+single alternative targeting the opportunities offered by modern deep learning seems to be missing. Overall, it seems
+still unlikely to develop a common deep learning framework for Neuroscience as a separate whole, but the engineering
+knowledge relevant and compressible into such framework would be common to other biomedical fields, and projects
+such as MONAI are strong candidates toward this goal. Instead, it seems achievable to deliver models and functions
+in a concerted way, restricted either to a sub-field or a data modality, based on the modularity of existent tools and the
+organizing possibilities of project initiation and management of open culture.
+6
+Conclusions
+Although a large and growing number of repositories offer code to build specific models, as published in experimental
+papers, these resources seldom aim to constitute proper libraries or frameworks for research or clinical practice. Both
+deep learning and neuroscience gain much value even from sophisticated proofs of concept. In parallel, organized
+packages are spreading and starting to provide and integrate pre-processing, training, testing and performance analyses
+of deep neural networks for neurological and biomedical research. This paper has offered both an historical and a
+technical context for the use of deep neural networks in Neuroinformatics, focusing on open-source tools that scientists
+can comprehend and adapt to their necessities. At the same time, this work underlines the value of the open culture and
+points to relevant institutions and platforms for neuroscientists. Although the aim is not restricted to making clinicians
+develop their own deep models without coding or Machine Learning background, as was the case in [90], the overall
+effect of these libraries and sources is to democratize deep learning applications and results, as well as standardizing
+such complex and varied models, supporting the research community in obtaining proper means to an end, and in
+envisioning then realizing collectively new projects and tools.
+Acknowledgments
+This work was supported by the "Department of excellence 2018-2022" initiative of the Italian Ministry of education
+(MIUR) awarded to the Department of Neuroscience - University of Padua.
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+arXiv:2301.00735v1  [math.DG]  2 Jan 2023
+FAILURE OF CURVATURE-DIMENSION CONDITIONS ON
+SUB-RIEMANNIAN MANIFOLDS VIA TANGENT ISOMETRIES
+LUCA RIZZI AND GIORGIO STEFANI
+Abstract. We prove that, on any sub-Riemannian manifold endowed with a positive
+smooth measure, the Bakry–Émery inequality for the corresponding sub-Laplacian,
+1
+2∆(∥∇u∥2) ≥ g(∇u, ∇∆u) + K∥∇u∥2,
+K ∈ R,
+implies the existence of enough Killing vector fields on the tangent cone to force the latter
+to be Euclidean at each point, yielding the failure of the curvature-dimension condition
+in full generality. Our approach does not apply to non-strictly-positive measures. In
+fact, we prove that the weighted Grushin plane does not satisfy any curvature-dimension
+condition, but, nevertheless, does admit an a.e. pointwise version of the Bakry–Émery
+inequality.
+As recently observed by Pan and Montgomery, one half of the weighted
+Grushin plane satisfies the RCD(0, N) condition, yielding a counterexample to gluing
+theorems in the RCD setting.
+1. Introduction and statements
+In the last twenty years, there has been an impressive effort in extending the concept of
+‘Ricci curvature lower bound’ to non-Riemannian structures, and even to general metric
+spaces equipped with a measure (metric-measure spaces, for short). We refer the reader
+to the ICM notes [3] for a survey of this line of research.
+There are two distinct points of view on the matter, traditionally known as the La-
+grangian and Eulerian approaches, respectively.
+The Lagrangian point of view is the one adopted by Lott–Villani and Sturm [36, 48,
+49]. In this formulation, Ricci curvature lower bounds are encoded by convexity-type
+inequalities for entropy functionals on the Wasserstein space. Such inequalities are called
+curvature-dimension conditions, CD(K, N) for short, where K ∈ R represents the lower
+bound on the curvature and N ∈ [1, ∞] stands for an upper bound on the dimension.
+The Eulerian point of view, instead, employs the metric-measure structure to define an
+energy form and, in turn, an associated diffusion operator. The notion of Ricci curvature
+lower bound is therefore encoded in the so-called Bakry–Émery inequality, BE(K, N) for
+short, for the diffusion operator, which can be expressed in terms of a suitable Gamma
+calculus, see the monograph [10].
+Thanks to several key contributions [4, 6, 7, 24], the Lagrangian and the Eulerian ap-
+proaches are now known to be essentially equivalent. In particular, CD(K, N) always
+Date: January 3, 2023.
+2020 Mathematics Subject Classification. Primary 53C17. Secondary 54E45, 28A75.
+Key words and phrases. Sub-Riemannian manifold, CD(K, ∞) condition, Bakry–Émery inequality,
+infinitesimally Hilbertian, Grushin plane, privileged coordinates.
+1
+
+2
+L. RIZZI AND G. STEFANI
+implies BE(K, N) in infinitesimal Hilbertian metric-measure spaces, as introduced in [25],
+while the converse implication requires further technical assumptions.
+Such synthetic theory of curvature-dimension conditions, besides being consistent with
+the classical notions of Ricci curvature and dimension on smooth Riemannian manifolds,
+is stable under pointed-measure Gromov–Hausdorff convergence. Furthermore, it yields
+a comprehensive approach for establishing all results typically associated with Ricci cur-
+vature lower bounds, like Poincaré, Sobolev, log-Sobolev and Gaussian isoperimetric in-
+equalities, as well as Brunn–Minkowski, Bishop–Gromov and Bonnet–Myers inequalities.
+1.1. The sub-Riemannian framework. Although the aforementioned synthetic cur-
+vature-dimension conditions embed a large variety of metric-measure spaces, a relevant
+and widely-studied class of smooth structures is left out—the family of sub-Riemmanian
+manifolds. A sub-Riemannian structure is a natural generalization of a Riemannian one,
+in the sense that its distance is induced by a scalar product that is defined only on a
+smooth sub-bundle of the tangent bundle, whose rank possibly varies along the manifold.
+See the monographs [2,40,45] for a detailed presentation.
+The first result in this direction was obtained by Driver–Melcher [23], who proved that
+an integrated version of the BE(K, ∞), the so-called pointwise gradient estimate for the
+heat flow, is false for the three-dimensional Heisenberg group.
+In [31], Juillet proved the failure of the CD(K, ∞) property for all Heisenberg groups
+(and even for the strictly related Grushin plane, see [32]). Later, Juillet [33] extended his
+result to any sub-Riemannian manifold endowed with a possibly rank-varying distribution
+of rank strictly smaller than the manifold’s dimension, and with any positive smooth
+measure, by exploiting the notion of ample curves introduced in [1]. The idea of [31,33]
+is to construct a counterexample to the Brunn–Minkowski inequality.
+The ‘no-CD theorem’ of [31] was extended to all Carnot groups by Ambrosio and the
+second-named author in [8, Prop. 3.6] with a completely different technique, namely, by
+exploiting the optimal version of the reverse Poincaré inequality obtained in [16].
+In the case of sub-Riemannian manifolds endowed with an equiregular distribution and
+a positive smooth measure, Huang–Sun [29] proved the failure of the CD(K, N) condition
+for all values of K ∈ R and N ∈ (1, ∞) contradicting a bi-Lipschitz embedding result.
+Very recently, in order to address the structures left out in [33], Magnabosco–Rossi [37]
+recently extended the ‘no-CD theorem’ to almost-Riemannian manifolds M of dimension 2
+or strongly regular. The approach of [37] relies on the localization technique developed by
+Cavalletti–Mondino [19] in metric-measure spaces.
+To complete the picture, we mention that several replacements for the Lott–Sturm–
+Villani curvature-dimension property have been proposed and studied in the sub-Rieman-
+nian framework in recent years. Far from being complete, we refer the reader to [11–15,38]
+for an account on the Lagrangian approach, to [17] concerning the Eulerian one, and finally
+to [47] for a first link between entropic inequalities and contraction properties of the heat
+flow in the special setting of metric-measure groups.
+Main aim. At the present stage, a ‘no-CD theorem’ for sub-Riemannian structures in
+full generality is missing, since the aforementioned approaches [8,23,29,31,33,37] either
+require the ambient space to satisfy some structural assumptions, or leave out the infinite
+dimensional case N = ∞.
+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+3
+The main aim of the present paper is to fill this gap by showing that (possibly rank-
+varying) sub-Riemannian manifolds do not satisfy any curvature bound in the sense of
+Lott–Sturm–Villani or Bakry–Émery when equipped with a positive smooth measure, i.e.,
+a Radon measure whose density in local charts with respect to the Lebesgue measure is
+a strictly positive smooth function.
+1.2. Failure of the Bakry–Émery inequality. The starting point of our strategy is
+the weakest curvature-dimension condition, as we now define.
+Definition 1.1 (Bakry–Émery inequality). We say that a sub-Riemannian manifold
+(M, d) endowed with a positive smooth measure m satisfies the Bakry–Émery BE(K, ∞)
+inequality, for K ∈ R, if
+1
+2 ∆(∥∇u∥2) ≥ g(∇u, ∇∆u) + K∥∇u∥2
+for all u ∈ C∞(M),
+(1.1)
+where ∆ is the corresponding sub-Laplacian, and ∇ the sub-Riemannian gradient.
+Our first main result is the following rigidity property for sub-Riemannian structures
+supporting the Bakry–Émery inequality (1.1).
+Theorem 1.2 (no-BE). Let (M, d) be a complete sub-Riemannian manifold endowed
+with a positive smooth measure m. If (M, d, m) satisfies the BE(K, ∞) inequality for some
+K ∈ R, then rank Dx = dim M at each x ∈ M, so that (M, d) is Riemannian.
+The idea behind our proof of Theorem 1.2 is to show that the metric tangent cone
+in the sense of Gromov [26] at each point of (M, d) is Euclidean. This line of thought is
+somehow reminiscent of the deep structural result for RCD(K, N) spaces, with K ∈ R and
+N ∈ (1, ∞), proved by Mondino–Naber [39]. However, differently from [39], Theorem 1.2
+provides information about the metric tangent cone at each point of the manifold. Showing
+that the distribution D is Riemannian at almost every point in fact would not be enough,
+as this would not rule out almost-Riemannian structures.
+Starting from (1.1), we first blow-up the sub-Riemannian structure and pass to its
+metric-measure tangent cone, showing that (1.1) is preserved with K = 0. Note that, in
+this blow-up procedure, the positivity of the density of m is crucial, since otherwise the
+resulting metric tangent cone would be endowed with the null measure.
+The resulting blown-up sub-Riemannian space is isometric to a homogeneous space
+of the form G/H, where G = exp g is the Carnot group associated to the underlying
+(finite-dimensional and stratified) Lie algebra g of bracket-generating vector fields, and
+H = exp h is its subgroup corresponding to the Lie subalgebra h of vector fields vanishing
+at the origin, see [18]. Of course, the most difficult case is when H is non-trivial, that is,
+the tangent cone is not a Carnot group.
+At this point, the key idea is to show that the Bakry–Émery inequality BE(K, ∞)
+implies the existence of special isometries on the tangent cone.
+Definition 1.3 (Sub-Riemannian isometries). Let M be a sub-Riemannian manifold,
+with distribution D and metric g. A diffeomorphism φ : M → M is an isometry if
+(φ∗D)|x = Dφ(x)
+for all x ∈ M,
+(1.2)
+and, furthermore, φ∗ is an orthogonal map with respect to g. We say that a smooth vector
+field V is Killing if its flow φV
+t is an isometry for all t ∈ R.
+
+4
+L. RIZZI AND G. STEFANI
+For precise definitions of g and h in the next statement, we refer to Section 2.4.
+Theorem 1.4 (Existence of Killing fields). Let (M, d) be a complete sub-Riemannian
+manifold equipped with a positive smooth measure m If (M, d, m) satisfies the BE(K, ∞)
+inequality for some K ∈ R, then, for the nilpotent approximation at any given point, there
+exists a vector space i ⊂ g1 such that
+g1 = i ⊕ h1
+(1.3)
+and every Y ∈ i is a Killing vector field.
+The existence of the space of isometries i forces the Lie algebra g to be commutative and
+of maximal rank, thus implying that the original manifold (M, d) was in fact Riemannian.
+Theorem 1.5 (Killing implies commutativity). If there exists a subspace i ⊂ g1 of Killing
+vector fields such that g1 = i ⊕ h1, then g is commutative.
+Theorem 1.5 states that, if a Carnot group contains enough horizontal symmetries, then
+it must be commutative. As it will be evident from its proof, Theorem 1.5 holds simply
+assuming that, for each V ∈ i, the flow φV
+t is pointwise distribution-preserving, namely it
+satisfies (1.2), without being necessarily isometries.
+1.3. Infinitesimal Hilbertianity. The Bakry–Émery inequality BE(K, ∞) in (1.1) is a
+consequence of the CD(K, ∞) condition as soon as the ambient metric-measure space is
+infinitesimal Hilbertian as defined in [25].
+Let (X, d) be a complete separable metric space, m be a locally bounded Borel mea-
+sure, and q ∈ [1, ∞). We let |Du|w,q ∈ Lq(X, m) be the minimal q-upper gradient of a
+measurable function u : X → R, see [5, Sec. 4.4]. We define the Banach space
+W1,q(X, d, m) = {u ∈ Lq(X, m) : |Du|w,q ∈ Lq(X, m)}
+with the norm
+∥u∥W1,q(X,d,m) =
+�
+∥u∥q
+Lq(X,m) + ∥|Du|w,q∥q
+Lq(X,m)
+�1/q .
+Definition 1.6 (Infinitesimal Hilbertianity). A metric measure space (X, d, m) is in-
+finitesimally Hilbertian if W1,2(X, d, m) is a Hilbert space.
+The infinitesimal Hilbertianity of sub-Riemannian structures has been recently proved
+in [35], with respect to any Radon measure.
+In particular, Theorem 1.2 immediately
+yields the following ‘no-CD theorem’ for sub-Riemannian manifolds, thus extending all
+the aforementioned results [8,23,29,31,33,37].
+Corollary 1.7 (no-CD). Let (M, d) be a complete sub-Riemannian manifold endowed
+with a positive smooth measure m. If (M, d, m) satisfies the CD(K, ∞) condition for some
+K ∈ R, then (M, d) is Riemannian.
+However, since the measure in Corollary 1.7 is positive and smooth, we can avoid to
+rely on the general result of [35], instead providing a simpler and self-contained proof
+of the infinitesimal Hilbertianity property. In particular, we prove the following result,
+which actually refines [35, Th. 5.6] in the case of smooth measures. In the following,
+HW1,q(M, m) denotes the sub-Riemannian Sobolev spaces (see Section 2.2).
+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+5
+Theorem 1.8 (Infinitesimal Hilbertianity). Let q ∈ (1, ∞). Let (M, d) be a complete sub-
+Riemannian manifold equipped with a positive smooth measure m. The following hold.
+(i) W1,q(M, d, m) = HW1,q(M, m), with |Du|w,q = ∥∇u∥ m-a.e. on M for all u ∈
+W1,q(M, d, m). In particular, taking q = 2, (M, d, m) is infinitesimally Hilbertian.
+(ii) If (M, d, m) satisfies the CD(K, ∞) condition for some K ∈ R, then the Bakry–
+Émery BE(K, ∞) inequality (1.1) holds on M.
+Note that Theorem 1.8 holds for less regular measures, see Remark 3.6.
+Remark 1.9 (The case of a.e. smooth measures). Theorem 1.8 can be adapted also to
+the case of a Borel and locally finite measure m which is smooth and positive only on Ω,
+where Ω ⊂ M is an open set with m(∂Ω) = 0. In this case, we obtain HW1,q(Ω, m) =
+W1,q(Ω, d, m), with |Du|w,q = ∥∇u∥ m-a.e. on Ω for all u ∈ W1,q(Ω, d, m). In particular,
+if m is smooth and positive out of a closed set Z, with m(Z) = 0, an elementary ap-
+proximation argument proves that (M, d, m) is infinitesimally Hilbertian and, if (M, d, m)
+satisfies the CD(K, ∞) condition for K ∈ R, then the Bakry-Émery BE(K, ∞) inequality
+(1.1) holds on M \Z. This is the case, for example, of the Grushin planes and half-planes
+with weighted measures of Section 1.5. The proof follows the same argument of the one of
+Theorem 1.8, exploiting the locality of the q-upper gradient, see for example [5, Sec. 8.2]
+and [25, Prop. 2.6], and similar properties for the distributional derivative.
+1.4. An alternative approach to the ‘no-CD theorem’. We mention an alternative
+proof of the ‘no-CD theorem’ for almost-Riemannian structures (i.e., sub-Riemannian
+structures that are Riemannian outside a closed nowhere dense singular set). The strategy
+relies on the Gromov-Hausdorff continuity of the metric tangent at interior points of
+geodesics in RCD(K, N) spaces, with N < ∞, proved by Deng in [22],
+For example, consider the standard Grushin plane (introduced in Section 1.5) equipped
+with a smooth positive measure. The curve γ(t) = (t, 0), t ∈ R, is a geodesic between
+any two of its point. The metric tangent at γ(t) is (isometric to) the Euclidean plane for
+every t ̸= 0, while it is (isometric to) the Grushin plane itself for t = 0. Since the Grushin
+plane cannot be bi-Lipschitz embedded into the Euclidean plane, the two spaces are at
+positive Gromov-Hausdorff distance, contradicting the continuity result.
+This strategy has a few drawbacks.
+On the one hand, it relies on the (non-trivial)
+machinery developed in [22].
+Consequently, this argument does not work in the case
+N = ∞. On the other hand, the formalization of this strategy for general almost-Rie-
+mannian structures requires certain quantitative bi-Lipschitz non-embedding results for
+almost-Riemannian structures into Euclidean spaces, which we are able to prove only
+under the same assumptions of [37].
+1.5. Weighted Grushin structures. When the density of the smooth measure is al-
+lowed to vanish, the ‘no-CD theorem’ breaks down. In fact, in this situation, the following
+two interesting phenomena occur:
+(A) the Bakry-Émery BE(K, ∞) inequality no longer implies the CD(K, ∞) condition;
+(B) there exist almost-Riemannian structures with boundary satisfying the CD(0, N)
+condition for N ∈ [1, ∞].
+
+6
+L. RIZZI AND G. STEFANI
+We provide examples of both phenomena on the so-called weighted Grushin plane. This
+is the sub-Riemannian structure on R2 induced by the family F = {X, Y }, where
+X = ∂x,
+Y = x ∂y,
+(x, y) ∈ R2.
+(1.4)
+The induced distribution D = span{X, Y } has maximal rank outside the singular region
+S = {x = 0} and rank 1 on S. Since [X, Y ] = ∂y on R2, the resulting sub-Riemannian
+metric space (R2, d) is Polish and geodesic. It is almost-Riemannian in the sense that, out
+of S, the metric is locally equivalent to the Riemannian one given by the metric tensor
+g = dx ⊗ dx + 1
+x2 dy ⊗ dy,
+x ̸= 0.
+(1.5)
+We endow the metric space (R2, d) with the weighted Lebesgue measure
+mp = |x|p dx dy,
+where p ∈ R is a parameter. The choice p = −1 corresponds to the Riemannian density
+volg = 1
+|x| dx dy,
+x ̸= 0,
+(1.6)
+so that
+mp = e−V volg,
+V (x) = −(p + 1) log |x|,
+x ̸= 0.
+(1.7)
+We call the metric-measure space Gp = (R2, d, mp) the (p-)weighted Grushin plane.
+We can now state the following result, illustrating phenomenon (A).
+Theorem 1.10. Let p ∈ R and let Gp = (R2, d, mp) be the weighted Grushin plane.
+(i) If p ≥ 0, then Gp does not satisfy the CD(K, ∞) property for all K ∈ R.
+(ii) If p ≥ 1, then Gp satisfies the BE(0, ∞) inequality (1.1) almost everywhere.
+To prove (i), we show that the corresponding Brunn–Minkowski inequality is violated.
+In fact, the case p = 0 is due to Juillet [32], while the case p > 0 can be achieved via a
+simple argument which was pointed out to us by J. Pan. Claim (ii), instead, is obtained
+by direct computations.
+Somewhat surprisingly, the weighted Grushin half -plane G+
+p —obtained by restricting
+the metric-measure structure of Gp to the (closed) half-plane [0, ∞)×R—does satisfy the
+CD(0, N) condition for sufficiently large N ∈ [1, ∞]. Precisely, we can prove the following
+result, illustrating phenomenon (B).
+Theorem 1.11. Let p ≥ 1. The weighted Grushin half-plane G+
+p satisfies the CD(0, N)
+condition if and only if N ≥ Np, where Np ∈ (2, ∞] is given by
+Np = (p + 1)2
+p − 1
++ 2,
+(1.8)
+with the convention that N1 = ∞. Furthermore, G+
+p is infinitesimally Hilbertian, and it
+is thus an RCD(0, N) space for N ≥ Np.
+While we were completing this work, Pan and Montgomery [41] observed that the spaces
+built in [20, 42] as Ricci limits are actually the weighted Grushin half-spaces presented
+above. Our construction and method of proof are more direct with respect to the approach
+of [20,42], and easily yield sharp dimensional bounds.
+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+7
+1.6. Counterexample to gluing theorems. We end this introduction with an inter-
+esting by-product of our analysis, in in connection with the so-called gluing theorems.
+Perelman’s Doubling Theorem [43, Sect. 5.2] states that a finite dimensional Alexan-
+drov space with a curvature lower bound can be doubled along its boundary yielding an
+Alexandrov space with same curvature lower bound and dimension. This result has been
+extended by Petrunin [44, Th. 2.1] to the gluing of Alexandrov spaces.
+It is interesting to understand whether these classical results hold true for general
+metric-measure spaces satisfying synthetic Ricci curvature lower bounds in the sense of
+Lott–Sturm–Villani. In [34], the gluing theorem was proved for CD(K, N) spaces with
+Alexandrov curvature bounded from below (while it is false for MCP spaces, see [46]).
+Here we obtain that, in general, the assumption of Alexandrov curvature bounded
+from below cannot be removed from the results in [34]. More precisely, Theorems 1.10
+and 1.11, and the fact that the metric-measure double of the Grushin half-plane G+
+p is Gp
+(see [46, Prop. 6]) yield the following corollary.
+Corollary 1.12 (Counterexample to gluing in RCD spaces). For all N ≥ 10, there exists
+a geodesically convex RCD(0, N) metric-measure space with boundary such that its metric-
+measure double does not satisfy the CD(K, ∞) condition for any K ∈ R.
+In [34, Conj. 1.6], the authors conjecture the validity of the gluing theorem for non-
+collapsed RCD(K, N), with N the Hausdorff dimension of the metric-measure space.
+As introduced in [21], a non-collapsed RCD(K, N) space is an infinitesimally Hilbertian
+CD(K, N) space with m = H N, where H N denotes the N-dimensional Hausdorff mea-
+sure of (X, d). Since the weighted half-Grushin spaces are indeed collapsed, Corollary 1.12
+also shows that the non-collapsing assumption cannot be removed from [34, Conj. 1.6].
+1.7. Acknowledgments. We wish to thank Michel Bonnefont for fruitful discussions
+and, in particular, for bringing some technical details in [23] that inspired the strategy of
+the proof of Theorem 1.2 to our attention.
+This work has received funding from the European Research Council (ERC) under the
+European Union’s Horizon 2020 research and innovation programme (grant agreement No.
+945655) and the ANR grant ‘RAGE’ (ANR-18-CE40-0012). The second-named author
+is member of the Istituto Nazionale di Alta Matematica (INdAM), Gruppo Nazionale
+per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), and is par-
+tially supported by the INdAM–GNAMPA 2022 Project Analisi geometrica in strutture
+subriemanniane, codice CUP_E55F22000270001.
+2. Preliminaries
+In this section, we introduce some notation and recall some results about sub-Rieman-
+nian manifolds and curvature-dimension conditions.
+2.1. Sub-Riemannian structures. For L ∈ N, we let F = {X1, . . . , XL} be a family
+of smooth vector fields globally defined on a smooth n-dimensional manifold M, n ≥ 2.
+The (generalized) sub-Riemannian distribution induced by the family F is defined by
+D =
+�
+x∈M
+Dx,
+Dx = span{X1|x, . . . , XL|x} ⊂ TxM,
+x ∈ M.
+(2.1)
+
+8
+L. RIZZI AND G. STEFANI
+Note that we do not require the dimension of Dx to be constant as x ∈ M varies, that is,
+we may consider rank-varying distributions. With a standard abuse of notation, we let
+Γ(D) = C∞-module generated by F.
+Notice that, for any smooth vector field V , it holds
+V ∈ Γ(D) =⇒ Vx ∈ Dx for all x ∈ M,
+but the converse is false in general. We let
+∥V ∥x = min
+�
+|u| : u ∈ RL such that V =
+L
+�
+i=1
+ui Xi|x, Xi ∈ F
+�
+(2.2)
+whenever V ∈ D and x ∈ M. The norm ∥ · ∥x induced by the family F satisfies the
+parallelogram law and, consequently, it is induced by a scalar product
+gx : Dx × Dx → R.
+An admissible curve is a locally Lipschitz in charts path γ : [0, 1] → M such that there
+exists a control u ∈ L∞([0, 1]; RL) such that
+˙γ(t) =
+L
+�
+i=1
+ui(t)Xi|γ(t)
+for a.e. t ∈ [0, 1].
+The length of an admissible curve γ is defined via the norm (2.2) as
+length(γ) =
+� 1
+0 ∥˙γ(t)∥γ(t) dt
+and the Carnot–Carathéodory (or sub-Riemannian) distance between x, y ∈ M is
+d(x, y) = inf{length(γ) : γ admissible with γ(0) = x, γ(1) = y}.
+We assume that the family F satisfies the bracket-generating condition
+TxM = {X|x : X ∈ Lie(F)}
+for all x ∈ M,
+(2.3)
+where Lie(F) is the smallest Lie subalgebra of vector fields on M containing F, namely,
+Lie(F) = span
+�
+[Xi1, . . . , [Xij−1, Xij]] : Xiℓ ∈ F, j ∈ N
+�
+.
+Under the assumption (2.3), the Chow–Rashevskii Theorem implies that d is a well-defined
+finite distance on M inducing the same topology of the ambient manifold.
+2.2. Gradient, sub-Laplacian and Sobolev spaces. The gradient of a function u ∈
+C∞(M) is the unique vector field ∇u ∈ Γ(D) such that
+g(∇u, V ) = du(V )
+for all V ∈ Γ(D).
+(2.4)
+One can check that ∇u can be globally represented as
+∇u =
+L
+�
+i=1
+Xiu Xi,
+with
+∥∇u∥2 =
+L
+�
+i=1
+(Xiu)2,
+(2.5)
+even if the family F is not linearly independent, see Corollary A.2 for a proof.
+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+9
+We equip the manifold M with a positive smooth measure m. The sub-Laplacian of a
+function u ∈ C∞(M) is the unique function ∆u ∈ C∞(M) such that
+�
+M g(∇u, ∇v) dm = −
+�
+M v ∆u dm
+(2.6)
+for all v ∈ C∞
+c (M). On can check that ∆u can be globally represented as
+∆u =
+L
+�
+i=1
+�
+X2
+i u + Xiu divm(Xi)
+�
+,
+(2.7)
+see Corollary A.2 for a proof. In (2.7), divmV is the divergence of the vector field V
+computed with respect to m, that is,
+�
+M v divm(V ) dm = −
+�
+M g(∇v, V ) dm
+for all v ∈ C∞
+c (M).
+For q ∈ [1, ∞), we say that u ∈ L1
+loc(M, m) has q-integrable distributional Xi-derivative
+if there exists a function Xiu ∈ Lq(M, m) such that
+�
+M vXiu dm =
+�
+M uX∗
+i v dm
+for all v ∈ C∞
+c (M),
+where X∗
+i v = −Xiv − v divm(Xi) denotes the adjoint action of Xi. We thus let
+HW1,q(M, m) = {u ∈ Lq(M, m) : Xiu ∈ Lq(M, m), i = 1, . . ., L}
+be the usual horizontal W1,q Sobolev space induced by the the family F and the measure m
+on M, endowed with the natural norm
+∥u∥HW1,q(M,m) =
+�
+∥u∥q
+Lq(M,m) + ∥∇u∥q
+Lq(M,m)
+�1/q
+for all u ∈ HW1,q(M, m), where ∇u =
+L
+�
+i=1
+Xiu Xi in accordance with (2.5) and
+∥∇u∥q
+Lq(M,m) =
+�
+M ∥∇u∥q dm.
+2.3. Privileged coordinates. Following [18,30], we introduce privileged coordinates, a
+fundamental tool in the description of the tangent cone of sub-Riemannian manifolds.
+Given a multi-index I ∈ {1, . . ., L}×i, i ∈ N, we let |I| = i be its length and we set
+XI = [XI1, [. . ., [XIi−1, XIi]]]].
+Accordingly, we define
+Di
+x = span{XI|x : |I| ≤ i}
+(2.8)
+and
+ki(x) = dim Di
+x
+for all x ∈ M and i ∈ N. In particular, D0
+x = {0} and D1
+x = Dx as in (2.1) for all x ∈ M.
+The spaces defined in (2.8) naturally yield the filtration
+{0} = D0
+x ⊂ D1
+x ⊂ · · · ⊂ Ds(x)
+x
+= TxM
+for all x ∈ M, where s = s(x) ∈ N is the step of the sub-Riemannian structure at the
+point x. We say that x ∈ M is a regular point if the dimension of each space Di
+y remains
+constant as y ∈ M varies in an open neighborhood of x, otherwise x is a singular point.
+
+10
+L. RIZZI AND G. STEFANI
+Definition 2.1 (Adapted and privileged coordinates). Let o ∈ M and let U ⊂ M be an
+open neighborhood of o. We say that the local coordinates given by a diffeomorphism
+z : U → Rn are adapted at o if they are centered at o, i.e. z(o) = 0, and ∂z1|0, . . ., ∂zki|0
+form a basis for Di
+o in these coordinates for all i = 1, . . ., s(o). We say that the adapted
+coordinate zi has weight wi = j if ∂zi|0 ∈ Dj
+o \ Dj−1
+o
+. Furthermore, we say that the coor-
+dinates z are privileged at o if they are adapted at o and, in addition, zi(x) = O(d(x, o)wi)
+for all x ∈ U and i = 1, . . ., n.
+Privileged coordinates exist in a neighborhood of any point, see [18, Th. 4.15].
+2.4. Nilpotent approximation. From now on, we fix a set of privileged coordinates
+z : U → Rn around a point o ∈ M in the sense of Definition 2.1.
+Without loss of
+generality, we identify the coordinate domain U ⊂ M with Rn and the base point o ∈ M
+with the origin 0 ∈ Rn. Similarly, the vector fields in F defined on U are identified with
+vector fields on Rn, and the restriction of the sub-Riemannian distance d to U is identified
+with a distance function on Rn, which is induced by the family F, for which we keep the
+same notation.
+On (Rn, F), we define a family of dilations, for λ ≥ 0, by letting
+dilλ : Rn → Rn,
+dilλ(z1, . . . , zn) = (λw1z1, . . . , λwnzn)
+for all z = (z1, . . . , zn) ∈ Rn, where the wi’s are the weights given by Definition 2.1. We
+say that a differential operator P is homogeneous of degree −d ∈ Z if
+P(f ◦ dilλ) = λ−d(Pf) ◦ dilλ
+for all λ > 0 and f ∈ C∞(Rn).
+(2.9)
+Note that the monomial zi is homogeneous of degree wi, while the vector field ∂zi is
+homogeneous of degree −wi, for i = 1, . . . , n. As a consequence, the differential operator
+zµ1
+1 · · · · · zµn
+n
+∂|ν|
+∂zν1
+1 · · · ∂zνn
+n
+,
+νi, µj ∈ N ∪ {0},
+is homogeneous of degree �n
+i=1 wi(µi − νi). For more details, see [18, Sec. 5].
+We can now introduce the new family
+�
+F =
+��
+X1, . . . , �
+XL
+�
+by defining
+�
+Xi = lim
+ε→0 Xε
+i ,
+Xε
+i = ε (dil1/ε)∗Xi,
+(2.10)
+for all i = 1, . . . , L, where (dil1/ε)∗ stands for the usual push-forward via the differential
+of the dilation map dil1/ε, see [18, Sec. 5.3]. The convergence in (2.10) can be actually
+made more precise, in the sense that
+Xε
+i = �
+Xi + Rε
+i,
+i = 1, . . ., L,
+where Rε
+i locally uniformly converges to zero as ε → 0, see [18, Th. 5.19].
+The family �
+F is a set of complete vector fields on Rn, homogeneous of degree −1, with
+polynomial coefficients, and can be understood as the ‘principal part’ of F upon blow-up
+by dilations. Since F satisfies the bracket-generating condition (2.3), also the new family
+�
+F is bracket-generating at all points of Rn, and thus induces a finite sub-Riemannian
+distance �d, see [18, Prop. 5.17]. The resulting n-dimensional sub-Riemannian structure
+(Rn, �
+F ) is called nilpotent approximation of (Rn, F) at 0 ∈ Rn.
+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+11
+The family �
+F =
+��
+X1, . . ., �
+XL
+�
+generates a finite-dimensional stratified Lie algebra
+g = Lie( �
+F ) = g1 ⊕ · · · ⊕ gs
+of step s = s(0) ∈ N, where the grading is given by the degree of the vector fields,
+according to the definition in (2.9), that is, the layer gi corresponds to vector fields
+homogeneous of degree −i with respect to dilations, see [18, Sec. 5.4].
+In particular,
+g1 = span
+��
+X1, . . . , �
+XL
+�
+, so that g is generated by its first stratum, namely,
+gj+1 = [g1, gj],
+∀j = 1, . . . , s − 1.
+(2.11)
+Finally, define the Lie subalgebra of vector fields vanishing at 0,
+h =
+��
+X ∈ g : �
+X|0 = 0
+�
+= h1 ⊕ · · · ⊕ hs,
+which inherits the grading from the one of g,
+hj+1 = [h1, hj],
+∀j = 1, . . ., s − 1.
+(2.12)
+It is a fundamental fact [18, Th. 5.21] that the nilpotent approximation (Rn, �
+F ) is diffeo-
+morphic to the homogeneous sub-Riemannian space G/H, where G is the Carnot group
+G = exp g (explicitly realized as the subgroup of the flows of the vector fields of g acting
+on Rn from the right) and H = exp h is the Carnot subgroup induced by h.
+In particular, if 0 ∈ Rn is a regular point, then H = {0}, and so the nilpotent approxi-
+mation (Rn, �
+F ) is diffeomorphic to the Carnot group G, see [18, Prop. 5.22].
+Recall that the smooth measure m on the original manifold M can be identified with
+a smooth measure on U ≃ Rn, for which we keep the same notation. In particular, m
+is absolutely continuous with respect to the n-dimensional Lebesgue measure L n on Rn,
+with m = ρ L n for some positive smooth function ρ: Rn → (0, ∞). The corresponding
+blow-up measure on the nilpotent approximation is naturally given by
+�m = lim
+ε→0 mε = ρ(0) L n,
+mε = εQ (dil1/ε)#m,
+in the sense of weak∗ convergence of measures in Rn, where
+Q =
+n
+�
+i=1
+i wi ∈ N
+is the so-called homogeneous dimension of (Rn, �
+F ) and (dil1/ε)# stands for the push-
+forward in the measure-theoretic sense via the dilation map dil1/ε. Consequently, without
+loss of generality, we can assume that ρ(0) = 1, thus endowing (Rn, �
+F ) with the n-
+dimensional Lebesgue measure.
+Notice that divL n �
+Xi = 0, for all i = 1, . . . , L, since
+each �
+Xi is homogeneous of degree −1. Hence, by (2.7), the sub-Laplacian of a function
+u ∈ C∞(Rn) can be globally represented as
+�∆u =
+L
+�
+i=1
+�
+X 2
+i u.
+(2.13)
+It is worth noticing that the metric space (Rn, �d ) induced by the nilpotent approxi-
+mation (Rn, �
+F ) actually coincides with the metric tangent cone at o ∈ M of the metric
+space (M, d) in the sense of Gromov [26], see [18, Th. 7.36] for the precise statement.
+
+12
+L. RIZZI AND G. STEFANI
+In fact, the sub-Riemmanian distance dε induced by the vector fields Xε
+i , i = 1, . . . , L,
+defined in (2.10) is uniformly converging to the distance �d on compact sets as ε → 0.
+It is not difficult to check that the family {(Rn, dε, mε, 0)}ε>0 of pointed metric-measure
+spaces converge to the pointed metric-measure space (Rn, �d, L n, 0) as ε → 0 in the pointed
+measure Gromov–Hausdorff topology, see [13, Sec. 10.3] for details.
+2.5. The curvature-dimension condition. We end this section by recalling the defi-
+nition of curvature-dimension conditions of introduced in [36,48,49].
+On a Polish (i.e., separable and complete) metric space (X, d), we let P(X) be the set
+of probability Borel measures on X and define the Wasserstein (extended) distance W2
+W2
+2(µ, ν) = inf
+��
+X×X d2(x, y) dπ : π ∈ Plan(µ, ν)
+�
+∈ [0, ∞],
+for µ, ν ∈ P(X), where
+Plan(µ, ν) = {π ∈ P(X × X) : (p1)#π = µ, (p2)#π = ν},
+where pi : X ×X → X, i = 1, 2, are the projections on each component and T#µ ∈ P(Y )
+denotes the push-forward measure given by any µ-measurable map T : X → X. The
+function W2 is a distance on the Wasserstein space
+P2(X) =
+�
+µ ∈ P(X) :
+�
+X d2(x, x0) dµ(x) < ∞ for some, and thus any, x0 ∈ X
+�
+.
+Note that (P2(X), W2) is a Polish metric space which is geodesic as soon as (X, d) is. In
+addition, letting Geo(X) be the set of geodesics of (X, d), namely, curves γ : [0, 1] → X
+such that d(γs, γt) = |s−t| d(γ0, γ1), for all s, t ∈ [0, 1], any W2-geodesic µ: [0, 1] → P2(X)
+can be (possibly non-uniquely) represented as µt = (et)♯ν for some ν ∈ P(Geo(X)), where
+et: Geo(X) → X is the evaluation map at time t ∈ [0, 1].
+We endow the metric space (X, d) with a non-negative Borel measure m such that
+m is finite on bounded sets and supp(m) = X.
+We define the (relative) entropy functional Entm : P2(X) → [−∞, +∞] by letting
+Entm(µ) =
+�
+X ρ log ρ dm
+if µ = ρm and ρ log ρ ∈ L1(X, m), while we set Entm(µ) = +∞ otherwise.
+Definition 2.2 (CD(K, ∞) property). We say that a metric-measure space (X, d, m)
+satisfies the CD(K, ∞) property if, for any µ0, µ1 ∈ P2(X) with Entm(µi) < +∞, i = 0, 1,
+there exists a W2-geodesic [0, 1] ∋ s �→ µs ∈ P2(X) joining them such that
+Entm(µs) ≤ (1 − s) Entm(µ0) + s Entm(µ1) − K
+2 s(1 − s) W2
+2(µ0, µ1)
+(2.14)
+for every s ∈ [0, 1].
+The geodesic K-convexity of Entm in (2.14) can be reinforced to additionally encode an
+upper bound on the dimension on the space, as recalled below. For N ∈ (1, ∞), we let
+SN(µ, m) = −
+�
+X ρ−1/N dµ,
+µ = ρm + µ⊥,
+be the N-Rényi entropy of µ ∈ P2(X) with respect to m, where µ = ρm + µ⊥ denotes
+the Radon–Nikodym decomposition of µ with respect to m.
+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+13
+Definition 2.3 (CD(K, N) property). We say that a metric-measure space (X, d, m)
+satisfies the CD(K, N) property for some N ∈ [1, ∞) if, for any µ0, µ1 ∈ P2(X) with
+µi = ρim, i = 0, 1, there exists a W2-geodesic [0, 1] ∋ s �→ µs ∈ P2(X) joining them, with
+µs = (es)♯ν for some ν ∈ P(Geo(X)) such that
+SN′(µs, m) ≤ −
+�
+Geo(X)
+�
+τ (1−s)
+K,N′ (d(γ0, γ1))ρ−1/N′
+0
+(γ0) + τ (s)
+K,N′(d(γ0, γ1))ρ−1/N′
+1
+(γ1)
+�
+dν(γ)
+for every s ∈ [0, 1], N′ ≥ N. Here τ (s)
+K,N is the model distortion coefficient, see [49, p. 137].
+Remark 2.4. The CD(0, N) corresponds to the convexity of the N′-Rényi entropy
+SN′(µs, m) ≤ (1 − s)SN′(µ0, m) + sSN′(µ1, m),
+for every s ∈ [0, 1] and N′ ≥ N, with µ0, µ1 ∈ P2(X) as in Definition 2.3.
+Remark 2.5. For a CD(K, N) metric-measure space, K and N represent a lower bound
+on the Ricci tensor and an upper bond on the dimension, respectively, and we have
+CD(K, N) =⇒ CD(K, N′)
+for all N′ ≥ N, N, N′ ∈ [1, ∞],
+CD(K, N) =⇒ CD(K′, N)
+for all K′ ≤ K, K, K′ ∈ R.
+In particular, the CD(K, ∞) condition (2.14) is the weakest of all the curvature-dimension
+conditions for fixed K ∈ R.
+3. Proofs
+We first deal with Theorems 1.4 and 1.5, from which Theorem 1.2 immediately follows.
+3.1. Proof of Theorem 1.4. We divide the proof in four steps.
+Step 1: passing to the nilpotent approximation via blow-up. Let (Rn, �
+F ) be the nilpotent
+approximation of (M, F) at some fixed point o ∈ M as explained in Section 2.4. Let
+u ∈ C∞
+c (M) and, without loss of generality, let us assume that supp u is contained in the
+domain of the privileged coordinates at o ∈ M. In particular, we identify u with a C∞
+c
+function on Rn. We now apply (1.1) to the dilated function
+uε = u ◦ dil1/ε ∈ C∞
+c (Rn),
+for ε > 0,
+and evaluate this expression at the point dilε(x) ∈ Rn. Exploiting the expressions in Corol-
+lary A.2, we get that
+L
+�
+i,j=1
+Xε
+i u
+�
+Xε
+ijju − Xε
+jjiu
+�
+− (Xε
+iju)2 + Rε
+i,j u ≤ 0,
+(3.1)
+where Xε
+i is as in (2.10) , Xijk = XiXjXk whenever i, j, k ∈ {1, . . . , L}, and Rε
+i,j is a
+reminder locally uniformly vanishing as ε → 0. Therefore, letting ε → 0 in (3.1), by the
+convergence in (2.10) we get
+L
+�
+i,j=1
+�
+Xiu
+��
+Xijju − �
+Xjjiu
+�
+−
+��
+Xiju
+�2 ≤ 0,
+(3.2)
+which is (1.1) with K = 0 for the nilpotent approximation (Rn, �
+F ).
+
+14
+L. RIZZI AND G. STEFANI
+Step 2: improvement via homogeneous structure. We now show that (3.2) implies a
+stronger identity, see (3.4) below, obtained from (3.2) by removing the squared term and
+replacing the inequality with an equality. Recall, in particular, the definition of weight of
+(privileged) coordinates in Definition 2.1. We take u ∈ C∞(Rn) of the form
+u = α + γ,
+where α and γ are homogeneous polynomial of weighted degree 1 and at least 3, respec-
+tively. Since XIα = 0 as soon as the multi-index satisfies |I| ≥ 2 (see [18, Prop. 4.10]),
+we can take the terms with lowest homogeneous degree in (3.2) to get
+L
+�
+i,j=1
+�
+Xiα
+��
+Xijjγ − �
+Xjjiγ
+�
+=
+L
+�
+i=1
+�
+Xiα
+��
+Xi, �∆
+�
+(γ) ≤ 0
+for all such α and γ. In the second equality, we used the fact that the sub-Laplacian �∆ is
+a sum of squares as in (2.13). Since α can be replaced with −α, we must have that
+L
+�
+i=1
+�
+Xiα
+��
+Xi, �∆
+�
+(γ) = 0.
+(3.3)
+Observing that �
+Xiα is homogeneous of degree 0, and thus a constant function, we can
+rewrite (3.3) as
+� L
+�
+i=1
+�
+Xiα �
+Xi, �∆
+�
+(γ) = 0,
+(3.4)
+which is the seeked improvement of (3.2).
+Step 3: construction of the space i ⊂ g1. Let Pn
+1 be the vector space of homogeneous
+polynomials of weighted degree 1 on Rn. Notice that
+Pn
+1 = span{zi | i = 1, . . . , k1},
+k1 = dim D|0,
+that is, Pn
+1 is generated by the monomials given by the coordinates of lowest weight. We
+now define a linear map φ: Pn
+1 → g1 by letting
+φ[α] = �
+∇α =
+L
+�
+i=1
+�
+Xiα �
+Xi
+for all α ∈ Pn
+1 (recall Corollary A.2). We claim that φ is injective. Indeed, if φ[α] = 0 for
+some α ∈ Pn
+1, then, by applying the operator φ[α] to the polynomial α, we get
+0 = φ[α](α) =
+� L
+�
+i=1
+�
+Xiα �
+Xi
+�
+(α) =
+L
+�
+i=1
+(�
+Xiα)2.
+Thus �
+Xiα = 0 for all i = 1, . . ., L.
+Hence α must have weighted degree at least 2.
+However, since α is homogeneous of weighted degree 1, we conclude that α = 0, proving
+that ker φ = {0}. We can thus define the subspace
+i = φ[Pn
+1] ⊂ g1.
+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+15
+By (3.4), any �
+X ∈ i is such that [�
+X, �∆](γ) = 0 for any homogeneous polynomial γ
+of degree at least 3. Exploiting the definitions given in Section 2.4, we observe that a
+differential operator P, homogeneous of weighted degree −d ∈ Z, has the form
+P =
+�
+µ,ν
+aµ,νzµ ∂|ν|
+∂zν ,
+(3.5)
+where µ = (µ1, . . . , µn), ν = (ν1, . . . , νn), µi, νj ∈ N ∪ {0}, aµ,ν ∈ R, and the weighted
+degree of every addend in (3.5) is equal to −d, namely, �n
+i=1(µi − νi)wi = −d.
+Thus, since �
+X and �∆ are homogeneous differential operators of order −1 and −2,
+respectively, then [�
+X, �∆] has order −3, see [18, Prop. 5.16]. It follows that [�
+X, �∆] = 0 as
+differential operator acting on C∞(Rn).
+We now show (1.3). Let us first observe that i ∩ h = {0}. Indeed, if φ[α] ∈ h for some
+α ∈ Pn
+1, that is, φ[α]|0 = 0, then �
+Xiα|0 = 0 for all i = 1, . . ., L. Since �
+Xiα is a constant
+function, this implies φ[α] = 0, as claimed. Therefore, since dim i = dim Pn
+1 = k1, we must
+have g1 = i ⊕ h1 thanks to Lemma 3.1 below.
+Lemma 3.1. With the same notation of Section 2.4, if g1 = v ⊕ h1, then dim v = k1.
+Proof. We claim that the dimension of v is preserved by evaluation at zero, that is,
+dim v|0 = dim v, where dim v|0 is the dimension of v|0 as a subspace of T0Rn, while
+dim v is the dimension of v as a subspace of g. Indeed, we have the trivial inequality
+dim v|0 ≤ dim v. On the other hand, if strict inequality holds, then v must contain non-
+zero vector fields vanishing at zero, contradicting the fact that v ∩ h = {0}. Therefore,
+since dim g1|0 = k1 and dim h1|0 = 0, we get dim v = dim v|0 = k1 as desired.
+□
+Step 4: proof of the Killing property. We have so far proved the existence of i such that
+g1 = i ⊕ h1, and such that any element Y ∈ i commutes with the sub-Laplacian �∆. We
+now show that all such Y is a Killing vector field.
+Let Y ∈ i. Since [Y, �∆] = 0, the induced flow φY
+s , for s ∈ R, commutes with �∆ when
+acting on smooth functions, that is,
+�∆(u ◦ φY
+s ) = ( �∆u) ◦ φY
+s
+(3.6)
+for all u ∈ C∞(Rn) and s ∈ R. Recall the sub-Riemannian Hamiltonian �
+H : T ∗Rn → R,
+�
+H(λ) = 1
+2
+L
+�
+i=1
+⟨λ, �
+Xi⟩2,
+(3.7)
+for all λ ∈ T ∗Rn. By (2.13), �
+H is the principal symbol of �∆. Thus, from (3.6) it follows
+�
+H ◦
+�
+φY
+s
+�∗ = �
+H,
+for all s ∈ R, where the star denotes the pull-back, and thus
+�
+φY
+s
+�∗ is a diffeomorphism
+on T ∗Rn. This means that φY
+s is an isometry, as we now show. Indeed, for any given
+x ∈ Rn, the restriction �
+H|T ∗
+x Rn is a quadratic form on T ∗
+xRn, so (φY
+s )∗ must preserve its
+kernel, that is,
+(φY
+s )∗ ker �
+H|T ∗
+φYs (x)Rn = ker �
+H|T ∗
+x Rn
+(3.8)
+
+16
+L. RIZZI AND G. STEFANI
+for all x ∈ Rn. By (3.7), it holds ker �
+H|T ∗
+x Rn = �
+D⊥
+x , where ⊥ denotes the annihilator of a
+vector space. By duality, from (3.8) we obtain that (φY
+s )∗ �
+Dx = �
+DφYs (x) for all x ∈ Rn as
+required by (1.2). Finally, for λ ∈ T ∗
+xM, let λ# ∈ Dx be uniquely defined by gx(λ#, V ) =
+⟨λ, V ⟩x for all V ∈ Dx, and notice that the map λ �→ λ# is surjective on Dx. Then it holds
+∥λ#∥2
+x = 2�
+H(λ), see Lemma A.1. Thus, since (φY
+s )∗ preserves �
+H, the map (φY
+s )∗ preserves
+the sub-Riemannian norm, and thus g. This means that φY
+s is an isometry, concluding
+the proof of Theorem 1.4.
+□
+3.2. Proof of Theorem 1.5. We claim that
+gj = hj
+for all j ≥ 2.
+(3.9)
+Note that (3.9) is enough to conclude the proof of Theorem 1.5, since, from (3.9) combined
+with (2.11) and (2.12), we immediately get that
+g = g1 ⊕ h2 ⊕ · · · ⊕ hs.
+In particular, we deduce that g|0 = g1|0, which in turn implies that g must be commu-
+tative, otherwise the bracket-generating condition would fail. To prove (3.9), we proceed
+by induction on j ≥ 2 as follows.
+Proof of the base case j = 2. We begin by proving the base case j = 2 in (3.9). To this
+aim, let �
+X ∈ i and �Y ∈ g1. By definition of Lie bracket, we can write
+�
+φ �
+X
+−s
+�
+∗
+�Y = s
+��
+X, �Y
+�
++ o(s)
+as s → 0,
+where φ �
+X
+s , for s ∈ R, is the flow of �
+X. Since g1|x = �
+D|x for all x ∈ Rn, and since �
+X is
+Killing (in particular (1.2) holds for its flow), we have that [�
+X, �Y ]|x ∈ �
+D|x for all x ∈ Rn.
+Since [�
+X, �Y ] ∈ g2 and so, in particular, [�
+X, �Y ] is homogeneous of degree −2, we have
+[�
+X, �Y ]|0 =
+�
+j : wj=2
+aj ∂zj|0,
+for some constants aj ∈ R. But we also must have that [�
+X, �Y ]|0 ∈ �
+D|0 and so, since
+�
+D|0 = span
+�
+∂zj : wj = 1
+�
+according to Definition 2.1, [�
+X, �Y ]|0 = 0, that is, [�
+X, �Y ] ∈ h. We thus have proved that
+[i, g1] ⊂ h2. In particular, since g1 = i ⊕ h1, we get
+[i, i] ⊂ h2
+and
+[i, h1] ⊂ h2,
+(3.10)
+from which we readily deduce (3.9) for j = 2.
+Proof of the induction step. Let us assume that (3.9) holds for some j ∈ N, j ≥ 2. Since
+g1 = i ⊕ h1, by the induction hypothesis we can write
+gj+1 = [g1, gj] = [g1, hj] = [i, hj] + [h1, hj] = [i, hj] + hj+1.
+We thus just need to show that [i, hj] ⊂ hj+1 for all j ∈ N with j ≥ 2. Note that we
+actually already proved the case j = 1 in (3.10). Again arguing by induction (taking
+j = 1 as base case), by the Jacobi identity and (3.10) we have
+[i, hj+1] = [i, [h1, hj]] = [h1, [hj, i]] + [hj, [i, h1]] ⊂ [h1, hj+1] + [hj, h2] = hj+2
+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+17
+as desired, concluding the proof of the induction step.
+□
+Remark 3.2 (Proof of Theorem 1.5 in the case h = {0}). The proof of Theorem 1.5 is
+much simpler if the nilpotent approximation (Rn, �
+F) is a Carnot group, i.e., h = {0}.
+Indeed, in this case, the base case j = 2 in (3.9) immediately implies that g2 = h2 = {0},
+which in turn gives g = g1, so that g is commutative.
+3.3. Proof of Theorem 1.8. In the following, we assume that the reader is familiar with
+the notions of upper gradient and of q-upper gradient, see [5] for the precise definitions.
+The next two lemmas are proved in [27] for sub-Riemannians structures on Rn equipped
+with the Lebesgue measure, and are immediately extended to the weighted case.
+Lemma 3.3. Let (M, d, m) be as in Theorem 1.8. If u ∈ C(M) and 0 ≤ g ∈ L1
+loc(M, m)
+be an upper gradient of u, then u ∈ HW1,1
+loc(M, m) with ∥∇u∥ ≤ g m-a.e. In particular, if
+u ∈ Lip(M, d), then ∥∇u∥ ≤ Lip(u).
+Proof. Without loss of generality we may assume that M = Ω ⊂ Rn is a bounded open
+set, the sub-Riemannian structure is induced by a family of smooth bracket-generating
+vector fields F = {X1, . . ., XL} on Ω and m = θL n, where θ: Ω → [0, ∞) is smooth
+and satisfies 0 < infΩ θ ≤ supΩ θ < ∞. Hence, L1(Ω, θL n) = L1(Ω, L n) as sets, with
+equivalent norms, so that 0 ≤ g ∈ L1
+loc(Ω, L n) is an upper gradient of u ∈ C(Ω). Hence,
+by [27, Th. 11.7], we get that u ∈ HW1,1
+loc(Ω, L n), with ∥∇u∥ ≤ g L n-a.e., and thus
+θL n-a.e., on Ω. By definition of distributional derivative, we can write
+�
+Ω v Xiu dx =
+�
+Ω u [−Xiv + div(Xi)v] dx,
+∀ v ∈ C1
+c (Ω), i = 1, . . . , L,
+where div denotes the Euclidean divergence.
+We apply the above formula with test
+function v = θw, for any w ∈ C1
+c (Ω), getting
+�
+Ω w Xiu θ dx =
+�
+Ω u
+�
+−Xiw + div(Xi)w + Xiθ
+θ w
+�
+θ dx,
+∀ w ∈ C1
+c (Ω), i = 1, . . . , L.
+The function within square brackets is the adjoint X∗
+i w with respect to the measure
+θL n. It follows that HW1,q(Ω, θL n) = HW1,q(Ω, L n) as sets, with equivalent norms. In
+particular, u ∈ W1,1
+D,loc(Ω, θL n) as desired.
+□
+Lemma 3.4 (Meyers–Serrin). Let (M, d, m) be as in Theorem 1.8 and let q ∈ [1, ∞).
+Then HW1,q(M, m) ∩ C∞(M) is dense in HW1,q(M, m).
+Proof. Up to a partition of unity and exhaustion argument, we can reduce to the case
+M = Ω ⊂ Rn is a bounded open set and m = θL n, where θ: Ω → [0, ∞) is as in the
+previous proof, so that HW1,q(Ω, L n) = HW1,q(Ω, θL n) as sets, with equivalent norms.
+In particular, we can assume that θ ≡ 1. This case is proved in [27, Th. 11.9].
+□
+Lemma 3.5. Let (M, d, m) be as in Theorem 1.8 and let q ∈ [1, ∞). If u ∈ HW1,q(M, m),
+then ∥∇u∥ is the minimal q-upper gradient of u.
+Proof. Let us first prove that ∥∇u∥ is a q-upper gradient of u. Indeed, by Lemma 3.4, we
+can find (uk)k∈N ⊂ HW1,q(M, m) ∩ C∞(M) such that uk → u in HW1,q(M, m) as k → ∞.
+
+18
+L. RIZZI AND G. STEFANI
+It is well-known that the sub-Riemannian norm of the gradient of a smooth function is
+an upper gradient, see [27, Prop. 11.6]. Thus, for uk it holds
+|uk(γ(1)) − uk(γ(0))| ≤
+�
+γ ∥∇uk∥ ds.
+Arguing as in [28, p. 179], using Fuglede’s lemma (see [28, Lem. 7.5 and Sec. 10]), we pass
+to the limit for k → ∞ in the previous equality, outside a q-exceptional family of curves.
+This proves that any Borel representative of ∥∇u∥ is a q-upper gradient of u.
+We now prove that ∥∇u∥ is indeed minimal. Let 0 ≤ g ∈ Lq(M, m) be any q-upper
+gradient of u. Arguing as in [28, p. 194], we can find a sequence (gk)k∈N ⊂ Lq(M, m) of
+upper gradients of u such that gk ≥ g for all k ∈ N and gk → g both pointwise m-a.e.
+on M and in Lq(M, m) as k → ∞. By Lemma 3.3, we thus must have that ∥∇u∥ ≤ gk
+m-a.e. on M for all k ∈ N. Hence, passing to the limit, we conclude that ∥∇u∥ ≤ g m-a.e.
+on M for any q-upper gradient g, concluding the proof.
+□
+We are now ready to deal with the proof of Theorem 1.8.
+Proof of (i). Recall that, here, q > 1. We begin by claiming that
+W1,q(M, d, m) ⊂ HW1,q(M, m)
+(3.11)
+isometrically, with ∥∇u∥ = |Du|w,q. Indeed, let u ∈ W1,q(M, d, m). By a well-known
+approximation argument, combining [5, Prop. 4.3, Th. 5.3 and Th. 7.4], we find (uk)k∈N ⊂
+Lip(M, d) ∩ W1,q(M, d, m) such that
+uk → u
+and
+|Duk|w,q → |Du|w,q
+in Lq(M, m).
+(3.12)
+Since uk ∈ Lip(M, d), by Lemma 3.3 we know that uk ∈ HW1,q(M, m).
+Hence, by
+Lemma 3.5, |Duk|w,q = ∥∇uk∥, and we immediately get that
+sup
+k∈N
+�
+M ∥∇uk∥q dm < ∞.
+Therefore, up to passing to a subsequence, (Xiuk)k∈N is weakly convergent in Lq(M, m),
+say Xiuk ⇀ αi ∈ Lq(M, m), for all i = 1, . . ., L. We thus get that u ∈ HW1,q(M, m) with
+Xiu = αi and thus ∇u =
+�L
+i=1 αiXi. By stability of q-upper gradients, [5, Th. 5.3 and
+Thm. 7.4], ∥∇u∥ is a q-upper gradient of u. By semi-continuity of the norm, we obtain
+�
+M ∥∇u∥q dm ≤ lim inf
+k→∞
+�
+M ∥∇uk∥q dm =
+�
+M |Du|q
+w,q dm,
+where we used (3.12). By definition of minimal q-upper gradient we thus get that ∥∇u∥ =
+|Du|w,q m-a.e., and the claimed inclusion in (3.11) immediately follows.
+We now observe that it also holds
+HW1,q(M, m) ∩ C∞(M) ⊂ W1,q(M, d, m),
+(3.13)
+with ∥∇u∥ = |Du|w,q. We just need to notice that, if u ∈ C∞(M), then ∥∇u∥ is an
+upper gradient of u, see [27, Prop. 11.6]. Therefore, by Lemma 3.3, ∥∇u∥ must coincide
+with the minimal q-upper gradient of u, i.e., ∥∇u∥ = |Du|w m-a.e., and (3.13) readily
+follows. In view of the isometric inclusions (3.11) and (3.13), and of the density provided
+by Lemma 3.4, this concludes the proof of (i).
+□
+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+19
+Proof of (ii). Let us assume that (M, d, m) satisfies the CD(K, ∞) property for some
+K ∈ R.
+By the previous point (i), we know that (M, d, m) satisfies the RCD(K, ∞)
+property. Consequently, since clearly C∞
+c (M) ⊂ W1,2(M, d, m) by (3.13), [6, Rem. 6.3]
+(even if the measure m is σ-finite, see [4, Sec. 7] for a discussion) implies that
+1
+2
+�
+M ∆v ∥∇u∥2 dm −
+�
+M v g(∇u, ∇∆u) dm ≥ K
+�
+M v ∥∇u∥2 dm
+for all u, v ∈ C∞
+c (M) with v ≥ 0 on M, from which we readily deduce (1.1).
+□
+Remark 3.6. The above proofs work for more general measures m. Namely, we can
+assume that, locally on any bounded coordinate neighborhood Ω ⊂ Rn, m = θL n with
+θ ∈ W1,1(Ω, L n) ∩ L∞(Ω, L n).
+In this case, the positivity of m corresponds to the
+requirement that θ is locally essentially bounded from below away from zero, in charts.
+3.4. Proof of Theorem 1.10. We prove the two points in the statement separately.
+Proof of (i). The case p = 0 has been already considered by Juillet in [32]. For p > 0,
+we can argue as follows. Let A0 = [−ℓ −1, −ℓ]×[0, 1] and A1 = [ℓ, ℓ + 1]×[0, 1] for ℓ > 0.
+We will shortly prove that the midpoint set
+A1/2 =
+�
+q ∈ R2 : ∃ q0 ∈ A0, ∃ q1 ∈ A1 with d(q, q0) = d(q, q1) = 1
+2 d(q0, q1)
+�
+satisfies
+A1/2 ⊂ [−1 − εℓ, 1 + εℓ] × [0, 1]
+(3.14)
+for some εℓ > 0, with εℓ ↓ 0 as ℓ → ∞. Since mp(A0) = mp(A1) ∼ ℓp as ℓ → ∞, we get
+�
+mp(A0) mp(A1) > mp(A1/2)
+for large ℓ > 0. This contradicts the logarithmic Brunn–Minkowski BM(0, ∞) inequality,
+which is a consequence of the CD(0, ∞) condition, see [50, Th. 30.7].
+To prove (3.14), let qi ∈ Ai, qi = (xi, yi), and let γ(t) = (x(t), y(t)), t ∈ [0, 1], be a
+geodesic such that γ(i) = qi, with i = 0, 1. We first note that
+min{y0, y1} ≤ y(t) ≤ max{y0, y1}
+for all t ∈ [0, 1],
+(3.15)
+since any curve that violates (3.15) can be replaced by a strictly shorter one satisfy-
+ing (3.15). In particular, we get that A1/2 ⊂ R × [0, 1]. Let us now observe that
+|xa − xb| ≤ d(a, b) ≤ |xa − xb| +
+|ya − yb|
+max{|xa|, |xb|}
+for all a = (xa, ya) and b = (xb, yb) with xa, xb ̸= 0. Therefore, if q = (x, y) ∈ A1/2, then
+|x − x0| ≤ d(q, q0) = 1
+2 d(q0, q1) ≤ ℓ + 1 + O(1/ℓ)
+and, similarly, |x − x1| ≤ ℓ + 1 + O(1/ℓ). Since x0 ∈ [−ℓ − 1, −ℓ] and x1 ∈ [ℓ, ℓ + 1], we
+deduce that |x| ≤ 1 + O(1/ℓ), concluding the proof of the claimed (3.14).
+□
+
+20
+L. RIZZI AND G. STEFANI
+Proof of (ii). Out of the negligible set {x = 0}, the metric g on Gp given by (1.5) is
+locally Riemannian. Recalling (1.6) and (1.7), the BE(K, ∞) inequality (1.1) is implied by
+the lower bound Ric∞,V ≥ K via Bochner’s formula, where Ric∞,V is the ∞-Bakry–Émery
+Ricci tensor of (R2, g, e−V volg), see [50, Ch. 14, Eqs. (14.36) – (14.51)]. By Lemma 3.7
+below, we have Ric∞,V ≥ 0 for all p ≥ 1, concluding the proof.
+□
+Lemma 3.7. Let p ∈ R and N > 2. The N-Bakry–Émery Ricci tensor of the Grushin
+metric (1.5), with weighted measure mp = |x|p dx dy, for all x ̸= 0 is
+RicN,V = p − 1
+x2
+g −(p + 1)2
+N − 2
+dx ⊗ dx
+x2
+,
+with the convention that 1/∞ = 0.
+Proof. The N-Bakry–Émery Ricci tensor of a n-dimensional weighted Riemannian struc-
+ture (g, e−V volg), for N > n, is given by
+RicN,V = Ricg + HessgV − dV ⊗ dV
+N − n ,
+(3.16)
+see [50, Eq. (14.36)]. In terms of the frame (1.4), the Levi-Civita connection is given by
+∇XX = ∇XY = 0,
+∇Y X = −1
+xY,
+∇Y Y = 1
+xX,
+whenever x ̸= 0. Recalling that, from (1.7), V (x) = −(p + 1) log |x|, for x ̸= 0, we obtain
+Ricg = − 2
+x2 g,
+HessgV = (p + 1)
+x2
+g,
+dV = −p + 1
+x
+dx,
+(3.17)
+whenever x ̸= 0. The conclusion thus follows by inserting (3.17) into (3.16).
+□
+3.5. Proof of Theorem 1.11. The statement is a consequence of the geodesic convexity
+of G+
+p and the computation of the N-Bakry–Émery curvature in Lemma 3.7. Since the
+proof uses quite standard arguments, we simply sketch its main steps.
+The interior of G+
+p , i.e., the open half-plane, can be regarded as a (non-complete)
+weighted Riemannian manifold with metric g as in (1.5) and weighted volume as in (1.7).
+Let µ0, µ1 ∈ P2(G+
+p ), µ0, µ1 ≪ mp, with bounded support contained in the Riemannian
+region {x > ε}, for some ε ≥ 0.
+Let (µs)s∈[0,1] be a W2-geodesic joining µ0 and µ1.
+By a well-known representation
+theorem (see [50, Cor. 7.22]), there exists ν ∈ P(Geo(G+
+p )), supported on the set
+Γ = (e0 × e1)−1(supp µ0 × supp µ1), such that µs = (es)♯ν for all s ∈ [0, 1]. Since the
+set {x ≥ ε} is a geodesically convex subset of the full Grushin plane Gp (by the same
+argument of [46, Prop. 5]), any γ ∈ Γ is contained for all times in the region {x > 0}.
+Therefore, Γ is a set of Riemannian geodesics contained in the weighted Riemannian struc-
+ture ({x > 0}, g, e−V volg). By Lemma 3.7, we have RicN,V ≥ 0 for all N ≥ Np, where Np
+is as in (1.8). At this point, a standard argument shows that the Rényi entropy is convex
+along Wasserstein geodesics joining µ0 with µ1, see the proof of [49, Th. 1.7] for example.
+The extension to µ0, µ1 ∈ P2(G+
+p ), with µ0, µ1 ≪ mp and compact support possibly
+touching the singular region {x = 0}, is achieved via a standard approximation argument.
+More precisely, one reduces to the previous case and exploits the stability of optimal
+transport [50, Th. 28.9] and the lower semi-continuity of the Rényi entropy [50, Th. 29.20].
+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+21
+Finally, the extension to general µ0, µ1 ∈ P2(G+
+p ) follows the routine argument outlined
+in [9, Rem. 2.12], which works when µs = (es)♯ν, s ∈ [0, 1], and ν is concentrated on a set
+of non-branching geodesics. This proves the ‘if’ part of the statement.
+The ‘only if’ part is also standard. The CD(0, N) condition for N > 2 implies that, on
+the Riemannian region {x > 0}, RicN,V ≥ 0, but this is false for N < Np.
+The fact that G+
+p is infinitesimally Hilbertian follows from Remark 1.9, by noting that
+mp is positive and smooth out of the closed set {x = 0}, which has zero measure. An
+alternative proof follows from the observation that G+
+p is a Ricci limit, see [42].
+□
+Appendix A. Gradient and Laplacian representations formulas
+For the reader’s convenience, in this appendix we provide a short proof of the repre-
+sentation formulas (2.5) and (2.7), in the rank-varying case.
+Lemma A.1. For λ ∈ T ∗M, let λ# ∈ D be uniquely defined by
+g(λ#, V ) = ⟨λ, V ⟩
+for all V ∈ D, where ⟨·, ·⟩ denotes the action of covectors on vectors. Then
+∥λ#∥2 =
+L
+�
+i=1
+�
+λ#, Xi
+�2.
+(A.1)
+As a consequence, if λ, µ ∈ T ∗M, then
+g(λ#, µ#) =
+L
+�
+i=1
+⟨λ, Xi⟩⟨µ, Xi⟩.
+(A.2)
+Proof. Given u ∈ RL, we set Xu = �L
+i=1 uiXi and define
+u∗ ∈ argmin
+�
+v �→ |v| : v ∈ RL, Xv = Xu
+�
+.
+In other words, for Xu ∈ D, u∗ is the element of minimal Euclidean norm such that
+Xu∗ = Xu. Note that, by definition, it holds ∥Xu∥ = |u∗|. We thus have
+∥λ#∥ = sup
+�
+g(λ#, X) : ∥X∥ = 1, X ∈ D
+�
+= sup
+�
+g(λ#, Xu) : |u∗| = 1, u ∈ RL�
+.
+We now claim that
+sup
+�
+g(λ#, Xu) : |u∗| = 1, u ∈ RL�
+= sup
+�
+g(λ#, Xu) : |u| = 1, u ∈ RL�
+.
+(A.3)
+Indeed, the inequality ≤ in (A.3) is obtained by observing that Xu = Xu∗ for any u ∈ RL.
+To prove the inequality ≥ in (A.3), we observe that, if u ∈ RL is such that |u| = 1 and
+0 < |u∗| < 1, then v = u/|u∗| satisfies |v∗| = 1 and gives
+g(λ#, Xv) > g(λ#, Xv) |u∗| = g(λ#, Xu).
+(A.4)
+Furthermore, if |u| = 1 and u∗ = 0, then Xu = 0 so also in this case we find v ∈ Rn with
+v∗ = 1 such that (A.4) holds. This ends the proof of the claimed (A.3). Hence, since
+g(λ#, Xu) =
+L
+�
+i=1
+g(λ#, Xi) ui,
+
+22
+L. RIZZI AND G. STEFANI
+we easily conclude that
+∥λ#∥ = sup
+�
+g(λ#, Xu) : |u| = 1, u ∈ RL�
+=
+�
+�
+�
+�
+L
+�
+i=1
+g(λ#, Xi)2,
+proving (A.1). Equality (A.2) then follows by polarization.
+□
+Corollary A.2. The following formulas hold:
+∇u =
+L
+�
+i=1
+Xiu Xi,
+(A.5)
+∆u =
+L
+�
+i=1
+�
+X2
+i u + Xiu divm(Xi)
+�
+,
+(A.6)
+g(∇u, ∇v) =
+L
+�
+i=1
+Xiu Xiv,
+(A.7)
+for all u, v ∈ C∞(M). In particular, ∥∇u∥ =
+�L
+i=1(Xiu)2 for all u ∈ C∞(M).
+Proof. We prove each formula separately.
+Proof of (A.5). Recalling the definition in (2.4), we can pick λ = du in (A.2) to get
+�
+du, µ#�
+= g(∇u, µ#) =
+L
+�
+i=1
+⟨du, Xi⟩⟨µ, Xi⟩
+=
+L
+�
+i=1
+Xiu ⟨µ, Xi⟩ = g
+�
+µ#,
+L
+�
+i=1
+Xiu Xi
+�
+whenever µ ∈ T ∗
+xM. Since the map #: T ∗
+xM → Dx is surjective, we immediately get (A.5).
+Proof of (A.6). Recall that
+divm(fX) = Xf + f divm(X)
+for any f ∈ C∞(M) and X ∈ Γ(TM). Hence, from the definition in (2.6), we can compute
+∆u = divm(∇u) =
+L
+�
+i=1
+divm(Xiu Xi) =
+L
+�
+i=1
+�
+X2
+i u Xi + Xiu divm(Xi)
+�
+,
+which is the desired (A.6).
+Proof of (A.7). Choosing λ = du and µ = dv in (A.2), we can compute
+g(∇u, ∇v) =
+L
+�
+i=1
+⟨du, Xi⟩ ⟨dv, Xi⟩ =
+L
+�
+i=1
+Xiu Xiv
+and the proof is complete.
+□
+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+23
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+
+FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS
+25
+(L. Rizzi) Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265,
+34136 Trieste (TS), Italy
+Email address: [email protected]
+(G. Stefani) Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265,
+34136 Trieste (TS), Italy
+Email address: [email protected] or [email protected]
+

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+Bayesian Learning for Dynamic Inference
+Aolin Xu
+Peng Guan
+Abstract
+The traditional statistical inference is static, in the sense that the estimate of the quantity
+of interest does not affect the future evolution of the quantity. In some sequential estimation
+problems however, the future values of the quantity to be estimated depend on the estimate of
+its current value. This type of estimation problems has been formulated as the dynamic inference
+problem. In this work, we formulate the Bayesian learning problem for dynamic inference,
+where the unknown quantity-generation model is assumed to be randomly drawn according to
+a random model parameter. We derive the optimal Bayesian learning rules, both offline and
+online, to minimize the inference loss. Moreover, learning for dynamic inference can serve as a
+meta problem, such that all familiar machine learning problems, including supervised learning,
+imitation learning and reinforcement learning, can be cast as its special cases or variants. Gaining
+a good understanding of this unifying meta problem thus sheds light on a broad spectrum of
+machine learning problems as well.
+1
+Introduction
+1.1
+Dynamic inference
+Traditional statistical estimation, or statistical inference in general is static, in the sense that the
+estimate of the quantity of interest does not affect the future evolution of the quantity. In some
+sequential estimation problems however, we do encounter the situation where the future value of the
+quantity to be estimated depends on the estimate of its current value. Examples include 1) stock
+price prediction by big investors, where the prediction of the tomorrow’s price of a stock affects
+tomorrow’s investment decision, which further changes the stock’s supply-demand status and hence
+its price the day after tomorrow; 2) interactive product recommendation, where the estimate of
+a user’s preference based on the user’s activity leads to certain product recommendations to the
+user, which would in turn shape the user’s future activity and preference; 3) behavior prediction in
+multi-agent systems, e.g. vehicles on the road, where the estimate of an adjacent vehicle’s intention
+based on its current driving situation leads to a certain action of the ego vehicle, which can change
+the future driving situation and intention of the adjacent vehicle. We may call such problems as
+dynamic inference, which is formulated and studied in depth in [1]. It is shown that the problem of
+dynamic inference can be converted to an Markov decision-making process (MDP), and the optimal
+estimation strategy can be derived through dynamic programming. We give a brief overview of the
+problem of dynamic inference in Section 2.
+1.2
+Learning for dynamic inference
+There are two major ingredients in dynamic inference: the probability transition kernels of the
+quantity of interest given each observation, and the probability transition kernels of the next
+1
+arXiv:2301.00032v1  [cs.LG]  30 Dec 2022
+
+observation given the current observation and the estimate of the current quantity of interest. We
+may call them the quantity-generation model and the observation-transition model, respectively.
+Solving the dynamic inference problem requires the knowledge of the two models. However, in most
+of the practically interesting situations, we do not have such knowledge. Instead, we either have a
+training dataset from which we can learn these models or we can learn them on-the-fly during the
+inference.
+In this work, we set up the learning problem in a Bayesian framework, and derive the optimal
+learning rules, both offline (Section 3) and online (Section 4), for dynamic inference under this
+framework. Specifically, we assume the unknown models are elements in some parametric families
+of probability transition kernels, and the unknown model parameters are randomly drawn according
+to some prior distributions. The goal is then to find an optimal Bayesian learning rule, which can
+return an estimation strategy that minimizes the inference loss. The approach we take toward this
+goal is converting the learning problem to an MDP with an augmented state, which consists of
+the current observation and a belief vector of the unknown parameters, and solving the MDP by
+dynamic programming over the augmented state space. The solution, though optimal, may still be
+computationally challenging unless the belief vector can be compactly represented. Nevertheless, it
+already has a greatly reduced search space compared to the original learning problem, and provides
+a theoretical basis for the design of more computationally efficient approximate solutions.
+Perhaps equally importantly, the problem of learning for dynamic inference can serve as a meta
+problem, such that almost all familiar learning problems can be cast as its special cases or variants.
+Examples include supervised learning, imitation learning, and reinforcement learning, including
+bandit and contextual bandit problems. For instance, the Bayesian offline learning for dynamic
+inference can be viewed as an extension of the behavior cloning method in imitation learning [2–4],
+in that it not only learns the demonstrator’s action-generation model, but simultaneously learns a
+policy based on the learned model to minimize the overall imitation error. As another instance, the
+quantity to be estimated in dynamic inference may be viewed as a latent variable of the loss function,
+so that the Bayesian online learning for dynamic inference can be viewed as Bayesian reinforcement
+learning [5–8], where an optimal policy is learned by estimating the unknown loss function. Learning
+for dynamic inference thus provides us with a unifying formulation of different learning problems.
+Having a good understanding of this problem is helpful for gaining better understandings of the
+other learning problems as well.
+1.3
+Relation to existing works
+The problem of dynamic inference and learning for dynamic inference appear to be new, but it
+can be viewed from different angles, and is related to a variety of existing problems. The most
+intimately related work is the original formulations of imitation learning [9]. The online learning for
+dynamic inference is closely related to ans subsumes Bayesian reinforcement learning. Some recent
+study on Bayesian reinforcement learning and interactive decision making include [10,11].
+A problem formulation with a similar spirit in a minimax framework appear recently in [12]. In
+that work, an adversarial online learning problem where the action in each round affects the future
+observed data is set up. It may be viewed as adversarial online learning for dynamic minimax
+inference, from our standpoint. The advantage of the Bayesian formulation is that all the variables
+under consideration, including the unknown model parameters, are generated from some fixed joint
+distribution, thus the optimality of learning can be defined and the optimal learning rule can be
+derived. On the contrary, with the adversarial formulation, only certain definitions of regret can be
+2
+
+studied.
+The overall optimality proof technique we adopt is similar to those used in solving partially
+observed MDP (POMDP) and Bayesian reinforcement learning over the augmented belief space
+[13,14]. Several proofs are adapted from the rigorous exposition of the optimality of the belief-state
+MDP reformulation of the POMDP [15].
+As mentioned in the previous subsection, Bayesian learning for dynamic inference can be viewed
+as a unifying formulation for Bayesian imitation learning and Bayesian reinforcement learning.
+These problems are surveyed in [16–18] for relevant imitation learning, and in [8,19–22] for relevant
+reinforcement learning.
+2
+Overview of dynamic inference
+2.1
+Problem formulation
+The problem of an n-round dynamic inference is to estimate n unknown quantities of interest
+Y n sequentially based on observations Xn, where in the ith round of estimation, Xi depends on
+the observation Xi−1 and the estimate �Yi−1 of Yi−1 in the previous round, while the quantity of
+interest Yi only depends on Xi, and the estimate �Yi of Yi can depend on everything available so
+far, namely (Xi, �Y i−1), through an estimator ψi as �Yi = ψi(Xi, �Y i−1). The sequence of estimators
+ψn = (ψ1, . . . , ψn) constitute an estimation strategy. We assume to know the distribution PX1 of the
+initial observation, and the probability transition kernels (KXi|Xi−1,�Yi−1)n
+i=2 and (KYi|Xi)n
+i=1. These
+distributions and ψn define a joint distribution of (Xn, Y n, �Y n), all the variables under consideration.
+The Bayesian network of the random variables in dynamic inference with a Markov estimation
+strategy, meaning that each estimator has the form ψi : X → �Y, is illustrated in Fig. 1.
+Figure 1: Bayesian network of the random variables under consideration with n = 4. Here we
+assume the estimates are made with Markov estimators, such that �Yi = ψi(Xi).
+The goal of dynamic inference can then be formally stated as finding an estimation strategy to
+minimize the accumulated expected loss over the n-rounds:
+arg min
+ψn
+E
+�
+n
+�
+i=1
+ℓ(Xi, Yi, �Yi)
+�
+,
+�Yi = ψi(Xi, �Y i−1)
+(1)
+where ℓ : X×Y× �Y → R is a loss function that evaluates the estimate made in each round. Compared
+with the traditional statistical inference under the Bayesian formulation, where the goal is to find
+an estimator ψ of a random quantity Y based on a jointly distributed observation X to minimize
+E[ℓ(Y, ψ(X))], we summarize the two distinctive features of dynamic inference in (1):
+3
+
+Yi
+Y2
+Y3
+Y4
+1
+X1
+★ X2
++ X3
+<Y1
+4
+<Y3
+4
+12
+14• The joint distribution of the pair (Xi, Yi) changes in each round in a controlled manner, as it
+depends on (Xi−1, �Yi−1);
+• The loss in each round is contextual, as it depends on Xi.
+2.2
+Optimal estimation strategy for dynamic inference
+It is shown in [1] that optimization problem in (1) is equivalent to
+arg min
+ψn
+E
+�
+n
+�
+i=1
+¯ℓ(Xi, �Yi)
+�
+,
+(2)
+where ¯ℓ(x, ˆy) ≜ E[ℓ(x, Y, ˆy)|X = x, �Y = ˆy], and for any realization (xi, ˆyi) of (Xi, �Yi), it can be
+computed as ¯ℓ(xi, ˆyi) = E[ℓ(xi, Yi, ˆyi)|Xi = xi]. With this reformulation, the unknown quantities Yi
+do not appear in the loss function any more, and the optimization problem becomes a standard
+MDP. The observations Xn become the states in this MDP, the estimates �Y n become the actions,
+the probability transition kernel KXi|Xi−1,�Yi−1 now defines the controlled state transition, and any
+estimation strategy ψn becomes a policy of this MDP. The goal becomes finding an optimal policy
+for this MDP to minimize the accumulated expected loss defined w.r.t. ¯ℓ. The solution to the MDP
+will be an optimal estimation strategy for dynamic inference.
+From the theory of MDP it is known that the optimal estimators (ψ∗
+1, . . . , ψ∗
+n) for the optimization
+problem in (2) can be Markov, meaning that ψ∗
+i can take only Xi as input, and the values of the
+optimal estimates ψ∗
+i (x) for i = 1, . . . , n and x ∈ X can be found via dynamic programming.
+Define the functions Q∗
+i : X × �Y → R and V ∗
+i : X → R recursively as Q∗
+n(x, ˆy) ≜ ¯ℓ(x, ˆy), V ∗
+i (x) ≜
+minˆy∈�Y Q∗
+i (x, ˆy) for i = n, . . . , 1, and Q∗
+i (x, ˆy) ≜ ¯ℓ(x, ˆy) + E[V ∗
+i+1(Xi+1)|Xi = x, �Yi = ˆy] for i =
+n − 1, . . . , 1. The optimal estimate to make in the ith round when Xi = x is then
+ψ∗
+i (x) ≜ arg min
+ˆy∈�Y
+Q∗
+i (x, ˆy).
+(3)
+It is shown that the estimators (ψ∗
+1, . . . , ψ∗
+n) defined in (3) achieve the minimum in (1). Moreover,
+For any i = 1, . . . , n and any initial distribution PXi,
+min
+ψi,...,ψn E
+�
+n
+�
+j=i
+ℓ(Xj, Yj, �Yj)
+�
+= E[V ∗
+i (Xi)],
+(4)
+with the minimum achieved by (ψ∗
+i , . . . , ψ∗
+n). As shown by the examples in [1], the implication of
+the optimal estimation strategy is that, in each round of estimation, the estimate to make is not
+necessarily the optimal single-round estimate in that round, but one which takes into account the
+accuracy in that round, and tries to steer the future observations toward those with which the
+quantities of interest tend to easy to estimate.
+3
+Bayesian offline learning for dynamic inference
+Solving dynamic inference requires the knowledge of the quantity-generation models (KYi|Xi)n
+i=1
+and the observation-transition models (KXi|Xi−1,�Yi−1)n
+i=2. In most of the practically interesting
+situations however, we may not have such knowledge. Instead we may have a training dataset from
+4
+
+which we can learn these models, or may learn them on-the-fly during inference. In this section and
+the next one, we study the offline learning and the online learning problems for dynamic inference
+respectively, with unknown quantity-generation models but known observation transition models.
+This is already a case of sufficient interest, as the observation-transition model in many problems, e.g.
+imitation learning, are available. The proof techniques we develop carry over to the case where the
+observation-transition models are also unknown. In that case, the solution will have the same form,
+but a further-augmented state with a belief vector of the observation-transition model parameter;
+and the belief update has two parts, separately for the parameters of the quantity-generation model
+and the observation transition model.
+Formally, in this section we assume that the initial distribution PX1 and the probability transition
+kernels (KXi|Xi−1,�Yi−1)n
+i=2 are still known, while the unknown KYi|Xi’s are the same element PY |X,W
+of a parametrized family of kernels {PY |X,w, w ∈ W} and the unknown parameter W is a random
+element of W with prior distribution PW . The training data Zm consists of m samples, and is drawn
+from some distribution PZm|W with W as a parameter. This setup is quite flexible, in that the Zm
+need not be generated in the same way as the data generated during inference. One example is a
+setup similar to imitation learning, where Zm = ((X′
+1, Y ′
+1), . . . , (X′
+m, Y ′
+m)) and
+PZm|W = PX′
+1KY ′
+1|X′
+1
+n
+�
+i=2
+KX′
+i|X′
+i−1,Y ′
+i−1KY ′
+i |X′
+i
+(5)
+with PX′
+1 = PX1, (KX′
+i|X′
+i−1,�Y ′
+i−1)n
+i=2 = (KXi|Xi−1,�Yi−1)n
+i=2, and KY ′
+i |X′
+i = KYi|Xi = PY |X,W for
+i = 1, . . . , n. With a training dataset, we can define the offline-learned estimation strategy for
+dynamic inference as follows.
+Definition 1. An offline-learned estimation strategy with an m-sample training dataset for an
+n-round dynamic inference is a sequence of estimators ψn
+m = (ψm,1, . . . , ψm,n), where ψm,i : (X ×
+�Y)m × Xi × �Yi−1 → �Y is the estimator for the ith round of estimation, which maps the dataset Zm
+as well as the past observations and estimates (Xi, �Y i−1) up to the ith round to an estimate �Yi of
+Yi, such that �Yi = ψm,i(Zm, Xi, �Y i−1), i = 1, . . . , n.
+Any specification of the above probabilistic models and an offline-learned estimation strategy
+determines a joint distribution of the random variables (W, Zm, Xn, Y n, �Y n) under consideration.
+The Bayesian network of the variables is shown in Fig. 2, where the training data is assumed to
+be generated in the imitation learning setup. A crucial observation from the Bayesian network is
+that W is conditionally independent of (Xn, �Y n) given Zm, as the quantities of interest Y n are not
+observed. In other words, given the training data, no more information about W can be gained
+during inference. We formally state this observation as the following lemma.
+Lemma 1. In offline learning for dynamic inference, the parameter W is conditionally independent
+of the observations and the estimates (Xn, �Y n) during inference given the training data Zm.
+Given an offline-learned estimation strategy ψn
+m for an n-round dynamic inference with an
+m-sample training dataset, we can define its inference loss as E
+� �n
+i=1 ℓ(Xi, Yi, �Yi)
+�. The goal of
+offline learning is to find an offline-learned estimation strategy to minimize the inference loss:
+arg min
+ψn
+m
+E
+�
+n
+�
+i=1
+ℓ(Xi, Yi, �Yi)
+�
+,
+with �Yi = ψm,i(Zm, Xi, �Y i−1).
+(6)
+5
+
+Figure 2: Bayesian network of the random variables in offline learning for dynamic inference with
+the imitation learning setup, with m = n = 4. Here we assume the estimates are made with Markov
+estimators, such that �Yi = ψm,i(Zm, Xi).
+3.1
+MDP reformulation
+3.1.1
+Equivalent expression of inference loss
+We first show that the inference loss in (6) can be expressed in terms of a loss function that does
+not take the unknown Yi as input.
+Theorem 1. For any offline-learned estimation strategy ψn
+m, its inference loss can be written as
+E
+�
+n
+�
+i=1
+ℓ(Xi, Yi, �Yi)
+�
+= E
+�
+n
+�
+i=1
+˜ℓ(πm, Xi, �Yi)
+�
+,
+(7)
+where πm(·) ≜ P[W ∈ ·|Zm] is the posterior distribution of the kernel parameter W given the training
+dataset Zm, and ˜ℓ : ∆ × X × �Y → R, with ∆ being the space of probability distributions on W, is
+defined as
+˜ℓ(π, x, ˆy) ≜
+�
+W
+�
+Y
+π(dw)PY |X,W (dy|x, w)ℓ(x, y, ˆy).
+(8)
+The proof is given in Appendix A. Theorem 1 states that the inference loss of an offline-learned
+estimation strategy ψn
+m is equal to
+J(ψn
+m) ≜ E
+�
+n
+�
+i=1
+˜ℓ(πm, Xi, �Yi)
+�
+,
+(9)
+with �Yi = ψm,i(Zm, Xi, �Y i−1). It follows that the offline learning problem in (6) can be equivalently
+written as
+arg min
+ψn
+m
+J(ψn
+m).
+(10)
+3.1.2
+(πm, Xi)n
+i=1 as a controlled Markov chain
+Next, we show that the sequence (πm, Xi)n
+i=1 appearing in (9) form a controlled Markov chain with
+�Y n as the control sequence. In other words, the tuple (πm, Xi+1) depends on the history (πm, Xi, �Y i)
+only through (πm, Xi, �Yi), as formally stated in the following lemma.
+6
+
+W
+Y1
+Y2
+Y?
+Y4
+Y1
+Y2
+Y3
+Y4
+1
+x1
+X2
+X3
+X4
+X1
+24
+X2
+X3
+X4
+<Y1
+2
+A3Lemma 2. Given any offline-learned estimation strategy ψn
+m, we have
+P
+�(πm, Xi+1) ∈ A × B
+��πm, Xi, �Y i� = 1{πm ∈ A}P
+�Xi+1 ∈ B|Xi, �Yi
+�
+(11)
+for any Borel sets A ⊂ ∆ and B ⊂ X, any realization of (πm, Xi, �Y i), and any i = 1, . . . , n − 1.
+The proof is given in Appendix B.
+3.1.3
+Optimality of Markov offline-learned estimators
+Furthermore, the next three lemmas will show that the search space of the minimization problem
+in (10) can be restricted to Markov offline-learned estimators ¯ψm,i : ∆ × X → Y, such that
+�Yi = ¯ψm,i(πm, Xi). We start with a generalization of Blackwell’s principle of irrelevant information.
+Lemma 3 (Generalized Blackwell’s principle of irrelevant information). For any fixed functions
+ℓ : Y × �Y → R and f : X → Y, the following equality holds:
+min
+g:X→�Y
+E
+�ℓ
+�f(X), g(X)
+�� = min
+g:Y→�Y
+E
+�ℓ
+�f(X), g(f(X))
+��.
+(12)
+Remark. The original Blackwell’s principle of irrelevant information, stating that for any fixed
+function ℓ : Y × �Y → R,
+min
+g:X×Y→�Y
+E
+�ℓ
+�Y, g(X, Y )
+�� = min
+g:Y→�Y
+E
+�ℓ
+�Y, g(Y )
+��,
+(13)
+can be seen as a special case of the above lemma.
+The proof of Lemma 3 is given in Appendix C. The first application of Lemma 3 is to prove that
+the last estimator of an optimal offline-learned estimation strategy can be replaced by a Markov
+one, which preserves the optimality.
+Lemma 4 (Last-round lemma for offline learning). Given any offline-learned estimation strategy
+ψn
+m, there exists a Markov offline-learned estimator ¯ψm,n : ∆ × X → �Y, such that
+J(ψm,1, . . . , ψm,n−1, ¯ψm,n) ≤ J(ψn
+m).
+(14)
+The proof is given in Appendix D. Lemma 3 can be further used to prove that whenever the last
+offline-learned estimator is Markov, the preceding estimator can also be replaced by a Markov one
+which preserves the optimality.
+Lemma 5 ((i − 1)th-round lemma for offline learning). For any i ≥ 2, given any offline-learned
+estimation strategy (ψm,1, . . . , ψm,i−1, ¯ψm,i) for an i-round dynamic inference with an m-sample
+training dataset, if the offline-learned estimator for the ith round of estimation is a Markov one
+¯ψm,i : ∆ × X → �Y, then there exists a Markov offline-learned estimator ¯ψm,i−1 : ∆ × X → �Y for the
+(i − 1)th round, such that
+J(ψm,1, . . . , ψm,i−2, ¯ψm,i−1, ¯ψm,i) ≤ J(ψm,1, . . . , ψm,i−1, ¯ψm,i).
+(15)
+The proof is given in Appendix E. With Lemma 4 and Lemma 5, we can prove the optimality of
+Markov offline-learned estimators, as given in Appendix F.
+Theorem 2. The minimum of J(ψn
+m) in (10) can be achieved by an offline-learned estimation
+strategy ¯ψn
+m with Markov estimators ¯ψm,i : ∆ × X → �Y, i = 1, . . . , n, such that �Yi = ¯ψm,i(πm, Xi).
+7
+
+3.1.4
+Conversion to MDP
+Theorem 1 and Theorem 2 with Lemma 2 imply that the original offline learning problem in (6) is
+equivalent to
+arg min
+ψn
+m
+E
+�
+n
+�
+i=1
+˜ℓ(πm, Xi, �Yi)
+�
+,
+�Yi = ψm,i(πm, Xi),
+(16)
+and the sequence (πm, Xi)n
+i=1 is a controlled Markov chain driven by �Y n. With this reformulation,
+we see that the offline learning problem becomes a standard MDP. The tuples (πm, Xi)n
+i=1 become
+the states in this MDP, the estimates �Y n become the actions, the probability transition kernel
+P (πm,Xi)|(πm,Xi−1),�Yi−1 now defines the controlled state transition, and any Markov offline-learned
+estimation strategy ψn
+m becomes a policy of this MDP. The goal of learning becomes finding the
+optimal policy of the MDP to minimize the accumulated expected loss defined w.r.t. ˜ℓ. The solution
+to this MDP will be an optimal offline-learned estimation strategy for dynamic inference.
+3.2
+Solution via dynamic programming
+3.2.1
+Optimal offline-learned estimation strategy
+From the theory of MDP it is known that the optimal policy for the MDP in (16), namely the
+optimal offline-learned estimation strategy, can be found via dynamic programming. To derive the
+optimal estimators, define the functions Q∗
+m,i : ∆ × X × �Y → R and V ∗
+m,i : ∆ × X → R for offline
+learning recursively for i = n, . . . , 1 as Q∗
+m,n(π, x, ˆy) ≜ ˜ℓ(π, x, ˆy), and
+V ∗
+m,i(π, x) ≜ min
+ˆy∈�Y
+Q∗
+m,i(π, x, ˆy),
+i = n, . . . , 1
+(17)
+Q∗
+m,i(π, x, ˆy) ≜ ˜ℓ(π, x, ˆy) + E[V ∗
+m,i+1(π, Xi+1)|Xi = x, �Yi = ˆy],
+i = n − 1, . . . , 1
+(18)
+with ˜ℓ is as defined in (8), and the conditional expectation in (18) is taken w.r.t. Xi+1. The optimal
+offline-learned estimate to make in the ith round when πm = π and Xi = x is then
+ψ∗
+m,i(π, x) ≜ arg min
+ˆy∈�Y
+Q∗
+m,i(π, x, ˆy).
+(19)
+3.2.2
+Minimum inference loss and loss-to-go
+For any offline-learned estimation strategy ψn
+m, we can define its loss-to-go in the ith round of
+estimation when πm = π and Xi = x as
+Vm,i(π, x; ψn
+m) ≜ E
+�
+n
+�
+j=i
+ℓ(Xj, Yj, �Yj)
+���πm = π, Xi = x
+�
+,
+(20)
+which is the conditional expected loss accumulated from the ith round to the final round when
+(ψm,i, . . . , ψm,n) are used as the offline-learned estimators, given that the posterior distribution of
+the kernel parameter W given the training dataset Zm is π and the observation in the ith round is x.
+The following theorem states that the offline-learned estimation strategy (ψ∗
+m,1, . . . , ψ∗
+m,n) derived
+from dynamic programming not only achieves the minimum inference loss over the n rounds, but
+also achieves the minimum loss-to-go in each round with any training dataset and any observation
+in that round.
+8
+
+Theorem 3. The offline-learned estimators (ψ∗
+m,1, . . . , ψ∗
+m,n) defined in (19) according to the recur-
+sion in (17) and (18) constitute an optimal offline-learned estimation strategy for dynamic inference,
+which achieves the minimum in (6). Moreover, for any Markov offline-learned estimation strategy
+ψn
+m, with ψm,i : ∆ × X → Y, its loss-to-go satisfies
+Vm,i(π, x; ψn
+m) ≥ V ∗
+m,i(π, x)
+(21)
+for all π ∈ ∆, x ∈ X and i = 1, . . . , n, where the equality holds if ψm,j(π, x) = ψ∗
+m,j(π, x) for all
+π ∈ ∆, x ∈ X and j ≥ i.
+The proof is given in Appendix G. A consequence of Theorem 3 is that in offline learning for
+dynamic inference, the minimum expected loss accumulated from the ith round to the final round
+can be expressed in terms of V ∗
+m,i, as stated in the following corollary.
+Corollary 1. In offline learning for dynamic inference, for any i and any initial distribution PXi,
+min
+ψm,i,...,ψm,n E
+�
+n
+�
+j=i
+ℓ(Xj, Yj, �Yj)
+�
+= E[V ∗
+m,i(πm, Xi)],
+(22)
+and the minimum is achieved by the estimators (ψ∗
+m,i, . . . , ψ∗
+m,n) defined in (19).
+4
+Bayesian online learning for dynamic inference
+In the setup of offline learning for dynamic inference, we assume that before the inference takes place,
+a training dataset Zm drawn from some distribution PZm|W is observed, and W can be estimated
+from Zm. In the online learning setup, we assume that there is no training dataset available before
+the inference; instead, during the inference, after an estimate �Yi is made in each round, the true
+value Yi is revealed, and W can be estimated on-the-fly in each round from all the observations
+available so far.
+Same as the offline learning setup, we assume that during inference, the initial distribution PX1
+and the probability transition kernels KXi|Xi−1,�Yi−1, i = 1, . . . , n are still known, while the unknown
+KYi|Xi’s are the same element PY |X,W of a parametrized family of kernels {PY |X,w, w ∈ W} and the
+unknown kernel parameter W is a random element of W with prior distribution PW . We can define
+the online-learned estimation strategy for dynamic inference as follows. Note that we overload the
+notations ψi as an online-learned estimator and Zi as (Xi, Yi) throughout this section.
+Definition 2. An online-learned estimation strategy for an n-round dynamic inference is a sequence
+of estimators ψn = (ψ1, . . . , ψn), where ψi : (X × Y)i−1 × �Yi−1 × X → �Y is the estimator in the ith
+round of estimation, which maps the past observations Zi−1 = (Xj, Yj)i−1
+j=1 and estimates �Y i−1 in
+addition to a new observation Xi to an estimate �Yi of Yi, such that �Yi = ψi(Zi−1, �Y i−1, Xi).
+The Bayesian network of all the random variables (W, Xn, Y n, �Y n) in online learning for dynamic
+inference is shown in Fig. 3.
+A crucial observation from the Bayesian network is that W is
+conditionally independent of (Xi, �Y i) given Zi−1, as stated in the following lemma.
+Lemma 6. In online learning for dynamic inference, in the ith round of estimation, the kernel
+parameter W is conditionally independent of the current observation Xi and the estimates �Y i up to
+the ith round given the past observations Zi−1.
+9
+
+Figure 3: Bayesian network of variables in online learning for dynamic inference, with n = 3.
+Same as the offline learning setup, given an online-learned estimation strategy ψn, we can define
+its inference loss as E
+� �n
+i=1 ℓ(Xi, Yi, �Yi)
+�. The goal of online learning for an n-round dynamic
+inference is to find an online-learned estimation strategy to minimize the inference loss:
+arg min
+ψn
+E
+�
+n
+�
+i=1
+ℓ(Xi, Yi, �Yi)
+�
+,
+with �Yi = ψi(Zi−1, �Y i−1, Xi).
+(23)
+4.1
+MDP reformulation
+4.1.1
+Equivalent expression of inference loss
+We first show that the inference loss in (23) can be expressed in terms of a loss function that does
+not take the unknown Yi as input.
+Theorem 4. For any online-learned estimation strategy ψn, its inference loss can be written as
+E
+�
+n
+�
+i=1
+ℓ(Xi, Yi, �Yi)
+�
+= E
+�
+n
+�
+i=1
+˜ℓ(πi, Xi, �Yi)
+�
+,
+(24)
+where πi(·) ≜ P[W ∈ ·|Zi−1] is the posterior distribution of the kernel parameter W given the past
+observations Zi−1 to the ith round, and ˜ℓ : ∆ × X × �Y → R, with ∆ being the space of probability
+distributions on W, is defined in the same way as in (8),
+˜ℓ(π, x, ˆy) =
+�
+W
+�
+Y
+π(dw)PY |X,W (dy|x, w)ℓ(x, y, ˆy).
+(25)
+The proof is given in Appendix H. Theorem 4 states that the inference loss of an online-learned
+estimation strategy ψn is equal to
+J(ψn) = E
+�
+n
+�
+i=1
+˜ℓ(πi, Xi, �Yi)
+�
+,
+with �Yi = ψi(Zi−1, �Y i−1, Xi).
+(26)
+It follows that the learning problem in (23) can be equivalently written as
+arg min
+ψn
+J(ψn).
+(27)
+10
+
+M
+Y
+Y2
+Y3
+4
++
+X1
+X2
+X3
+44.1.2
+(πi, Xi)n
+i=1 as a controlled Markov chain
+Next, we show that the sequence (πi, Xi)n
+i=1 appearing in (26) form a controlled Markov chain
+with �Y n as the control sequence. In other words, the tuple (πi+1, Xi+1) depends on the history
+(πi, Xi, �Y i) only through (πi, Xi, �Yi), as formally stated in the following lemma.
+Lemma 7. There exists a function f : ∆ × X × Y → ∆, such that given any learned estimation
+strategy ψn, we have
+P
+�(πi+1, Xi+1) ∈ A × B
+��πi, Xi, �Y i� =
+�
+W
+�
+Y
+πi(dw)PY |X,W (dyi|Xi, w)P[f(πi, Xi, yi) ∈ A]P
+�Xi+1 ∈ B|Xi, �Yi
+�
+(28)
+for any Borel sets A ⊂ ∆ and B ⊂ X, any realization of (πi, Xi, �Y i), and any i = 1, . . . , n − 1.
+Lemma 7 is proved in Appendix I, based on the auxiliary lemma below proved in Appendix J.
+Lemma 8. For a generic random tuple (T, U, V ) ∈ T×U×V that forms a Markov chain T −U −V ,
+we have
+P
+�V ∈ A
+��PV |U(·|U) = p, T ∈ B
+� = p(A)
+(29)
+for any Borel sets A ∈ V and B ∈ T, and any probability distribution p on V.
+4.1.3
+Optimality of Markov online-learned estimators
+The next two lemmas will show that the search space of the minimization problem in (27) can
+be restricted to Markov online-learned estimators ¯ψi : ∆ × X → Y, such that �Yi = ¯ψi(πi, Xi). In
+parallel to the discussion of the offline learning, we first prove that the last estimator of an optimal
+online-learned estimation strategy can be replaced by a Markov one, which preserves the optimality.
+Lemma 9 (Last-round lemma for online learning). Given any online-learned estimation strategy
+ψn, there exists a Markov online-learned estimator ¯ψn : ∆ × X → �Y, such that
+J(ψ1, . . . , ψn−1, ¯ψn) ≤ J(ψn).
+(30)
+The proof is given in Appendix K. We further prove that whenever the last online-learned
+estimator is Markov, the preceding estimator can be replaced by a Markov one which preserves the
+optimality.
+Lemma 10 ((i − 1)th-round lemma for online learning). For any i ≥ 2, given any online-learned
+estimation strategy (ψ1, . . . , ψi−1, ¯ψi) for an i-round dynamic inference, if the last estimator is a
+Markov one ¯ψi : ∆ × X → �Y, then there exists a Markov onlined-learned estimator ¯ψi−1 : ∆ × X → �Y
+for the (i − 1)th round, such that
+J(ψ1, . . . , ψi−2, ¯ψi−1, ¯ψi) ≤ J(ψ1, . . . , ψi−1, ¯ψi).
+(31)
+The proof is given in Appendix L. With Lemma 4 and Lemma 5, we can prove the optimality of
+Markov online-learned estimators, as given in Appendix M.
+Theorem 5. The minimum of J(ψn) in (27) can be achieved by a online-learned estimation strategy
+¯ψn with Markov online-learned estimators ¯ψi : ∆ × X → �Y, such that �Yi = ¯ψi(πi, Xi).
+11
+
+4.1.4
+Conversion to MDP
+Theorem 4 and Theorem 5 with Lemma 7 imply that the original online learning problem in (23) is
+equivalent to
+arg min
+ψn
+E
+�
+n
+�
+i=1
+˜ℓ(πi, Xi, �Yi)
+�
+,
+�Yi = ψi(πi, Xi)
+(32)
+and the sequence (πi, Xi)n
+i=1 is a controlled Markov chain driven by �Y n. With this reformulation,
+we see that the online learning problem becomes a standard MDP. The tuples (πi, Xi)n
+i=1 become
+the states in this MDP, the estimates �Y n become the actions, the probability transition kernel
+P (πi,Xi)|(πi−1,Xi−1),�Yi−1 now defines the controlled state transition, and any Markov online-learned
+estimation strategy ψn becomes a policy of this MDP. The goal of online learning becomes finding
+the optimal policy of the MDP to minimize the accumulated expected loss defined w.r.t. ˜ℓ. The
+solution to this MDP will be an optimal online-learned estimation strategy for dynamic inference.
+4.2
+Solution via dynamic programming
+4.2.1
+Optimal online-learned estimation strategy
+From the theory of MDP it is known that the optimal policy for the MDP in (32), namely the
+optimal online-learned estimation strategy, can be found via dynamic programming. To derive
+the optimal estimators, define the functions Q∗
+i : ∆ × X × �Y → R and V ∗
+i : ∆ × X → R for online
+learning recursively for i = n, . . . , 1 as Q∗
+n(π, x, ˆy) ≜ ˜ℓ(π, x, ˆy), and
+V ∗
+i (π, x) ≜ min
+ˆy∈�Y
+Q∗
+i (π, x, ˆy),
+i = n, . . . , 1
+(33)
+Q∗
+i (π, x, ˆy) ≜ ˜ℓ(π, x, ˆy) + E[V ∗
+i+1(πi+1, Xi+1)|πi = π, Xi = x, �Yi = ˆy], i = n − 1, . . . , 1
+(34)
+with ˜ℓ is as defined in (8), and the conditional expectation in (34) is taken w.r.t. (πi+1, Xi+1). The
+optimal online-learned estimate to make in the ith round when πi = π and Xi = x is then
+ψ∗
+i (π, x) ≜ arg min
+ˆy∈�Y
+Q∗
+i (π, x, ˆy).
+(35)
+4.2.2
+Minimum inference loss and loss-to-go
+For any online-learned estimation strategy ψn, we can define its loss-to-go in the ith round of
+estimation when πi = π and Xi = x as
+Vi(π, x; ψn) ≜ E
+�
+n
+�
+j=i
+ℓ(Xj, Yj, �Yj)
+���πi = π, Xi = x
+�
+,
+(36)
+which is the conditional expected loss accumulated from the ith round to the final round when
+(ψi, . . . , ψn) are used as the learned estimators, given that in the ith round the posterior distribution
+of the kernel parameter W given the past observations Zi−1 is π and the observation Xi is x.
+The following theorem states that the online-learned estimation strategy (ψ∗
+1, . . . , ψ∗
+n) derived from
+dynamic programming not only achieves the minimum inference loss over the n rounds, but also
+achieves the minimum loss-to-go in each round with any past and current observations in that
+round.
+12
+
+Theorem 6. The online-learned estimators (ψ∗
+1, . . . , ψ∗
+n) defined in (35) according to the recursion
+in (33) and (34) constitute an optimal online-learned estimation strategy for dynamic inference,
+which achieves the minimum in (23). Moreover, for any Markov online-learned estimation strategy
+ψn, with ψi : ∆ × X → Y, its loss-to-go satisfies
+Vi(π, x; ψn) ≥ V ∗
+i (π, x)
+(37)
+for all π ∈ ∆, x ∈ X and i = 1, . . . , n, where the equality holds if ψj(π, x) = ψ∗
+j (π, x) for all π ∈ ∆,
+x ∈ X and j ≥ i.
+The proof is given in Appendix N. A consequence of Theorem 6 is that in online learning for
+dynamic inference, the minimum expected loss accumulated from the ith round to the final round
+can be expressed in terms of V ∗
+i , as stated in the following corollary.
+Corollary 2. In online learning for dynamic inference, for any i and any initial distribution PXi,
+min
+ψi,...,ψn E
+�
+n
+�
+j=i
+ℓ(Xj, Yj, �Yj)
+�
+= E[V ∗
+i (πi, Xi)],
+(38)
+and the minimum is achieved by the estimators (ψ∗
+i , . . . , ψ∗
+n) defined in (35).
+A
+Proof of Theorem 1
+For each i = 1, . . . , n, we have
+E
+�ℓ(Xi, Yi, �Yi)
+��Zm, Xi, �Y i−1�
+=
+�
+Y
+PYi|Zm,Xi,�Y i−1(dy)ℓ(Xi, y, �Yi)
+(39)
+=
+�
+W
+�
+Y
+PW|Zm,Xi,�Y i−1(dw)PYi|Zm,Xi,�Y i−1,W=w(dy)ℓ(Xi, y, �Yi)
+(40)
+=
+�
+W
+�
+Y
+πm(dw)PY |X,W (dy|Xi, w)ℓ(Xi, y, �Yi)
+(41)
+=˜ℓ(πm, Xi, �Yi),
+(42)
+where (39) is due to the fact that Xi and �Yi are determined by (Zm, Xi, �Y i−1); and (41) follows
+from the fact that W is conditionally independent of (Xi, �Y i−1) given Zm as stated in Lemma 1,
+and the fact that Yi is conditionally independent of (Zm, Xi−1, �Y i−1) given (Xi, W). With the
+above equality and the fact that
+E
+�
+n
+�
+i=1
+ℓ(Xi, Yi, �Yi)
+�
+=
+n
+�
+i=1
+E
+�E[ℓ(Xi, Yi, �Yi)|Zm, Xi, �Y i−1]
+�,
+(43)
+we obtain (7).
+13
+
+B
+Proof of Lemma 2
+For any offline-learned estimation strategy ψn
+m, any Borel sets A ⊂ ∆ and B ⊂ X, and any realization
+of (πm, Xi, �Y i),
+P
+�(πm, Xi+1) ∈ A × B
+��πm, Xi, �Y i� = P
+�πm ∈ A
+��πm
+�P
+�Xi+1 ∈ B|πm, Xi, �Y i�
+(44)
+= 1{πm ∈ A}P
+�Xi+1 ∈ B|Xi, �Yi
+�
+(45)
+where the second equality is due to the fact that Xi+1 is conditionally independent of (πm, Xi−1, �Y i−1)
+given (Xi, �Yi). This proves the claim, and we can see that the right side of (11) only depends on
+(πm, Xi, �Yi).
+C
+Proof of Lemma 3
+. The left side of (12) is the Bayes risk of estimating f(X) based on X, defined w.r.t. the loss
+function ℓ, which can be written as Rℓ(f(X)|X); while the right side of (12) is the Bayes risk of
+estimating f(X) based on f(X) itself, also defined w.r.t. the loss function ℓ, which can be written as
+Rℓ(f(X)|f(X)). It follows from a data processing inequality of the generalized conditional entropy
+that
+Rℓ(f(X)|X) ≤ Rℓ(f(X)|f(X)),
+(46)
+as f(X) − X − f(X) form a Markov chain. If follows from the same data processing inequality that
+Rℓ(f(X)|X) ≥ Rℓ(f(X)|f(X)),
+(47)
+as X − f(X) − f(X) also form a Markov chain. Hence Rℓ(f(X)|X) = Rℓ(f(X)|f(X)), which proves
+the claim.
+D
+Proof of Lemma 4
+The inference loss of ψn
+m can be written as
+J(ψn
+m) = E
+� n−1
+�
+i=1
+˜ℓ
+�(πm, Xi), �Yi
+��
++ E
+�˜ℓ
+�(πm, Xn), ψm,n(Zm, Xn, �Y n−1)
+��.
+(48)
+Since the first expectation in (48) does not depend on ψm,n, it suffices to show that there exists a
+learned estimator ¯ψm,n : ∆ × X → �Y, such that
+E
+�˜ℓ
+�(πm, Xn), ¯ψm,n(πm, Xn)
+�� ≤ E
+�˜ℓ
+�(πm, Xn), ψm,n(Zm, Xn, �Y n−1)
+��.
+(49)
+The existence of such an estimator is guaranteed by Lemma 3, as (πm, Xn) is a function of
+(Zm, Xn, �Y n−1).
+14
+
+E
+Proof of Lemma 5
+The inference loss of the given (ψm,1, . . . , ψm,i−1, ¯ψm,i) is
+J(ψm,1, . . . , ψm,i−1, ¯ψm,i) = E
+� i−2
+�
+j=1
+˜ℓ
+�(πm, Xj), �Yj
+��
++
+E
+�˜ℓ
+�(πm, Xi−1), �Yi−1
+��+
+E
+�˜ℓ
+�(πm, Xi), ¯ψm,i(πm, Xi)
+��.
+(50)
+Since the first expectation in (50) does not depend on ψm,i−1, it suffices to show that there exists a
+learned estimator ¯ψm,i−1 : ∆ × X → �Y, such that
+E
+�˜ℓ
+�(πm, Xi−1), ¯ψm,i−1(πm, Xi−1)
+�� + E
+�˜ℓ
+�(πm, ¯Xi), ¯ψm,i(πm, ¯Xi)
+��
+≤E
+�˜ℓ
+�(πm, Xi−1), �Yi−1
+�� + E
+�˜ℓ
+�(πm, Xi), ¯ψm,i(πm, Xi)
+��,
+(51)
+where ¯Xi on the left side is the observation in the ith round when the Markov offline-learned
+estimator ¯ψm,i−1 is used in the (i − 1)th round. To get around with the dependence of Xi on ψm,i−1,
+we write the second expectation on the right side of (51) as
+E
+�E
+�˜ℓ
+�(πm, Xi), ¯ψm,i(πm, Xi)
+���πm, Xi−1, �Yi−1
+��
+(52)
+and notice that the conditional expectation E
+�˜ℓ
+�(πm, Xi), ¯ψi(πm, Xi)
+���πm, Xi−1, �Yi−1
+� does not
+depend on ψi−1. This is because the conditional distribution of (πm, Xi) given (πm, Xi−1, �Yi−1)
+is solely determined by the probability transition kernel P Xi|Xi−1,�Yi−1, as shown in the proof of
+Lemma 2 stating that (πm, Xi)n
+i=1 is a controlled Markov chain with �Y n as the control sequence. It
+follows that the right side of (51) can be written as
+E
+�˜ℓ
+�(πm, Xi−1), �Yi−1
+� + E
+�˜ℓ
+�(πm, Xi), ¯ψm,i(πm, Xi)
+���πm, Xi−1, �Yi−1
+��
+=E
+�g
+�πm, Xi−1, �Yi−1
+��
+(53)
+=E
+�g
+�πm, Xi−1, ψm,i−1(Zm, Xi−1, �Y i−2)
+��
+(54)
+for a function g that does not depend on ψm,i−1. Since (πm, Xi−1) is a function of (Zm, Xi−1, �Y i−2),
+it follows from Lemma 3 that there exists a learned estimator ¯ψm,i−1 : ∆ × X → �Y, such that
+E
+�g
+�πm, Xi−1, ψm,i−1(Zm, Xi−1, �Y i−2)
+��
+(55)
+≥E
+�g
+�πm, Xi−1, ¯ψm,i−1(πm, Xi−1)
+��
+(56)
+=E
+�˜ℓ
+�(πm, Xi−1), ¯ψm,i−1(πm, Xi−1)
+�+
+E
+�˜ℓ
+�(πm, ¯Xi), ¯ψm,i(πm, ¯Xi)
+���πm, Xi−1, ¯ψm,i−1(πm, Xi−1)
+��
+(57)
+=E
+�˜ℓ
+�(πm, Xi−1), ¯ψm,i−1(πm, Xi−1)
+�� + E
+�˜ℓ
+�(πm, ¯Xi), ¯ψm,i(πm, ¯Xi)
+��,
+(58)
+which proves (51) and the claim.
+15
+
+F
+Proof of Theorem 2
+Picking an optimal offline-learned estimation strategy ψn
+m, we can first replace its last estimator by
+a Markov one that preserves the optimality of the strategy, which is guaranteed by Lemma 4. Then,
+for i = n, . . . , 2, we can repeatedly replace the (i − 1)th estimator by a Markov one that preserves
+the optimality of the previous strategy, which is guaranteed by Lemma 5 and the additive structure
+of the inference loss as in (9). Finally we obtain an offline-learned estimation strategy consisting
+of Markov estimators that achieves the same inference loss as the originally picked offline-learned
+estimation strategy.
+G
+Proof of Theorem 3
+The first claim stating that the offline-learned estimation strategy (ψ∗
+m,1, . . . , ψ∗
+m,n) achieves the
+minimum in (6) follows from the equivalence between (6) and the MDP in (16), and from the
+well-known optimality of the solution derived from dynamic programming to MDP.
+The second claim can be proved via backward induction. Consider an arbitrary Markov offline-
+learned estimation strategy ψn
+m with ψm,i : ∆ × X → Y, based on which the learned estimates during
+inference are made.
+• In the final round, for all π ∈ ∆ and x ∈ X,
+Vm,n(π, x; ψn
+m) = ˜ℓ(π, x, ψm,n(π, x))
+(59)
+≥ V ∗
+m,n(π, x),
+(60)
+where (59) is due to the definitions of Vm,n in (20) and ˜ℓ in (8); and (60) is due to the definition
+of V ∗
+m,n in (17), while the equality holds if ψm,n(π, x) = ψ∗
+m,n(π, x).
+• For i = n − 1, . . . , 1, suppose (21) holds in the (i + 1)th round. We first show a self-recursive
+expression of Vm,i(π, x; ψn
+m):
+Vm,i(π, x; ψn
+m) = E
+�
+n
+�
+j=i
+ℓ(Xj, Yj, �Yj)
+���πm = π, Xi = x
+�
+(61)
+= E[ℓ(Xi, Yi, �Yi)|πm = π, Xi = x] + E
+�
+n
+�
+j=i+1
+ℓ(Xj, Yj, �Yj)
+���πm = π, Xi = x
+�
+(62)
+= E
+�E[ℓ(Xi, Yi, �Yi)| �Yi, πm = π, Xi = x]
+��πm = π, Xi = x
+�+
+E
+�
+E
+�
+n
+�
+j=i+1
+ℓ(Xj, Yj, �Yj)
+���Xi+1, πm = π, Xi = x
+������πm = π, Xi = x
+�
+(63)
+= E
+�˜ℓ(π, x, �Yi)
+��πm = π, Xi = x
+�+
+E
+�
+E
+�
+n
+�
+j=i+1
+ℓ(Xj, Yj, �Yj)
+���πm = π, Xi+1
+������πm = π, Xi = x
+�
+(64)
+= ˜ℓ(π, x, ψm,i(π, x)) + E
+�Vm,i+1(π, Xi+1; ψn
+m)|πm = π, Xi = x
+�
+(65)
+16
+
+where the second term of (64) follows from the fact that Xi is conditionally independent of
+(Xn
+i+1, Y n
+i+1, �Y n
+i+1) given (πm, Xi+1), which is a consequence of the assumption that the offline-
+learned estimators are Markov and the specification of the joint distribution of (Zm, Xn, Y n, �Y n)
+in the setup of the offline learning problem, and can be seen from Fig. 2. Then,
+Vm,i(π, x; ψn
+m) ≥ ˜ℓ(π, x, ψm,i(π, x)) + E
+�V ∗
+m,i+1(π, Xi+1)|πm = π, Xi = x
+�
+(66)
+= ˜ℓ(π, x, ψm,i(π, x)) + E
+�V ∗
+m,i+1(π, Xi+1)|πm = π, Xi = x, �Yi = ψm,i(π, x)
+�
+(67)
+= ˜ℓ(π, x, ψm,i(π, x)) + E
+�V ∗
+m,i+1(π, Xi+1)|Xi = x, �Yi = ψm,i(π, x)
+�
+(68)
+= Q∗
+m,i(π, x, ψm,i(π, x))
+(69)
+≥ V ∗
+m,i(π, x)
+(70)
+where (66) follows from the inductive assumption; (67) follows from the fact that �Yi is determined
+given πm = π and Xi = x; (68) follows from the fact that Xi+1 is independent of πm given
+(Xi, �Yi); and the final inequality with the equality condition follow from the definitions of V ∗
+m,i
+and ψ∗
+m,i in (17) and (19).
+This proves the second claim.
+H
+Proof of Theorem 4
+For each i = 1, . . . , n, we have
+E
+�ℓ(Xi, Yi, �Yi)
+��Zi−1, �Y i−1, Xi
+�
+=
+�
+Y
+PYi|Zi−1,�Y i−1,Xi(dy)ℓ(Xi, y, �Yi)
+(71)
+=
+�
+W
+�
+Y
+PW|Zi−1,�Y i−1,Xi(dw)PYi|Zi−1,�Y i−1,Xi,W=w(dy)ℓ(Xi, y, �Yi)
+(72)
+=
+�
+W
+�
+Y
+πi(dw)PY |X,W (dy|Xi, w)ℓ(Xi, y, �Yi)
+(73)
+=˜ℓ(πi, Xi, �Yi),
+(74)
+where (71) is due to the fact that Xi and �Yi are determined by (Zi−1, �Y i−1, Xi); and (73) follows
+from the fact that W is conditionally independent of ( �Y i−1, Xi) given Zi−1 as a consequence of
+Lemma 6, and the fact that Yi is conditionally independent of (Zi−1, �Y i−1) given (Xi, W). With
+the above equality and the fact that
+E
+�
+n
+�
+i=1
+ℓ(Xi, Yi, �Yi)
+�
+=
+n
+�
+i=1
+E
+�E[ℓ(Xi, Yi, �Yi)|Zi−1, �Y i−1, Xi]
+�,
+(75)
+we obtain (24).
+17
+
+I
+Proof of Lemma 7
+We first show that πi+1 can be determined by (πi, Xi, Yi). To see it, we express πi+1 as
+PW|Zi = PW,Zi|Zi−1/PZi|Zi−1
+(76)
+= PW|Zi−1PXi|W,Zi−1PYi|Xi,W,Zi−1/PZi|Zi−1
+(77)
+= πiPXi|Xi−1,�Yi−1PYi|Xi,W /PZi|Zi−1
+(78)
+=
+πiPYi|Xi,W
+�
+W πi(dw′)PYi|Xi,W=w′
+(79)
+where (78) follows from the facts that 1) �Yi−1 is determined by Zi−1, and Xi is conditionally
+independent of (W, Zi−2, Yi−1) given (Xi−1, �Yi−1); and 2) Yi is conditionally independent of Zi−1
+given (Xi, W). It follows that πi+1 can be written as
+πi+1 = f(πi, Xi, Yi)
+(80)
+for a function f that maps
+�πi(·), Xi, Yi
+� to πi+1(·) ∝ πi(·)PY |X,W (Yi|Xi, ·).
+With (80), for any online-learned estimation strategy ψn, any Borel sets A ⊂ ∆ and B ⊂ X, and
+any realization of (πi, Xi, �Y i), we have
+P
+�(πi+1, Xi+1) ∈ A × B
+��πi, Xi, �Y i�
+=
+�
+Y
+P
+�dyi
+��πi, Xi, �Y i�P
+�(πi+1, Xi+1) ∈ A × B
+��πi, Xi, �Y i, Yi = yi
+�
+(81)
+=
+�
+Y
+P
+�dyi
+��πi, Xi, �Y i�P
+�f(πi, Xi, yi) ∈ A]P
+�Xi+1 ∈ B
+��Xi, �Yi
+�
+(82)
+=
+�
+Y
+�
+W
+P
+�dw
+��πi, Xi, �Y i�P
+�dyi
+��πi, Xi, �Y i, W = w
+�P
+�f(πi, Xi, yi) ∈ A]P
+�Xi+1 ∈ B
+��Xi, �Yi
+�
+(83)
+=
+�
+Y
+�
+W
+πi(dw)PY |X,W (dyi|Xi, w)P[f(πi, Xi, yi) ∈ A]P
+�Xi+1 ∈ B|Xi, �Yi
+�,
+(84)
+where (82) follows from (80) and the fact that Xi+1 is conditionally independent of (Zi−1, Yi, �Y i−1)
+given (Xi, �Yi); and (84) follows from 1) Lemma 8 and the fact that W is conditionally independent
+of (Zi−1, Xi, �Y i) given Zi−1, as a consequence of Lemma 6, and 2) the fact that Yi is conditionally
+independent of (Zi−1, �Y i) given (Xi, W).
+This proves the Lemma 7, and we see that the right side of (28) only depends on (πi, Xi, �Yi).
+J
+Proof of Lemma 8
+Given a probability distribution p on V, let Up ≜ {u ∈ U : PV |U(·|u) = p}. Then, for any Borel sets
+A ∈ V and B ∈ T,
+P
+�V ∈ A
+��PV |U(·|U) = p, T ∈ B
+� = P
+�V ∈ A, PV |U(·|U) = p, T ∈ B
+�
+P
+�PV |U(·|U) = p, T ∈ B
+�
+(85)
+=
+�
+Up PU(du)PV |U(A|u)PT|U(B|u)
+�
+Up PU(du)PT|U(B|u)
+(86)
+= p(A),
+(87)
+18
+
+where (86) follows from the definition of Up and the assumption that T and V are conditionally
+independent given U; and (87) follows from the fact that PV |U(A|u) = p(A) for all u ∈ Up.
+K
+Proof of Lemma 9
+The inference loss of ψn
+i can be written as
+J(ψn) = E
+� n−1
+�
+i=1
+˜ℓ
+�(πi, Xi), �Yi
+��
++ E
+�˜ℓ
+�(πn, Xn), ψn(Zn−1, �Y n−1, Xn)
+��.
+(88)
+Since the first expectation in (88) does not depend on ψn, it suffices to show that there exists a
+Markov online-learned estimator ¯ψn : ∆ × X → �Y, such that
+E
+�˜ℓ
+�(πn, Xn), ¯ψn(πn, Xn)
+�� ≤ E
+�˜ℓ
+�(πn, Xn), ψn(Zn−1, �Y n−1, Xn)
+��.
+(89)
+The existence of such an estimator is guaranteed by Lemma 3, as (πn, Xn) is a function of
+(Zn−1, �Y n−1, Xn).
+L
+Proof of Lemma 10
+The proof is given in Appendix L. The inference loss of the given (ψ1, . . . , ψi−1, ¯ψi) is
+J(ψ1, . . . , ψi−1, ¯ψi) = E
+� i−2
+�
+j=1
+˜ℓ
+�(πj, Xj), �Yj
+��
++
+E
+�˜ℓ
+�(πi−1, Xi−1), �Yi−1
+��+
+E
+�˜ℓ
+�(πi, Xi), ¯ψi(πi, Xi)
+��.
+(90)
+Since the first expectation in (90) does not depend on ψi−1, it suffices to show that there exists a
+Markov online-learned estimator ¯ψi−1 : ∆ × X → �Y, such that
+E
+�˜ℓ
+�(πi−1, Xi−1), ¯ψi−1(πi−1, Xi−1)
+�� + E
+�˜ℓ
+�(πi, ¯Xi), ¯ψi(πi, ¯Xi)
+��
+≤E
+�˜ℓ
+�(πi−1, Xi−1), �Yi−1
+�� + E
+�˜ℓ
+�(πi, Xi), ¯ψi(πi, Xi)
+��,
+(91)
+where ¯Xi on the left side is the observation in the ith round when the Markov estimator ¯ψi−1 is
+used in the (i − 1)th round. To get around with the dependence of Xi on ψi−1, we write the second
+expectation on the right side of (91) as
+E
+�E
+�˜ℓ
+�(πi, Xi), ¯ψi(πi, Xi)
+���πi−1, Xi−1, �Yi−1
+��
+(92)
+and notice that the conditional expectation E
+�˜ℓ
+�(πi, Xi), ¯ψi(πi, Xi)
+���πi−1, Xi−1, �Yi−1
+� does not de-
+pend on ψi−1. This is because the conditional distribution of (πi, Xi) given (πi−1, Xi−1, �Yi−1) is
+solely determined by the probability transition kernels PYi−1|Xi−1,W and P Xi|Xi−1,�Yi−1, as shown in
+the proof of Lemma 7 stating that (πi, Xi)n
+i=1 is a controlled Markov chain driven by �Y n. It follows
+19
+
+that the right side of (91) can be written as
+E
+�˜ℓ
+�(πi−1, Xi−1), �Yi−1
+� + E
+�˜ℓ
+�(πi, Xi), ¯ψi(πi, Xi)
+���πi−1, Xi−1, �Yi−1
+��
+=E
+�g
+�πi−1, Xi−1, �Yi−1
+��
+(93)
+=E
+�g
+�πi−1, Xi−1, ψi−1(Zi−2, �Y i−2, Xi−1)
+��
+(94)
+for a function g that does not depend on ψi−1. Since (πi−1, Xi−1) is a function of (Zi−2, �Y i−2, Xi−1),
+it follows from Lemma 3 that there exists a learned estimator ¯ψi−1 : ∆ × X → �Y, such that
+E
+�g
+�πi−1, Xi−1, ψi−1(Zi−2, �Y i−2, Xi−1)
+��
+(95)
+≥E
+�g
+�πi−1, Xi−1, ¯ψi−1(πi−1, Xi−1)
+��
+(96)
+=E
+�˜ℓ
+�(πi−1, Xi−1), ¯ψi−1(πi−1, Xi−1)
+�+
+E
+�˜ℓ
+�(πi, ¯Xi), ¯ψi(πi, ¯Xi)
+���πi−1, Xi−1, ¯ψi−1(πi−1, Xi−1)
+��
+(97)
+=E
+�˜ℓ
+�(πi−1, Xi−1), ¯ψi−1(πi−1, Xi−1)
+�� + E
+�˜ℓ
+�(πi, ¯Xi), ¯ψi(πi, ¯Xi)
+��,
+(98)
+which proves (91) and the claim.
+M
+Proof of Theorem 5
+Picking an optimal online-learned estimation strategy ψn, we can first replace its last estimator by
+a Markov one that preserves the optimality of the strategy, which is guaranteed by Lemma 9. Then,
+for i = n, . . . , 2, we can repeatedly replace the (i − 1)th estimator by a Markov one that preserves
+the optimality of the previous strategy, which is guaranteed by Lemma 10 and the additive structure
+of the inference loss as in (26). Finally we obtain an online-learned estimation strategy consisting
+of Markov online-learned estimators that achieves the same inference loss as the originally picked
+online-learned estimation strategy.
+N
+Proof of Theorem 6
+The first claim stating that the online-learned estimation strategy (ψ∗
+1, . . . , ψ∗
+n) achieves the minimum
+in (23) follows from the equivalence between (23) and the MDP in (32), and from the well-known
+optimality of the solution derived from dynamic programming to MDP.
+The second claim can be proved via backward induction. Consider an arbitrary Markov online-
+learned estimation strategy ψn with ψi : ∆ × X → Y, based on which the learned estimates are
+made. For any pair (i, j) such that 1 ≤ i ≤ j ≤ n,
+E
+�ℓ(Xj, Yj, �Yj)
+��πi, Xi
+�
+=E
+�E[ℓ(Xj, Yj, �Yj)|πj, Xj, πi, Xi]
+��πi, Xi
+�
+(99)
+=E
+� �
+W
+P(dw|πj, Xj, πi, Xi)
+�
+Y
+P(dyj|πj, Xj, πi, Xi, W = w)ℓ(Xj, yj, �Yj)
+���πi, Xi
+�
+(100)
+=E
+� �
+W
+�
+Y
+πj(dw)PY |X,W (dyj|Xj, w)ℓ(Xj, yj, �Yj)
+���πi, Xi
+�
+(101)
+=E
+�˜ℓ(πj, Xj, �Yj)
+��πi, Xi
+�
+(102)
+20
+
+where (100) follows from the fact that �Yj is determined by (πj, Xj); (101) follows from 1) Lemma 8
+and the fact that W is conditionally independent of (Zi−1, Xi, Xj) given Zj−1, and 2) Yj is
+conditionally independent of Zj−1 given (Xj, W); and (102) follows from the definition of ˜ℓ in (8).
+With the above identity, the loss-to-go defined in (36) can be rewritten as
+Vi(π, x; ψn) = E
+�
+n
+�
+j=i
+˜ℓ(πj, Xj, �Yj)
+���πi = π, Xi = x
+�
+,
+i = 1, . . . , n.
+(103)
+Now we can proceed with proving the second claim via backward induction.
+• In the final round, for all π ∈ ∆ and x ∈ X,
+Vn(π, x; ψn) = ˜ℓ(π, x, ψn(π, x))
+(104)
+≥ V ∗
+n (π, x),
+(105)
+where (104) is due to (102) with i = j = n; and (105) is due to the definition of V ∗
+n in (33), while
+the equality holds if ψn(π, x) = ψ∗
+n(π, x).
+• For i = n − 1, . . . , 1, suppose (37) holds in the (i + 1)th round. We first show a self-recursive
+expression of Vi(π, x; ψn):
+Vi(π, x; ψn)
+= E
+�
+n
+�
+j=i
+˜ℓ(πj, Xj, �Yj)
+���πi = π, Xi = x
+�
+(106)
+= E[˜ℓ(πi, Xi, �Yi)|πi = π, Xi = x] + E
+�
+n
+�
+j=i+1
+˜ℓ(πj, Xj, �Yj)
+���πi = π, Xi = x
+�
+(107)
+= ˜ℓ(π, x, ψi(π, x)) + E
+�
+E
+�
+n
+�
+j=i+1
+˜ℓ(πj, Xj, �Yj)
+���πi+1, Xi+1, πi = π, Xi = x
+������πi = π, Xi = x
+�
+(108)
+= ˜ℓ(π, x, ψi(π, x)) + E
+�
+E
+�
+n
+�
+j=i+1
+˜ℓ(πj, Xj, �Yj)
+���πi+1, Xi+1
+������πi = π, Xi = x
+�
+(109)
+= ˜ℓ(π, x, ψi(π, x)) + E
+�Vi+1(πi+1, Xi+1; ψn)|πi = π, Xi = x
+�
+(110)
+where the second term of (109) follows from the fact that �Yi+1 is determined by (πi+1, Xi+1),
+and the fact that (πj, Xj)n
+j=i+1 is conditionally independent of (πi, Xi) given (πi+1, Xi+1, �Yi+1)
+as guaranteed by Lemma 7. Then,
+Vi(π, x; ψn) ≥ ˜ℓ(π, x, ψi(π, x)) + E
+�V ∗
+i+1(πi+1, Xi+1)|πi = π, Xi = x
+�
+(111)
+= ˜ℓ(π, x, ψi(π, x)) + E
+�V ∗
+i+1(πi+1, Xi+1)|πi = π, Xi = x, �Yi = ψi(π, x)
+�
+(112)
+= Q∗
+i (π, x, ψi(π, x))
+(113)
+≥ V ∗
+i (π, x)
+(114)
+where (111) follows from the inductive assumption; (112) follows from the fact that �Yi is
+determined given (πi, Xi); (113) follows from the definition of Q∗
+i in (34); and the final inequality
+with the equality condition follow from the definitions of V ∗
+i and ψ∗
+i in (33) and (35).
+This proves the second claim.
+21
+
+Acknowledgement
+The authors would like to thank Prof. Maxim Raginsky for the encouragement of looking into
+dynamic aspects of statistical problems, and Prof. Lav Varshney for helpful discussions on this work.
+References
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+author email: [email protected], [email protected]
+23
+

2NAyT4oBgHgl3EQfbve8/content/tmp_files/2301.00270v1.pdf.txt

@@ -0,0 +1,1966 @@
+ULTRAPROP: Principled and Explainable Propagation on
+Large Graphs
+Meng-Chieh Lee
+[email protected]
+Pittsburgh, USA
+Carnegie Mellon University
+Shubhranshu Shekhar
+[email protected]
+Pittsburgh, USA
+Carnegie Mellon University
+Jaemin Yoo
+[email protected]
+Pittsburgh, USA
+Carnegie Mellon University
+Christos Faloutsos
+[email protected]
+Pittsburgh, USA
+Carnegie Mellon University
+ABSTRACT
+Given a large graph with few node labels, how can we (a) identify
+the mixed network-effect of the graph and (b) predict the unknown
+labels accurately and efficiently? This work proposes Network
+Effect Analysis (NEA) and ULTRAPROP, which are based on two
+insights: (a) the network-effect (NE) insight: a graph can exhibit
+not only one of homophily and heterophily, but also both or none
+in a label-wise manner, and (b) the neighbor-differentiation (ND)
+insight: neighbors have different degrees of influence on the target
+node based on the strength of connections.
+NEA provides a statistical test to check whether a graph ex-
+hibits network-effect or not, and surprisingly discovers the ab-
+sence of NE in many real-world graphs known to have heterophily.
+ULTRAPROP solves the node classification problem with notable
+advantages: (a) Accurate, thanks to the network-effect (NE) and
+neighbor-differentiation (ND) insights; (b) Explainable, precisely
+estimating the compatibility matrix; (c) Scalable, being linear
+with the input size and handling graphs with millions of nodes;
+and (d) Principled, with closed-form formula and theoretical guar-
+antee. Applied on eight real-world graph datasets, ULTRAPROP
+outperforms top competitors in terms of accuracy and run time,
+requiring only stock CPU servers. On a large real-world graph
+with 1.6M nodes and 22.3M edges, ULTRAPROP achieves ≥ 9×
+speedup (12 minutes vs. 2 hours) compared to most competitors.
+ACM Reference Format:
+Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Falout-
+sos. 2023. ULTRAPROP: Principled and Explainable Propagation on
+Large Graphs. In Under Submission. ACM, New York, NY, USA, 12 pages.
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+1
+INTRODUCTION
+Given a large, undirected, and unweighted graph with few la-
+beled nodes, how can we infer the labels of remaining unlabeled
+nodes, often without node features? Node classification is often
+employed to infer labels on large real-world graphs, since manual
+labeling is expensive and time-consuming. For example, in social
+networks with millions of users, identifying even a fraction (say
+5%) of users’ groups is prohibitive, which limits the application
+of methods that assume a large fraction of labels are given. More-
+over, node features are frequently missing in real-world graphs.
+For those methods that require node features in classification,
+Under Submission, ,
+2023. ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+they create the features based on the graph [9, 12, 13], such as
+using the one-hot encoding of node degree.
+Previous works on node classification have two main limita-
+tions. First, they ignore the complex network-effect of real-world
+graphs and understand their characteristic as either homophily or
+heterophily. The co-existing case of homophily and heterophily,
+which we call X-ophily in this work, has been neglected. Sec-
+ond, they either a) ignore the different influences of neighboring
+nodes during inference or b) require extensive computation to
+give dynamic weights to the adjacency matrix. In this work, we
+address these two challenges and consider the dynamic and com-
+plex relationships between neighboring nodes with two insights
+network-effect and neighbor-differentiation for designing an ac-
+curate and efficient approach for node classification.
+NE (network-effect): The first goal is to analyze the network-
+effect of a graph (i.e., homophily, heterophily, or any combination
+which we call X-ophily) in a principled and class-conditional way.
+That is, a single graph can have homophily and heterophily at
+the same time between different pairs of classes. The challenge
+is usually avoided in literature: inference-based methods assume
+that the relationship is given by domain experts [10]; deep graph
+models either assume homophily [16, 39] or misidentify graphs
+having no NE as heterophily graphs [23, 41].
+ND (neighbor-differentiation): The second goal is to approx-
+imate different influence levels of neighboring nodes effectively.
+Existing works require extensive computation to measure the
+influence levels in node classification. For instance, HOLS [8]
+solves ND by mining 𝑘−cliques, while listing all the instances
+is time-consuming; Graph Attention Network (GAT) [35] learns
+more than one relationship for each neighbor, while heavily rely-
+ing on the node features.
+We provide an informal definition of the problem:
+INFORMAL PROBLEM 1.
+• Given an undirected and unweighted graph
+– with few labeled nodes,
+– without node features,
+• Infer the labels of all the remaining nodes
+– accurately under any types of network effects,
+– explaining the predictions to human experts,
+– efficiently in large-scale graphs with scalability.
+Our solutions: We propose Network Effect Analysis (NEA),
+an algorithm to statistically test NE of a real-world graph with
+only a few observed node labels. NEA analyzes the relationships
+between all pairs of different classes in an efficient manner. In
+arXiv:2301.00270v1  [cs.SI]  31 Dec 2022
+
+Under Submission, ,
+Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos
+Label 
+2400x6 
+Node ID
+Label ID
+Adjacency
+2400x2400
+Input
+Estimated
+Compatibility Matrix
+Homophily
+Heterophily
+Output
+Compatibility 
+6x6
+(a) NE: Compatibility Matrix Estimation
+R3
+R1
+R2
+X
+B4
+B3
+B2
+B1
+B8
+B7
+B6
+B5
+1.66
+1.96
+Proposed 
+Embedding Space
+Predicts Blue
+predicts Red
+?
+B1-4
+R1-3
+X
+(b) ND: Neighbor Differentiation
+0.5
+1.0
+1.5
+2.0
+# of Edges
+1e7
+0
+2000
+4000
+6000
+Run Time (s)
+UltraProp
+GCN
+HOLS
+11x
+4x
+(c) Scalability
+Figure 1: ULTRAPROP is Effective, Explainable, and Scalable. (a) Thanks to Network Effect Formula, ULTRAPROP ex-
+plains the dataset by precisely estimating the compatibility matrix, observing both heterophily and homophily. (b) Thanks
+to “Emphasis” Matrix, ULTRAPROP predicts the label of the gray node X correctly, while LINBP fails. (c) ULTRAPROP is
+fast and scales linearly with the number of edges. See Introduction for more details.
+Figure 2, we show that surprisingly many large public datasets
+known as heterophily graphs do not have NE at all.
+We then propose ULTRAPROP, a principled approach using
+both insights of NE and ND to conduct accurate node classifi-
+cation on large graphs with explainability. The explainability is
+built upon the combination of influential neighbors (ND) and the
+compatibility matrix that we carefully and automatically estimate
+(see Lemma 3). Figure 1 illustrates the advantages of ULTRA-
+PROP. Figure 1a shows how ULTRAPROP provides explanation
+by estimating a compatibility matrix from only 5% of node labels:
+the interrelations of classes imply that the first half follows het-
+erophily, while the other half follows homophily. Figure 1b shows
+that ULTRAPROP predicts the different influences of neighbors
+correctly by ND, where the central vertex X is closer to the red
+nodes R1, R2, and R3 in the embedding space, as it participates
+in a closely-knit community with them. Finally, Figure 1c shows
+the linear scalability of ULTRAPROP with the number of edges.
+It is 9× faster than most of the competitors, and requires only 12
+minutes on a large real-world graph with over 22M edges.
+In summary, the advantages of ULTRAPROP are
+(1) Accurate, thanks to the precise estimation of the compati-
+bility matrix, and the reliable measurement of the different
+importance of neighbors,
+(2) Explainable, interpreting the datasets with estimated com-
+patibility matrices, which work for homophily, heterophily,
+or any combination – X-ophily,
+(3) Scalable, scaling linearly with the input size,
+(4) Principled, providing a tight bound of convergence for
+the random walks, and the closed-form formula for the
+compatibility matrix (see Lemma 2 and 3).
+Reproducibility: Our implemented source code and prepro-
+cessed datasets will be published once the paper is accepted.
+2
+BACKGROUND AND RELATED WORK
+We introduce preliminaries, and related works on label propaga-
+tion and node embedding. Table 1 presents qualitative compari-
+son of state-of-the-art approaches against our proposed method
+ULTRAPROP. No competitor fulfills all the specs in Table 1.
+Notation. Let 𝐺 be an undirected and unweighted graph with
+𝑛 nodes and 𝑚 edges with 𝑨 as the adjacency matrix. 𝑨𝑖𝑗 = 1
+indicates that nodes 𝑖, 𝑗 are connected by an edge. Each node 𝑖 has
+a unique label 𝑙(𝑖) ∈ {1, 2, . . . ,𝑐}, where 𝑐 denotes the number
+of classes. Let 𝑬 ∈ R𝑛×𝑐 be the initial belief matrix containing
+the prior information, i.e., the labeled nodes. 𝑬𝑖𝑘 = 1 if 𝑙(𝑖) = 𝑘,
+and the rest entries of the 𝑖𝑡ℎ row are filled up with zeros. For
+the nodes without labels, all the entries corresponding to those
+nodes are set to 1/𝑐. 𝑯 ∈ R𝑐×𝑐 is a row-normalized compatibility
+matrix where 𝑯𝑘𝑙 denotes the relative influence of class 𝑙 on
+class 𝑘. The residual of a matrix around 𝑘 is denoted as ˆ𝒀 and is
+defined as ˆ𝒀 = 𝒀 −𝑘 × 1 where 𝒀 is centered 1 around 𝑘, and 1 is
+matrix of ones.
+2.1
+Label Propagation
+Belief Propagation. Belief Propagation (BP) is a popular method
+for label inference in graphs [10, 18, 28]. FABP [18] and LINBP
+[10] accelerate BP by approximating the final belief assignment
+from BP. In particular, LINBP approximates the final belief as:
+ˆ𝑩 = ˆ𝑬 + 𝑨ˆ𝑩 ˆ𝑯,
+(1)
+where ˆ𝑩 is a residual final belief matrix, initialized with all zeros,
+𝑨 is the adjacency matrix. The compatibility matrix 𝑯 and initial
+beliefs 𝑬 are centered around 1/𝑐 to ensure convergence.
+Higher-Order Propagation Methods. HOLS [8] leverages
+higher-order graph structures, i.e. 𝑘−cliques. It propagates the
+labels by incorporating the weights from higher-order cliques.
+However, mining cliques is computationally intensive, and pro-
+hibitive for large graphs.
+2.2
+Embedding Methods
+Traditional Embedding Methods. Numerous embedding meth-
+ods [5, 7, 29] have been proposed to capture neighborhood sim-
+ilarity and role of nodes in the graph. Chen et al. [5] propose a
+random walk based generalized embedding method to capture
+non-linear relations among nodes. Similarly, Pixie [7] utilizes
+localized random walk based on node features. Further, [29] intro-
+duced a generalized method that derives the matrix closed forms
+of different graph embedding methods.
+1A matrix “centered around” 𝑘 has all its entries close to 𝑘 and the average of the
+entries is exactly 𝑘.
+
+ULTRAPROPLINBPULTRAPROPULTRAPROP: Principled and Explainable Propagation on Large Graphs
+Under Submission, ,
+Table 1: ULTRAPROP matches all specs, while competitors
+miss one or more of the properties. Each property corre-
+sponds to a contribution in Introduction. ‘?’ indicates that
+it is unclear from the original paper.
+Property
+Method
+BP [10, 18]
+HOLS [8]
+General GNNs [16, 17]
+Attention GNNs [15, 35]
+Heterophily GNNs [2, 6]
+ULTRAPROP
+Contr. (1): Handling NE
+?
+�
+Contr. (1): Handling ND
+�
+�
+�
+Contr. (2): Explainable
+�
+Contr. (3): Scalable
+�
+�
+�
+Contr. (4): Principled
+�
+�
+�
+Deep Graph Models. Graph Convolutional Networks (GCN) [16]
+employ approximate spectral convolutions to incorporate neigh-
+borhood information. APPNP [17] utilizes personalized PageR-
+ank to leverage the local information and a larger neighborhood.
+To account for ND, Graph Attention Networks (GAT) [15, 35] al-
+low for assigning importance weights to neighborhoods. However,
+attention GNNs require node features, and need many learnable
+parameters, making it infeasible for large graphs. MIXHOP [2]
+makes no assumption of homophily, and mixes powers of the
+adjacency matrix to incorporate more than 1-hop neighbors in
+each layer. H2GCN [41] is built on three key designs to better
+learn the structure of heterophily graphs; nevertheless, it requires
+too much memory and thus is not able to handle large graphs.
+GPR-GNN [6] allows the learnable weights to be negative during
+propagation with Generalized PageRank. LINKX [23] introduces
+multiple large heterophily datasets, but it is not applicable to
+graphs without node features. [26] empirically evaluates the per-
+formance of GNNs on small heterophily datasets (≤ 10K nodes).
+However, most of the conclusions are made based on the evalua-
+tions where the node features are used. While deep graph models
+have been shown to be state-of-the-art methods, it relies on node
+features and is not scalable without GPU. Further, it is hard to
+supply explanations or provide theoretical analysis.
+3
+PROPOSED METHOD PART I – “NEA”
+Given a graph with few node labels, how can we identify what are
+the classes that a node with a specific class connects to? In other
+words, how can we find whether the graph exhibits X-ophily – ho-
+mophily, heterophily, or even none? We propose Network Effect
+Analysis (NEA), a statistical approach to identify the network-
+effect (NE) in a graph. It leads to interesting discovery that many
+widely used heterophily graphs exhibit no NE.
+3.1
+Network Effect Analysis (NEA)
+Previous works on identifying NE of a graph [23, 41] have two
+main limitations. First, when a class connects to all existing
+classes uniformly, they misunderstand this non-homophily class
+as heterophily, which should be considered as having no NE. Sec-
+ond, they require the labels of most nodes in a graph, even though
+Data: Edges E and priors P
+Result: 𝑝-value table 𝑭
+/* edges with both nodes in priors
+*/
+1 Extract E
+′ such that (𝑖, 𝑗) ∈ E,𝑖, 𝑗 ∈ P ∀(𝑖, 𝑗) ∈ E
+′;
+2 𝑻 ← 𝑶𝑐×𝑐;
+// test statistic table
+/* do 𝜒2 test for 𝐵 times
+*/
+3 for 𝑏1 = 1, ..., 𝐵 do
+4
+for 𝑐1 = 1, ...,𝑐 do
+5
+for 𝑐2 = 𝑐1 + 1, ...,𝑐 do
+6
+𝑽 ← 𝑶2×2;
+// contingency table
+7
+Shuffle(E
+′);
+// sampling
+8
+for (𝑖, 𝑗) ∈ E
+′ do
+9
+if 𝑙(𝑖) = 𝑐1 and 𝑙(𝑗) = 𝑐1 then
+10
+𝑽11 ← 𝑽11 + 2;
+11
+else if (𝑙(𝑖) = 𝑐1 and 𝑙(𝑗) = 𝑐2) or
+12
+(𝑙(𝑖) = 𝑐2 and 𝑙(𝑗) = 𝑐1) then
+13
+𝑽21 ← 𝑽21 + 1;
+14
+𝑽12 ← 𝑽12 + 1;
+15
+else if 𝑙(𝑖) = 𝑐2 and 𝑙(𝑗) = 𝑐2 then
+16
+𝑽22 ← 𝑽22 + 2;
+17
+if �2
+𝑖=1
+�2
+𝑗=1 𝑽𝑖𝑗 > 250 then
+18
+Break;
+19
+end
+20
+end
+/* record statistics of class pairs
+*/
+21
+𝑇 = 𝜒2-Test-Statistic(𝑽);
+22
+𝑻𝑐1𝑐2 ← 𝑻𝑐1𝑐2 +𝑇/𝐵;
+23
+𝑻𝑐2𝑐1 ← 𝑻𝑐2𝑐1 +𝑇/𝐵;
+24
+end
+25
+end
+26 end
+27 Compute 𝑝-value table 𝑭𝑐×𝑐 with average statistics in 𝑻;
+28 Return 𝑭;
+Algorithm 1: Network Effect Analysis (NEA)
+in most real-world node classification tasks only a few node la-
+bels are observed. We propose NEA to address such limitations.
+Before introducing NEA, we provide two propositions:
+PROPOSITION 1. Given a graph and a class 𝑐𝑖, if the nodes
+with class 𝑐𝑖 tend to connect uniformly to the nodes with all
+classes 1, ...,𝑐 equally, then class 𝑐𝑖 has no NE.
+PROPOSITION 2. If all classes 𝑐𝑖 = 1, ...,𝑐 in a graph have no
+NE, then this graph has no NE.
+We separate heterophily graphs from those with no NE by the
+propositions. In heterophily graphs, the nodes of a specific class
+are likely to be connected to the nodes of other classes, such as
+in bipartite graphs that connect different classes of nodes. In this
+case, knowing the label of a node gives meaningful information
+about the labels about its neighbors. On the other hand, if a graph
+has no NE, every node has equal probabilities for more than one
+class even after we consider the structural information from its
+neighbors, which is useless to infer its true label.
+To analyze whether a specific class 𝑐𝑖 has NE or not, we use
+𝜒2 test to identify whether there exists a statistically significant
+contingency between the classes. Given two classes 𝑐1 and 𝑐2, the
+
+Under Submission, ,
+Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos
+1
+2
+Class ID
+1
+2
+Class ID
+Edge Counting
+105
+106
+2 × 105
+3 × 105
+4 × 105
+6 × 105
+1
+2
+Class ID
+1
+2
+Class ID
+p-value Table
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+(a) “Genius”: No NE
+1
+2
+Class ID
+1
+2
+Class ID
+Edge Counting
+6.6 × 105
+6.7 × 105
+6.8 × 105
+6.9 × 105
+7 × 105
+7.1 × 105
+7.2 × 105
+7.3 × 105
+7.4 × 105
+1
+2
+Class ID
+1
+2
+Class ID
+p-value Table
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+(b) “Penn94”: No NE
+1
+2
+Class ID
+1
+2
+Class ID
+Edge Counting
+3 × 106
+3.2 × 106
+3.4 × 106
+3.6 × 106
+3.8 × 106
+4 × 106
+1
+2
+Class ID
+1
+2
+Class ID
+p-value Table
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+(c) “Twitch”: No NE
+1
+2
+3
+4
+5
+Class ID
+1
+2
+3
+4
+5
+Class ID
+Edge Counting
+105
+3 × 104
+4 × 104
+6 × 104
+2 × 105
+1
+2
+3
+4
+5
+Class ID
+1
+2
+3
+4
+5
+Class ID
+p-value Table
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+(d) “arXiv-Year”: X-ophily with Weak NE
+1
+2
+3
+4
+5
+Class ID
+1
+2
+3
+4
+5
+Class ID
+Edge Counting
+105
+2 × 105
+3 × 105
+4 × 105
+6 × 105
+1
+2
+3
+4
+5
+Class ID
+1
+2
+3
+4
+5
+Class ID
+p-value Table
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+(e) “Patent-Year”: Heterophily with Weak NE
+1
+2
+Class ID
+1
+2
+Class ID
+Edge Counting
+107
+8 × 106
+9 × 106
+1.1 × 107
+1.2 × 107
+1.3 × 107
+1
+2
+Class ID
+1
+2
+Class ID
+p-value Table
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+(f) “Pokec-Gender”: Heterophily with Strong NE
+Figure 2: NEA discovers that real-world heterophily graphs do not necessarily have network-effect (NE). For each dataset,
+we report the edge counting on the left, and the 𝑝-value table output from NEA on the right. We have a case of X-ophily, e.g.
+in “arXiv-Year”, class 1 is homophily, and the rest are heterophily.
+input to the test is 2 × 2 contingency table with counts of edges
+where nodes of each edge ∈ {𝑐1,𝑐2}.
+NULL HYPOTHESIS 1. Edges are equally likely to exhibit
+homophily and heterophilly.
+Algorithm 1 presents the procedure for the proposed NEA.
+A practical challenge is that if the numbers in the table are too
+large, 𝑝-value becomes extremely small and meaningless [24].
+However, sampling for only a single round can be unstable and
+output very different results. To address this, we combine 𝑝-
+values from different random sampling by Universal Inference
+[38]. We firstly sample edges to add to the contingency table
+until the frequency is above a specified threshold, and compute
+the 𝜒2 test statistic for each class pair. Next, following Universal
+Inference, we repeat the procedure for random samples of edges
+for 𝐵 rounds and average the statistics. At last, we use the average
+statistics to compute the 𝑝-value table 𝑭𝑐×𝑐 of 𝜒2 tests.
+It is worth noting that, NEA is robust to the noisy edges, thanks
+to the random sampling. It also works well given either a few or
+many node labels. Given only a few observations, 𝜒2 test works
+well enough when the frequency in the contingency table are only
+at least 5; given many observations, the sampling and combining
+trick ensures the correctness of 𝑝-value.
+We give observations based on the result of NEA:
+OBSERVATION 1. If a class accepts all the null hypotheses in
+Algorithm 1, then this class has no NE.
+We then extend Observation 1 to an extreme case:
+OBSERVATION 2. If all classes in a graph obey Observation 1,
+the node classification problem is unsolvable under our setting.
+3.2
+Discoveries
+For each dataset, we equally sample 5% of node labels and com-
+pute the 𝑝-value table by Algorithm 1. This is because a) only
+a few labels are observed in most node classification tasks, and
+thus it is natural to make the same assumption in this analysis,
+and b) our NEA can correctly analyze NE even from partial ob-
+servations. We set 𝐵 = 1000 to output stable results. Based on
+Observation 2, here is our surprising discovery:
+DISCOVERY 1 (NO NE). “Genius”, “Penn94”, and “Twitch”
+have no NE, exhibiting neither homophily nor heterophily.
+“Genius” [22], “Penn94” [34], and “Twitch” [31] have been
+widely used in previous works [21, 23, 25, 27, 36, 40]. In “Ge-
+nius” (Figure 2a), we see that both classes 1 and 2 tend to connect
+to class 1. This makes the class 2 indistinguishable by the graph
+structure. NEA thus accepts the null hypothesis and identifies
+that there exists no statistically significant difference. This means
+that the edges have the same probabilities to be homophily and
+heterophily. We can see a similar phenomenon in “Penn94” (Fig-
+ure 2b). “Twitch” (Figure 2c) is not considered as a homophily
+graph because the effect is too weak, where the scales on the
+color bar are very close. However, it is not a heterophily graph
+as well, where NEA correctly identifies that every class tends to
+connect to both classes near-uniformly.
+We further analyzed three more datasets:
+DISCOVERY 2 (WEAK AND STRONG NE). “Arxiv” and
+“Patent-Year” exhibit weak NE; and “Pokec-Gender” exhibits
+strong NE.
+The “arXiv-Year” and “Patent-Year” datasets (Figure 2d and 2e)
+have weak NE, where one of the classes accepts more than one
+null hypothesis. “Pokec-Gender” (Figure 2f) shows strong NE,
+where the estimated 𝑝-value is 0.008. These three datasets will
+later be used in our experiments.
+4
+PROPOSED METHOD PART II –
+ULTRAPROP
+We propose ULTRAPROP, our approach for accurate node classifi-
+cation. Algorithm 2 shows the algorithm of ULTRAPROP. In line
+1, given an adjacency matrix 𝑨 and rank 𝑑, we make “Emphasis”
+Matrix 𝑨∗ (in Section 4.1) to handle the neighbor-differentiation
+(ND). To handle network-effect (NE), we estimate the compati-
+bility matrix ˆ𝑯∗ from 𝑨∗ in line 2 (in Section 4.2). In line 3 to 7,
+we initialize and propagate the beliefs ˆ𝑩 iteratively through 𝑨∗
+until they converge. In each iteration, we aggregate the beliefs of
+neighbors in ˆ𝑩, weighted by the values in 𝑨∗. This aims to draw
+attention to the neighbors that are more structurally important.
+
+ULTRAPROP: Principled and Explainable Propagation on Large Graphs
+Under Submission, ,
+Data: Adjacency matrix 𝑨, initial belief ˆ𝑬, priors P, and
+decomposition rank 𝑑
+Result: Final belief 𝑩
+1 𝑨∗ ← “Emphasis”-Matrix(𝑨,𝑑);
+2 ˆ𝑯∗ ← Compatibility-Matrix-Estimation(𝑨∗, ˆ𝑬, P);
+/* propagation
+*/
+3 ˆ𝑩(0) ← 𝑶𝑛×𝑐,𝑡 ← 0;
+4 while inferences changed and
+� | ˆ𝑩(𝑡+1)− ˆ𝑩(𝑡) |
+𝑛𝑐
+>
+1
+lg𝑛𝑐 do
+5
+ˆ𝑩(𝑡+1) ← ˆ𝑬 + 𝑓 𝑨∗ ˆ𝑩(𝑡) ˆ𝑯∗;
+6
+𝑡 ← 𝑡 + 1;
+7 end
+8 Return 𝑩 ← ˆ𝑩(𝑡) + 1
+𝑐 ;
+Algorithm 2: ULTRAPROP
+The interrelations between classes is handled by multiplying with
+ˆ𝑯∗. We further include an early stopping criterion in line 4 for
+more efficient propagation.
+4.1
+“Emphasis” Matrix
+To incorporate the idea of ND, where neighbors have different
+importances, we propose to replace the unweighted adjacency
+matrix 𝑨 with a weighted one. The weight of edge (𝑖, 𝑗) reflects
+the influence of node 𝑖 for 𝑗. We present an efficient solution to
+weigh 𝑨 without using any node labels. It firstly embeds nodes
+into structure-aware representations via random walks, and then
+measures their similarities via distances in the embedding space.
+Structure-Aware Node Representation. We represent nodes
+in 𝑑-dimensional vector space efficiently using Singular Value
+Decomposition (SVD) on the high-order proximity matrix of the
+graph and capture information from pairwise connections. To
+fast approximate the higher-order proximity matrix, we utilize
+random walks described in Algorithm 3 from line 1 to 8. Given a
+proximity matrix 𝑾 ′, 𝑾 ′
+𝑖𝑗 records the number of times we visit
+node 𝑗 if we start a random walk from node 𝑖. Each neighbor has
+the same probability of being visited in the unweighted graphs,
+where only those structurally important neighbors are visited
+more frequently.
+To theoretically justify why it works, we prove that the neigh-
+bor distribution for each node converges after a number of trials:
+LEMMA 1 (CONVERGENCE OF REGULAR RANDOM WALKS).
+With probability 1−𝛿, the error 𝜖 between the approximated distri-
+bution and the true one for a node walking to its 1-hop neighbor
+by a regular random walk of length 𝐿 with 𝑀 trials is less than
+𝜖 ≤ ⌈(𝐿 − 1)/2⌉
+𝐿
+√︂
+log (2/𝛿)
+2𝐿𝑀
+(2)
+PROOF. Omitted for brevity. Proof in Supplementary A.1.
+■
+To further make the estimation converge faster, we use non-
+backtracking random walk. Given the start node 𝑠 and walk length
+𝐿, its function is defined as follows:
+W(𝑠, 𝐿) =
+�
+(𝑤0 = 𝑠, ...,𝑤𝐿)
+𝑤𝑙 ∈ 𝑁 (𝑤𝑙−1), ∀𝑙 ∈ [1, 𝐿]
+𝑤𝑙−1 ≠ 𝑤𝑙+1, ∀𝑙 ∈ [1, 𝐿 − 1] ,
+(3)
+Data: Adjacency matrix 𝑨, number of trials 𝑀, number
+of steps 𝐿, and dimension 𝑑
+Result: Emphasis matrix 𝑨∗
+1 𝑾 ′ ← 𝑶𝑛×𝑛;
+/* approximate proximity matrix by random walk
+*/
+2 for node 𝑖 in 𝐺 do
+3
+for 𝑚 = 1, ..., 𝑀 do
+4
+for 𝑗 ∈ W(𝑖, 𝐿) do
+5
+𝑾 ′
+𝑖𝑗 ← 𝑾 ′
+𝑖𝑗 + 1;
+6
+end
+7
+end
+8 end
+/* masking, degree normalization and logarithm
+*/
+9 𝑾𝑛×𝑛 ← log (𝑫−1(𝑾 ′ ◦ 𝑨));
+// proximity matrix
+10 𝑼𝑛×𝑑, 𝚺𝑑×𝑑, 𝑽𝑇
+𝑑×𝑛 ← SVD(𝑾,𝑑);
+// embedding
+11 Weigh 𝑨∗
+𝑛×𝑛, where 𝑨∗
+𝑖𝑗 = S(𝑼𝑖, 𝑼 𝑗), ∀{𝑖, 𝑗|𝑨𝑖𝑗 = 1};
+12 Return 𝑨∗;
+Algorithm 3: “Emphasis” Matrix
+where 𝑁 (𝑖) denotes the neighbors of node 𝑖. Thus, with the same
+𝐿 and 𝑀, we improve Lemma 1 to have a tighter bound of 𝜖:
+LEMMA 2 (CONVERGENCE OF NON-BACKTRACKING RAN-
+DOM WALKS). With the same condition as in Lemma 1, the error
+𝜖 by a non-backtracking random walks is less than
+𝜖 ≤ ⌈(𝐿 − 1)/3⌉
+𝐿
+√︂
+log (2/𝛿)
+2𝐿𝑀
+(4)
+PROOF. Omitted for brevity. Proof in Supplementary A.1.
+■
+For example, when using regular random walks of length
+𝐿 = 4 with 𝑀 = 30 trials, the estimated error by Lemma 1 with
+probability 95% is about 6.2%. Nevertheless, if we instead use
+non-backtracking random walks, the error is reduced to 3.1%,
+which is 2× lower than the one by regular walks, indicating that
+the approximated distribution converges well to the true one.
+In Algorithm 3 line 9, an element-wise multiplication by 𝑨
+is done to keep the approximation of 1-hop neighbor for each
+node, which sufficiently supplies necessary information as well
+as keeps the resulting matrix sparse. We use the inverse of the
+degree matrix 𝑫−1 to reduce the influence of nodes with large de-
+grees. This prevents them from dominating the pairwise distance
+by containing more elements in their rows. The element-wise
+logarithm aims to rescale the distribution in 𝑾, in order to en-
+large the difference between smaller structures. We use SVD for
+efficient rank-𝑑 decomposition of the sparse proximity matrix
+𝑾. We multiply the left-singular vectors 𝑼 by the corresponding
+squared eigenvalues
+√
+𝚺 to correct the scale.
+Node Similarity. To estimate the node similarity, we compute
+the distance of nodes in the embedding space. The intuition is that
+the nodes that are closer in the embedding space should be better
+connected with higher-order structures. Given the aforementioned
+embedding 𝑼, the node similarity function S is:
+S(𝑼𝑖, 𝑼 𝑗) = 𝑒−D(𝑼 𝑖𝑘,𝑼 𝑗𝑘),
+(5)
+where 𝑒 ≈ 2.718 denotes Euler’s number. Equation 5 is a universal
+law proposed by Shepard [32], connecting the similarity with
+distance via an exponential function. While the function D can
+
+Under Submission, ,
+Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos
+be any distance metric, we use Euclidean because it is empirically
+shown to work well. Negative exponential distribution is used
+to bound the similarity from 0 to 1, which is close to 0 if the
+distance is too large. Given 𝑨 and 𝑼, “Emphasis” Matrix 𝑨∗
+with weighted edges estimated by S is defined in line 11. Since
+S(𝑼𝑖, 𝑼 𝑗) = S(𝑼 𝑗, 𝑼𝑖), 𝑨∗ is still a symmetric matrix. This is a
+convenient property, which is later used for the fast computation
+of the spectral radius (see Lemma 4).
+4.2
+Compatibility Matrix Estimation
+A compatibility matrix contains the class-wise strength of edges
+and is important for properly inferring the node labels. In this
+subsection, we show how to turn compatibility matrix estimation
+into an optimization problem by introducing our closed-form
+formula, which overcomes the defect of edge counting. We then
+illustrate how we conquer several practical challenges to give a
+precise and fast estimation.
+Why NOT Edge Counting. The naive way to estimate com-
+patibility matrix is via counting labeled edges. However, it is
+inaccurate and has limitations: 1) rare labels will get neglected,
+and 2) being noisy or biased due to few labeled nodes in real
+graphs. The result is even more unreliable if the given labels are
+imbalanced. Figure 3 is an example that edge counting fails if we
+upsample 10× labels for only class 1. This occurs commonly in
+practice, since we have only partial labels in node classification
+tasks, and becomes fatal if the observed distribution is different
+from the true one.
+Closed-Form Formula. In Equation 1, if we initialize the final
+belief with the initial one, and omit the addition of the initial
+belief for the iterative propagation purpose, we have:
+ˆ𝑩 = 𝑨ˆ𝑬 ˆ𝑯
+(6)
+Our goal is to estimate the compatibility matrix ˆ𝑯 of a given
+graph, so that the difference between belief propagated by the
+given priors and the final belief is minimized. To solve this, we
+firstly derive the closed-form solution of Equation 6 based on our
+proposed Network Effect Formula:
+LEMMA 3 (NETWORK EFFECT FORMULA). Given adja-
+cency matrix 𝑨 and initial and final beliefs ˆ𝑬 and ˆ𝑩, the closed-
+form solution of vectorized compatibility matrix vec( ˆ𝑯) is:
+vec( ˆ𝑯) = (𝑿𝑇 𝑿)−1𝑿𝑇𝒚,
+(7)
+where 𝑿 = 𝑰𝑐×𝑐 ⊗ (𝑨ˆ𝑬) and 𝒚 = vec( ˆ𝑩).
+PROOF. Omitted for brevity. Proof in Supplementary A.2.
+■
+Although the final belief matrix ˆ𝑩 is not available before we
+run actual propagation on the graph, we can replace it by 𝒚 =
+vec( ˆ𝑬), and extract the ones that are corresponding to the priors
+P. In other words, we change the problem into minimizing the
+difference between initial belief of each node 𝑖 ∈ P by the initial
+beliefs of its neighbors in the priors P, i.e., 𝑁 (𝑖) ∩ P. Intuitively,
+neighbors should be able to estimate the belief for the node. The
+optimization problem can then be formulated as follows:
+min
+ˆ𝑯
+∑︁
+𝑖 ∈P
+𝑐∑︁
+𝑢=1
+ˆ𝑬𝑖𝑢 − (
+𝑐∑︁
+𝑘=1
+∑︁
+𝑗 ∈𝑁 (𝑖)∩P
+ˆ𝑬 𝑗𝑘 ˆ𝑯𝑘𝑙)
+(8)
+With the help of Network Effect Formula, the optimization prob-
+lem can then be solved by regression.
+1
+2
+3
+4
+5
+6
+Class ID
+1
+2
+3
+4
+5
+6
+Class ID
+Edge Counting
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a) Balanced Prior
+1
+2
+3
+4
+5
+6
+Class ID
+1
+2
+3
+4
+5
+6
+Class ID
+Edge Counting
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b) Imbalanced Prior
+Figure 3:
+Edge counting can not handle imbalanced case.
+Class 1 is upsampled in this example.
+Data: Emphasis Matrix 𝑨∗, initial belief ˆ𝑬, and priors P
+Result: Estimated compatibility matrix ˆ𝑯∗
+1 𝒊 ← ∅;
+// indices only related to priors
+2 for 𝑝 ∈ P do
+3
+for 𝑗 = 1, ...,𝑐 do
+4
+𝒊 ← 𝒊 ∪ {𝑝 + (𝑗 − 1) ∗ 𝑐};
+5
+end
+6 end
+7 𝑿 ← (𝑰𝑐×𝑐 ⊗ (𝑨∗ ˆ𝑬));
+// feature matrix
+8 𝒚 ← vec( ˆ𝑬);
+// target vector
+9 ˆ𝑯∗ ← 𝑅𝑖𝑑𝑔𝑒𝐶𝑉 (𝑿 [𝒊],𝒚[𝒊]);
+10 Return row-normalize(max ( ˆ𝑯∗, 0));
+Algorithm 4: Compatibility Matrix Estimation
+Practical Challenges and Solutions. Network Effect Formula
+allows us to estimate the compatibility matrix by solving this op-
+timization problem, but there still exists two practical challenges
+that need to be addressed.
+First, with few labels, it is difficult to properly separate them
+into training and validation sets for the regression. We thus use
+ridge regression with leave-one-out cross-validation (RidgeCV)
+instead of the traditional linear regression. This allows us to fully
+exploit the observations without having a bias caused by random
+splits of training and validation sets. Moreover, the regularization
+effect of ridge regression makes the compatibility matrix more
+robust to noisy observations. It is noteworthy that the additional
+computational cost of RidgeCV is negligible.
+Next, the compatibility matrix estimated with the adjacency
+matrix 𝑨 is easily interfered with by noisy neighbors, i.e., weakly-
+connected pairs. To address this issue, we use our proposed “Em-
+phasis” Matrix 𝑨∗ instead (see Section 4.1), to pay attention to
+the labels of neighbors that are structurally important. Since the
+rows of the estimated matrix 𝑯 do not sum to one in this ap-
+proach, we filter out the negative values and normalize the sum
+of each row to one. This is done safely, since the negative values
+represent negligible relationships between nodes.
+Algorithm. The overall process of estimation is shown in Al-
+gorithm 4. We extract the indices that are corresponding to the
+priors after the Kronecker product and vectorization in line 2 to
+7. The optimization is then conducted in line 8 to 10 to estimate
+the compatibility matrix ˆ𝑯∗. The negative value filtering and row
+normalization is done on line 11.
+
+ULTRAPROP: Principled and Explainable Propagation on Large Graphs
+Under Submission, ,
+4.3
+Theoretical Analysis
+Convergence Guarantee. To ensure the convergence of propa-
+gation, we introduce a scaling factor multiplied to it during the
+iterations. The exact convergence of ULTRAPROP is as follows:
+LEMMA 4 (EXACT CONVERGENCE). The criterion for the
+exact convergence of ULTRAPROP is:
+ULTRAPROP exactly converges ⇔ 0 < 𝑓 <
+1
+𝜌(𝑨∗) ,
+(9)
+where 𝜌(·) denotes the spectral radius of the given matrix.
+PROOF. Omitted for brevity. Proof in Supplementary A.3.
+■
+A smaller scaling factor leads to a faster convergence, never-
+theless, distorts the results. In ULTRAPROP, we recommend a
+large eigenvalue close to 1, setting 𝑓 = 0.9/𝜌(𝑨∗) as a reason-
+able default. Since 𝑨∗ is built to be symmetric and sparse (see
+Section 4.1), the computation of the spectral radius can be done
+efficiently.
+Complexity Analysis. ULTRAPROP uses sparse matrix repre-
+sentation of graphs. The time complexity is given as:
+LEMMA 5. ULTRAPROP scales linearly on the input size. the
+time complexity of ULTRAPROP is at most
+𝑂(𝑚),
+(10)
+and the space complexity is at most
+𝑂(max (𝑚,𝑛 · 𝐿 · 𝑀) + 𝑛 · 𝑐2).
+(11)
+PROOF. Omitted for brevity. Proof in Supplementary A.4.
+■
+5
+EXPERIMENTS
+In this section, we aims to answer the following questions.
+Q1. Accuracy: How well does ULTRAPROP work on real-world
+graphs as compared to the baselines?
+Q2. Scalability: How does the running-time of ULTRAPROP
+scale w.r.t. graph size?
+Q3. Explainability: How to explain the results of ULTRAPROP?
+Experimental Setup
+Datasets. We focus on large graphs and include eight graph
+datasets with at least 22.5K nodes (details in Supplementary B.1)
+in our evaluation. The statistics of datasets are shown in Table 2
+and 3. For each dataset, we sample only a few node labels as
+initial beliefs. We do this for five times and report the average
+and standard deviation to omit the biases.
+“Synthetic” is the enlarged version of the graph shown in
+Figure 1, which contains both heterophily and homophily NE.
+Noisy edges are injected in the background, and the dense blocks
+are constructed by randomly generating higher-order structures.
+Baselines. We compare ULTRAPROP with five state-of-the-art
+baselines and separate them into four groups: General GNNs:
+GCN [16], and APPNP [17]. Heterophily GNN: MIXHOP [2],
+and GPR-GNN [6]. BP-based methods: HOLS [8]. Our pro-
+posed methods: ULTRAPROP-Hom and ULTRAPROP. ULTRA-
+PROP-Hom is ULTRAPROP using identity matrix as compatibility
+matrix, which assumes homophily and does not handle NE. The
+details of baselines are given in Supplementary B.2.
+Experimental Settings. For deep graph models, since we fo-
+cus on the graph without node features, the node degrees are
+transformed into one hot encoding and used as the node fea-
+tures, which is suggested and implemented by several studies
+(e.g. GraphSAGE and PyTorch Geometric) [9, 12, 13]. The de-
+tails of hyperparameters are given in Supplementary B.3. To give
+fair comparisons on run time, all the experiments are run on the
+same machine, which is a stock Linux server with 3.2GHz Intel
+Xeon CPU. In Section 5.2, we further investigate how much the
+extra cost is, if a more powerful and but more expensive machine
+is used.
+5.1
+Q1 - Accuracy
+In Table 2 and 3, we report the accuracy and wall-clock time for
+each method. We highlight the top three from dark to light by
+,
+and
+denoting the first, second and third place.
+OBSERVATION 3. ULTRAPROP wins on X-ophily, heterophily
+and homophily datasets.
+X-ophily and Heterophily. In Table 2, ULTRAPROP outper-
+forms all the competitors significantly by more than 34.4% and
+12.8% accuracy on the “Synthetic” and “Pokec-Gender” datasets,
+respectively. These datasets have strong NE, thus ULTRAPROP
+boosts the accuracy owing to precise estimations of compatibility
+matrix. The success in “Synthetic” further demonstrates its ability
+to handle the dataset with X-ophily. Heterophily GNNs, namely
+MIXHOP and GPR-GNN, all fail to predict correctly, giving
+results close to random guessing. With homophily assumption,
+General GNNs and BP-based methods also perform poorly.
+Both “arXiv-Year” and “Patent-Year” datasets are shown to
+only have weak NE (in Section 3.2), thus resulting in relatively
+low accuracy for all methods compared with the other two datasets
+with strong NE. Even so, ULTRAPROP still outperforms the
+competitors by estimating a reasonable compatibility matrix. In
+“arXiv-Year”, ULTRAPROP receives the second place by running
+74.6× faster than MIXHOP. In “Patent-Year”, only ULTRAPROP,
+APPNP and MIXHOP are able to give accuracy higher than
+random guessing, which is 26.1%.
+In the cases that ULTRAPROP is faster than ULTRAPROP-
+Hom is because of both the low cost of compatibility matrix
+estimation, and the lower spectral radius of ˆ𝑯∗, leading to a
+faster convergence while propagating.
+Homophily. In Table 3, ULTRAPROP-Hom outperforms all
+the competitors on two homophily datasets, namely “GitHub”
+and “Pokec-Locality”. ULTRAPROP performs similarly to UL-
+TRAPROP-Hom, indicating its generalizability to the homophily
+datasets by estimating near-identity matrices. In addition, ULTRA-
+PROP-Hom gives competitive results with HOLS on the other
+two homophily datasets “Facebook” and “arXiv-Category”, while
+being 84.9× and 5.7× faster than HOLS respectively. General
+GNNs rely heavily on node features for inference which explains
+their poor performance.
+OBSERVATION 4. Our optimizations makes difference.
+We evaluate the effect of different compatibility matrices – (i)
+ULTRAPROP-EC conducts edge counting on the labels of adja-
+cent nodes in the priors, instead of using our Network Effect
+Formula, and (ii) ULTRAPROP-A uses the adjacency matrix in-
+stead of “Emphasis” Matrix to estimate the compatibility matrix
+
+Under Submission, ,
+Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos
+Table 2: ULTRAPROP wins on X-ophily and Heterophily datasets. Accuracy, running time, and speedup are reported. Win-
+ners and runner-ups in
+,
+and
+.
+Dataset
+Synthetic
+Pokec-Gender
+arXiv-Year
+Patent-Year
+# of Nodes / Edges / Classes
+1.2M / 34.0M / 6
+1.6M / 22.3M / 2
+169K / 1.2M / 5
+1.3M / 4.3M / 5
+Label Fraction
+4%
+0.4%
+4%
+4%
+NE Strength
+Strong
+Strong
+Weak
+Weak
+NE Type
+X-ophily
+Heterophily
+X-ophily
+Heterophily
+Method
+Accuracy (%)
+Time (s)
+Speedup Accuracy (%)
+Time (s)
+Speedup Accuracy (%)
+Time (s)
+Speedup Accuracy (%)
+Time (s)
+Speedup
+GCN
+16.7±0.0
+3456
+4.7×
+51.8±0.1
+2906
+3.9×
+35.3±0.1
+132
+3.3×
+26.0±0.0
+894
+3.3×
+APPNP
+18.6±1.1
+7705
+10.4×
+50.9±0.3
+6770
+9.1×
+33.5±0.2
+423
+10.6×
+27.5±0.2
+2050
+7.6×
+MIXHOP
+16.7±0.0
+58391
+79.0×
+53.4±1.2
+53871
+72.7×
+39.6±0.1
+2983
+74.6×
+26.8±0.1
+18787
+70.1×
+GPR-GNN
+18.9±1.2
+7637
+10.3×
+50.7±0.2
+6699
+9.0×
+30.1±1.4
+400
+10.0×
+25.3±0.1
+2034
+7.6×
+HOLS
+46.1±0.1
+1672
+2.3×
+54.4±0.1
+8552
+11.5×
+34.1±0.3
+566
+14.2×
+23.6±0.0
+510
+1.9×
+ULTRAPROP-Hom
+45.7±0.1
+726
+1.0×
+56.9±0.1
+736
+1.0×
+37.0±0.3
+44
+1.0×
+24.1±0.0
+316
+1.2×
+ULTRAPROP
+80.5±0.0
+739
+1.0×
+67.2±0.1
+742
+1.0×
+38.9±0.3
+42
+1.0×
+28.6±0.1
+268
+1.0×
+Table 3: ULTRAPROP wins on Homophily datasets. Accuracy, running time, and speedup are reported. Winners and runner-
+ups in
+,
+and
+.
+Dataset
+Facebook
+GitHub
+arXiv-Category
+Pokec-Locality
+# of Nodes / Edges / Classes
+22.5K / 171K / 4
+37.7K / 289K / 2
+169K / 1.2M / 40
+1.6M / 22.3M / 10
+Label Fraction
+4%
+4%
+0.4%
+0.4%
+Method
+Accuracy (%)
+Time (s)
+Speedup Accuracy (%)
+Time (s)
+Speedup Accuracy (%)
+Time (s)
+Speedup Accuracy (%)
+Time (s)
+Speedup
+GCN
+67.0±0.8
+12
+3.0×
+81.0±0.6
+28
+2.5×
+24.5±0.6
+209
+1.7×
+17.3±0.4
+4002
+3.3×
+APPNP
+50.5±2.2
+46
+10.5×
+74.2±0.0
+73
+6.6×
+17.7±1.3
+993
+8.1×
+16.8±1.7
+11885
+9.7×
+MIXHOP
+69.2±0.7
+296
+73.5×
+77.8±1.3
+526
+47.8×
+23.6±0.5
+3029
+24.8×
+16.9±0.3
+52139
+43.9×
+GPR-GNN
+51.9±1.5
+47
+11.8×
+74.1±0.1
+75
+6.8×
+18.4±1.2
+1016
+8.3×
+30.0±2.0
+11959
+9.7×
+HOLS
+86.0±0.4
+934
+84.9×
+80.8±0.5
+126
+11.5×
+52.0±0.5
+692
+5.7×
+63.7±0.3
+8139
+6.6×
+ULTRAPROP-Hom
+84.7±0.5
+4
+1.0×
+81.7±0.7
+11
+1.0×
+49.5±1.2
+124
+1.0×
+65.4±0.3
+1270
+1.0×
+ULTRAPROP
+84.7±0.5
+4
+1.0×
+81.7±0.7
+11
+1.0×
+48.4±2.5
+122
+1.0×
+64.6±1.0
+1231
+1.0×
+Table 4: Ablation Study: Estimating compatibility matrix by
+the proposed “Emphasis” Matrix is essential. Accuracy (%)
+is reported in the table.
+Datasets
+NE Strength
+ULTRAPROP-Hom ULTRAPROP-EC ULTRAPROP-A
+ULTRAPROP
+Synthetic
+Strong
+77.7±0.0
+68.0±0.1
+77.4±0.0
+80.5±0.0
+Pokec-Gender
+56.9±0.1
+64.9±0.2
+64.8±0.2
+67.2±0.1
+arXiv-Year (imba.)
+Weak
+37.0±0.3
+36.5±1.0
+35.7±0.6
+38.4±0.0
+Patent-Year (imba.)
+24.1±0.0
+24.0±0.9
+28.7±0.1
+28.7±0.0
+Table 5: ULTRAPROP is thrifty. AWS total dollar amount ($)
+is reported in the table. The blue and red fonts denote run-
+ning a single experiment by t3.small and p3.2xlarge, respec-
+tively. Accuracy (%) is reported in Table 2 and 3.
+Datasets
+ULTRAPROP
+GCN
+Pokec-Gender
+$ 0.28 (1.0×)
+$ 12.61 (45.0×)
+Pokec-Locality
+$ 0.47 (1.0×)
+$ 13.66 (29.1×)
+in Algorithm 4. To demonstrate effectiveness of our proposed
+estimation over edge counting, we upsample 5% labels to the
+class with the fewest labels in the datasets with weak NE, which
+are class 2 in “arXiv-Year” and class 1 in “Patent-Year”. We use
+the original labels for propagation in the imbalanced datasets.
+In Table 4, we find that ULTRAPROP outperforms all its vari-
+ants in four datasets. In the datasets with strong NE, ULTRAPROP
+shows its robustness to the structural noises and gives better re-
+sults. In the imbalanced datasets, while ULTRAPROP-EC brings
+its vulnerability to light, ULTRAPROP stays with high accuracy.
+This study highlights the importance of a precise compatibility
+matrix estimation, as well as forming it into an optimization
+problem by our Network Effect Formula as shown in Lemma 3.
+Furthermore, we compare ULTRAPROP with LINBP to dis-
+play its advantages in Figure 4. In Figure 4a, the accuracy gap
+between them indicates the necessity of precisely estimating the
+compatibility matrix. In figure 4b, owing to “Emphasis” Matrix,
+ULTRAPROP-Hom improves the accuracy in all homophily cases
+100
+101
+102
+103
+Run Time
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+Accuracy
+LinBP
+UltraProp
+Synthetic
+Pokec-Gender
+arXiv-Year
+Patent-Year
+Facebook
+GitHub
+arXiv-Category
+Pokec-Locality
+(a) Run Time vs. Accuracy
+Synthetic
+Pokec-Gender
+arXiv-Year
+Patent-Year
+Facebook
+GitHub
+arXiv-Category
+Pokec-Locality
+0.2
+0.4
+0.6
+0.8
+Accruacy
+LinBP
+UltraProp-Hom
+UltraProp
+1.8x
+(b) Accuracy
+Figure 4: Ablation Study: ULTRAPROP wins. It provides the
+best trade-off between accuracy and running time compared
+with LINBP.
+compared with LINBP; owing to both “Emphasis” Matrix and
+Network Effect Formula, ULTRAPROP improves the accuracy
+in all cases while adding negligible penalty on run time, provid-
+ing the best trade-off compared with LINBP. ULTRAPROP per-
+forming similarly to ULTRAPROP-Hom on homophily datasets,
+indicates that it correctly estimates near-identity matrices.
+
+ULTRAPROP: Principled and Explainable Propagation on Large Graphs
+Under Submission, ,
+5.2
+Q2 - Scalability
+We vary the edge number in “Pokec-Gender” and plot against
+the wall-clock running time for ULTRAPROP in Figure 1c, in-
+cluding both training and inference time. As there is no good
+way to sample the graph [19], and also it is prohibitive to use
+graph generator with million nodes, we try our best to ensure the
+connectivity by continuously removing the nodes in the graph,
+until the number of edges is no greater than the target. Note that
+ULTRAPROP scales linearly as expected from Lemma 5.
+Not only ULTRAPROP is scalable and linear, but it is also
+thrifty, achieving up to 45× savings in dollar cost. It requires only
+CPU, while comparable speeds by competitors, require GPUs.
+Table 5 shows the estimated cost, assuming that we use a small
+CPU machine for ULTRAPROP, and a GPU machine for GCN.
+Details of computation are provided in Supplementary B.4.
+5.3
+Q3 - Explainability
+OBSERVATION 5. ULTRAPROP estimated the correct compat-
+ibility matrices.
+We illustrate that the estimations of compatibility matrix by
+Network Effect Formula are precise in Figure 5, so as to inter-
+preting the interrelations of classes extremely well. The inter-
+relations of shown estimated compatibility matrices are similar
+to the ones of edge counting in Figure 2, while being more ro-
+bust to the noisy neighbors, namely, weakly connected ones. For
+“Synthetic”, ULTRAPROP gives the exact answer that we use
+to generate the dataset. For “Pokec-Gender”, ULTRAPROP suc-
+cessfully estimates that people tend to connect to the ones with
+opposite gender. This corresponds to the fact that people incline
+to have more opposite gender interactions during their reproduc-
+tive age [11], where the average ages of male and female in the
+dataset are 25.4 and 24.2, respectively. Although “arXiv-Year”
+and “Patent-Year” do not have strong NE, ULTRAPROP still gives
+an estimated compatibility matrices making much sense in the
+real world, where the papers and patents only cite to the ones
+whose published dates are relatively close to them. We omit the re-
+sults on homophily datasets, for brevity. In all cases ULTRAPROP
+resulted in an near-identity compatibility matrix, as expected,
+supported by giving similar results as ULTRAPROP-Hom, which
+uses identity matrix as compatibility matrix.
+6
+CONCLUSIONS
+We firstly presented Network Effect Analysis (NEA) to identify
+whether a graph exhibit network-effect or not, and surprisingly dis-
+cover the absence of it in many real-world graphs known to have
+heterophily. Next, we present ULTRAPROP to solve node classi-
+fication based two insights, network-effect (NE) and neighbor-
+differentiation (ND), which has the following advantages:
+(1) Accurate: thanks to the precise compatibility matrix esti-
+mation by NE, and ND that weighs important neighbors.
+(2) Explainable: it interprets interrelations of classes with the
+estimated compatibility matrix.
+(3) Scalable: it scales linearly with the input size.
+(4) Principled: it provides provable guarantees (Lemma 1, 2
+and 4) and closed-form solution (Lemma 3).
+Applied on real-world million-scale graph datasets with over
+22M edges, ULTRAPROP only requires 12 minutes on a stock
+1
+2
+3
+4
+5
+6
+Class ID
+1
+2
+3
+4
+5
+6
+Class ID
+Est. Comp. Matrix
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(a)
+“Synthetic”: X-ophily with
+Strong NE
+1
+2
+Class ID
+1
+2
+Class ID
+Est. Comp. Matrix
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+(b) “Pokec-Gender”: Heterophily
+with Strong NE
+1
+2
+3
+4
+5
+Class ID
+1
+2
+3
+4
+5
+Class ID
+Est. Comp. Matrix
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+(c)
+“arXiv-Year”: X-ophily with
+Weak NE
+1
+2
+3
+4
+5
+Class ID
+1
+2
+3
+4
+5
+Class ID
+Est. Comp. Matrix
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+(d)
+“Patent-Year”: Heterophily
+with Weak NE
+Figure 5: ULTRAPROP is explainable. The estimated com-
+patibility matrices are similar to the edge counting matrix
+(in Figure 2), while being robust to the noises.
+CPU-machine, and outperforms recent baselines on accuracy, as
+well as on speed (≥ 9×).
+Reproducibility: Our implemented source code and prepro-
+cessed datasets will be published once the paper is accepted.
+
+Under Submission, ,
+Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos
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+ULTRAPROP: Principled and Explainable Propagation on Large Graphs
+Under Submission, ,
+A
+PROOF
+A.1
+Proof of Lemma 1 and 2
+PROOF. For a 𝐿-steps random walk sequence 𝑆 with 𝑀 trials,
+the sequence length |𝑆| is 𝐿𝑀. We define the random variable 𝑋,
+denoting the probability of node 𝑖 will walk to its 𝑗-th neighbor:
+𝑋 = P(node 𝑖 walks to 𝑁 (𝑖)𝑗) =
+�|𝑆 |
+𝑘=1 1(𝑁 (𝑖)𝑗 = 𝑆𝑘)
+|𝑆|
+,
+(12)
+where P denotes the probability and 1 denotes the indicator. With
+regular random walk in the graph without self-loops, the random
+variable 𝑋 is upper-bounded by ⌈(𝐿−1)/2⌉
+𝐿
+. We can thus apply
+Hoeffding’s inequality:
+P(| ˆ𝜇|𝑆 | − 𝜇| ≥ 𝜖) ≤ 2 exp
+−2𝐿3𝑀𝑡2
+⌈(𝐿 − 1)/2⌉2 ,
+(13)
+where ˆ𝜇|𝑆 | denotes the sampled mean of the given random vari-
+able, and 𝜇 denotes the expectation. Let 𝛿 = 2 exp
+−2𝐿3𝑀𝑡2
+⌈(𝐿−1)/2⌉2 ,
+with probability 1 − 𝛿, the error 𝜖 is:
+𝜖 = | ˆ𝜇|𝑆 | − 𝜇| ≤ ⌈(𝐿 − 1)/2⌉
+𝐿
+√︂
+log (2/𝛿)
+2𝐿𝑀
+(14)
+With the help of non-backtracking random walk [3], we can
+further shrink the upper bound of 𝑋 into ⌈(𝐿−1)/3⌉
+𝐿
+. Now, let
+𝛿 = 2 exp
+−2𝐿3𝑀𝑡2
+⌈(𝐿−1)/3⌉2 , with probability 1 − 𝛿, the error 𝜖 can thus
+be improved to:
+𝜖 = | ˆ𝜇|𝑆 | − 𝜇| ≤ ⌈(𝐿 − 1)/3⌉
+𝐿
+√︂
+log (2/𝛿)
+2𝐿𝑀
+(15)
+■
+A.2
+Proof of Lemma 3
+PROOF. In the beginning, we introduce two necessary nota-
+tions. vec(·) denotes the vectorization operator:
+vec(𝑿) = [𝑿11, · · · , 𝑿𝑚1, 𝑿12, · · · , 𝑿𝑚2, · · · , 𝑿𝑚𝑛]⊤,
+(16)
+where 𝑿 is an 𝑚 ×𝑛 matrix, and 𝑿𝑖𝑗 denotes the element of 𝑿 on
+the 𝑖-th row and the 𝑗-th column. Next, the Knronecker product
+of given two 𝑚 × 𝑛 matrices 𝑿 and 𝒀 is:
+𝑿 ⊗ 𝒀 =
+
+𝑿11𝒀
+𝑿12𝒀
+· · ·
+𝑿1𝑛𝒀
+𝑿21𝒀
+𝑿22𝒀
+· · ·
+𝑿2𝑛𝒀
+...
+...
+...
+...
+𝑿𝑚1𝒀
+𝑿𝑚2𝒀
+· · ·
+𝑿𝑚𝑛𝒀
+
+(17)
+The idea of this proof is to reformulate the equation in order to
+derive the final result by the closed formula of Linear Regression.
+We firstly show two well-known equations that will be used in
+our proof. Given the features 𝑿 and target 𝒚, the closed formula
+of the weights 𝑾 of Linear Regression is:
+𝑾 = (𝑿𝑇 𝑿)−1𝑿𝑇𝒚.
+(18)
+The famous property of the mixed Kronecker matrix-vector prod-
+uct [4] is also used:
+vec(𝑩𝑽𝑨𝑇 ) = (𝑨 ⊗ 𝑩)𝒗,
+(19)
+where the matrix 𝑽 = vec−1(𝒗) is the result of the inverse of the
+vectorization operator on 𝒗.
+To begin the derivation, we vectorize Equation 6 into:
+vec( ˆ𝑩) = vec((𝑨ˆ𝑬) ˆ𝑯𝑰𝑐×𝑐),
+(20)
+where 𝑰𝑐×𝑐 is a 𝑐 × 𝑐 identity matrix. The trick here, which is the
+key of this proof, is to multiply one more identity matrix by ˆ𝑯.
+Therefore, we use Equation 19 to reformulate the equation to:
+vec( ˆ𝑩) = (𝑰𝑐×𝑐 ⊗ (𝑨ˆ𝑬))vec( ˆ𝑯)
+(21)
+By letting 𝑿 = 𝑰𝑐×𝑐 ⊗ (𝑨ˆ𝑬) and 𝒚 = vec( ˆ𝑩) in Equation 18, we
+can then derive the closed-form solution of vectorized compati-
+bility matrix as follows:
+vec( ˆ𝑯) = (𝑿𝑇 𝑿)−1𝑿𝑇𝒚
+(22)
+■
+A.3
+Proof of Lemma 4
+PROOF. ULTRAPROP exactly converges if and only if
+𝜌(𝑨∗)𝜌( ˆ𝑯∗) < 1. However, the compatibility matrix 𝑯∗ is row-
+normalized, so the largest eigenvalue 𝜌(𝑯∗) = 1 is a constant,
+and is less than one after centering. Thus, the scaling factor 𝑓
+multiplied to the propagation (in Algorithm 2 line 5) should be in
+the range of (0,
+1
+𝜌 (𝑨∗) ) to meet the criterion of exact convergence.
+■
+A.4
+Proof of Lemma 5
+PROOF. In the neighbor-differentiation phase, for each ran-
+dom walk, each node visits at most 𝐿·𝑀 unique nodes, so the max-
+imum number of non-zero elements in 𝑾 is either 𝑛 · 𝐿 · 𝑀 if we
+have not walked through all the edges, or 𝑚 otherwise. The time
+complexity of SVD on 𝑾 then takes 𝑂(𝑑 · max (𝑚,𝑛 · 𝐿 · 𝑀)). In
+the network-effect phase, the time complexity for the Fisher’s ex-
+act test is 𝑂(max (𝑪)), where max (𝑪) is a constant bounded by
+500 in our algorithm. Therefore, network-effect analysis takes
+𝑂(|𝒆
+′| · 𝑐2). For the regression, since there are 𝑐 sets of pa-
+rameters are independent, we can separate the problem into 𝑐
+tasks, where each contains 𝑐 features and |𝒑| samples. Thus
+the complexity can be reduced to 𝑂(|𝒑| · 𝑐3), and the efficient
+leave-one-out cross-validation only needs to be done once. In the
+propagation phase, it takes at most 𝑂(𝑚 + 𝑛) for sparse matrix
+multiplication to run 𝑡 iterations. Thus, the time complexity is
+𝑂(𝑑 max (𝑚,𝑛 · 𝐿 · 𝑀) + |𝒑| ·𝑐3 +𝑚). However, in practice, 𝑐, |𝒑|
+and 𝑡 are usually small constants which are negligible, and 𝑚
+is usually much larger than them. Therefore, keeping only the
+dominating terms, the time complexity is approximately 𝑂(𝑚).
+𝑾 contains at most max (𝑚,𝑛 · 𝐿 · 𝑀) non-zero elements. The
+Kronecker product at most contains 𝑛 · 𝑐2 non-zero elements. ˆ𝑩
+and ˆ𝑯 contain at most 𝑛 ·𝑐 and 𝑐2 non-zero elements, respectively.
+Thus, the space complexity is 𝑂(max (𝑚,𝑛 · 𝐿 · 𝑀) + 𝑛 · 𝑐2).
+■
+B
+REPRODUCIBILITY
+B.1
+Datasets
+• “Pokec-Gender” [33] is an online social network in Slo-
+vakia. [23] re-labels the nodes by users’ genders instead.
+• “arXiv-Year” [14] is a citation network between all Com-
+puter Science arXiv papers. [23] re-labels the nodes by the
+posted years.
+• “Patent-Year” [20] is the patent citation network from
+1980 to 1985. [23] re-labels the nodes by the application
+year, bucketized into five consecutive 3-year ranges.
+• “Synthetic” is a graph enlarged by the one in Figure 1. It
+contains both heterophily and homophily network-effect.
+
+Under Submission, ,
+Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos
+Table 6: Hyperparameters for Deep Graph Models
+Method
+Hyperparameters
+GCN
+lr=0.01, wd=0.0005, hidden=16, dropout=0.5
+APPNP
+lr=0.002, wd=0.0005, hidden=64, dropout=0.5, K=10, alpha=0.1
+MIXHOP
+lr=0.01, wd=0.0005, cutoff=0.1, layers1=[200, 200, 200], layers2=[200, 200, 200]
+GPR-GNN
+lr=0.002, wd=0.0005, hidden=64, dropout=0.5, K=10, alpha=0.1
+Noisy edges are randomly injected in the background, and
+the dense blocks are constructed by randomly creating
+higher-order structures.
+• “Facebook” [30] is a page-to-page network of verified
+Facebook sites. Nodes are labeled by the categories such
+as politicians and companies.
+• “GitHub” [30] is a social network of developers in June
+2019. Nodes are labeled as web or a machine learning
+developer.
+• “arXiv-Category” [37] is the same dataset as the arXiv-
+Year dataset. Nodes are labeled by the primary categories.
+• “Pokec-Locality” [33] is the same dataset as the Pokec-
+Gender dataset. Nodes are labeled by the uses’ localities.
+B.2
+Baselines
+• GCN2 [16] is a well-known deep graph model, learning
+and aggregating the weights of two-hop neighbors.
+• APPNP4 [17] utilizes personalized PageRank to leverage
+the local information and a larger neighborhood.
+• MIXHOP3 [2] mixes powers of the adjacency matrix to
+incorporate more than 1-hop neighbors in each layer.
+• GPR-GNN4 [6] allows the learnable weights to be nega-
+tive during propagation with Generalized PageRank.
+• HOLS5 [8] is a label propagation method with attention,
+by increasing the importance of a neighbor if they appear
+in the same motif at the same time.
+B.3
+Hyperparameters
+For ULTRAPROP and ULTRAPROP-Hom, we use random walks
+of length 4 with 10 trials except GitHub, arXiv-Category and
+Pokec-Locality datasets, where we use 30 trials. The decompo-
+sition rank is set to be 128, which is empirically shown to be
+enough in the embedding tasks. The weights of HOLS for differ-
+ent motifs are set to be equal. For the deep graph models, under
+the setting that the given labels are very few, it is impossible to
+separate a validation set. We then train them for a fixed number
+of epochs (i.e. 200 epochs), which is usually sufficient enough for
+them to converge. All the fully connected layers are replaced by
+the sparse version in order to fit into memory. Both adjacency ma-
+trices and features are normalized and turn into sparse matrices if
+needed. For other hyperparameters, we use the default settings
+given by the authors, and give the details in Table 6.
+2https://github.com/tkipf/pygcn
+3https://github.com/benedekrozemberczki/MixHop-and-N-GCN
+4https://github.com/jianhao2016/GPRGNN
+5https://github.com/dhivyaeswaran/hols
+B.4
+Scalability
+We select machines provided by AWS with comparable specs as
+we use for the experiments. For CPU machine, we select t3.small
+with 3.3GHz CPU and 2GB RAM, which is faster than ours, and
+costs $0.023 per hour. For GPU machine, we select p3.2xlarge
+with a V100 GPU, which costs $3.06 per hour. According to [1],
+it is 0.89 slower than the RTX A6000 GPU we use on running
+PyTorch. The running time of GCN on “Pokec-Gender” and
+“Pokec-Locality” are 673 and 730 seconds, respectively. Using the
+provided information, the results in Table 5 can be computed.
+

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+ORBIT: A Unified Simulation Framework for
+Interactive Robot Learning Environments
+Mayank Mittal1,2, Calvin Yu3, Qinxi Yu3, Jingzhou Liu1,3, Nikita Rudin1,2, David Hoeller1,2,
+Jia Lin Yuan3, Pooria Poorsarvi Tehrani3, Ritvik Singh1,3, Yunrong Guo1, Hammad Mazhar1,
+Ajay Mandlekar1, Buck Babich1, Gavriel State1, Marco Hutter2, Animesh Garg1,3
+Fig. 1: ORBIT framework provides a large set of robots, sensors, rigid and deformable objects, motion generators, and teleoperation
+interfaces. Through these, we aim to simplify the process of defining new and complex environments, thereby providing a common
+platform for algorithmic research in robotics and robot learning.
+Abstract— We present ORBIT, a unified and modular frame-
+work for robot learning powered by NVIDIA Isaac Sim. It
+offers a modular design to easily and efficiently create robotic
+environments with photo-realistic scenes and fast and accurate
+rigid and deformable body simulation. With ORBIT, we provide
+a suite of benchmark tasks of varying difficulty– from single-
+stage cabinet opening and cloth folding to multi-stage tasks
+such as room reorganization. To support working with diverse
+observations and action spaces, we include fixed-arm and
+mobile manipulators with different physically-based sensors
+and motion generators. ORBIT allows training reinforcement
+learning policies and collecting large demonstration datasets
+from hand-crafted or expert solutions in a matter of minutes
+by leveraging GPU-based parallelization. In summary, we offer
+an open-sourced framework that readily comes with 16 robotic
+platforms, 4 sensor modalities, 10 motion generators, more than
+20 benchmark tasks, and wrappers to 4 learning libraries. With
+this framework, we aim to support various research areas,
+including representation learning, reinforcement learning, imi-
+tation learning, and task and motion planning. We hope it helps
+establish interdisciplinary collaborations in these communities,
+and its modularity makes it easily extensible for more tasks
+and applications in the future. For videos, documentation, and
+code: https://isaac-orbit.github.io/.
+I. INTRODUCTION
+The recent surge in machine learning has led to a paradigm
+shift in robotics research. Methods such as reinforcement
+learning (RL) have shown incredible success in challenging
+problems such as quadrupedal locomotion [1], [2], [3] and
+in-hand manipulation [4], [5]. However, learning techniques
+1 NVIDIA, 2 ETH Z¨urich, 3 University of Toronto, Vector Institute.
+Correspondence: [email protected], [email protected].
+require a wealth of training data, which is often challenging
+and expensive to obtain at scale on a physical system. This
+makes simulators an appealing alternative for developing
+systems safely, efficiently, and cost-effectively.
+An ideal robot simulation framework needs to provide fast
+and accurate physics, high-fidelity sensor simulation, diverse
+asset handling, and easy-to-use interfaces for integrating new
+tasks and environments. However, existing frameworks often
+make a trade-off between these aspects depending on their
+target application. For instance, simulators designed mainly
+for vision, such as Habitat [18] or ManipulaTHOR [16], offer
+decent rendering but simplify low-level interaction intricacies
+such as grasping. On the other hand, physics simulators for
+robotics, such as IsaacGym [17] or Sapien [15], provide fast
+and reasonably accurate rigid-body contact dynamics but do
+not include physically-based rendering (PBR), deformable
+objects simulation or ROS [19] support out-of-the-box.
+In this work, we present ORBIT an open-source frame-
+work, built on NVIDIA Isaac Sim [20], for intuitive designing
+of environments and tasks for robot learning with photo-
+realistic scenes and state-of-the-art physics simulation. Its
+modular design supports various robotic applications, such as
+reinforcement learning (RL), learning from demonstrations
+(LfD), and motion planning. Through careful design of inter-
+faces, we aim to support learning for a diverse range of robots
+and tasks, allowing operation at different levels of observa-
+tion (proprioception, images, pointclouds) and action spaces
+(joint space, task space). To ensure high-simulation through-
+put, we leverage hardware-accelerated robot simulation, and
+include GPU implementations for motion generation and
+arXiv:2301.04195v1  [cs.RO]  10 Jan 2023
+
+TABLE I: Comparison between different simulation frameworks and ORBIT. The check (✓) and cross (X) denote presence or absence of
+the feature. In Robotic Platforms column, M stands for manipulator. In Scene Authoring column, G stands for game-based designing,
+M for mesh-scan scenes, and P for procedural-generation.
+Vectorization
+Supported Dynamics
+Sensors
+Robotic Platforms
+Name
+Physics Engine
+Renderer
+CPU
+GPU
+Rigid
+Cloth
+Soft
+Fluid
+PBR
+Tracing
+RGBD
+Semantic
+LiDAR
+Contact
+Fixed-M
+Mobile-M
+Legged
+Scene
+Authoring
+MetaWorld [6]
+MuJoCo
+OpenGL
+✓
+X
+✓
+X
+X
+X
+X
+X
+X
+X
+X
+✓
+X
+X
+P
+RoboSuite [7]
+MuJoCo
+OpenGL, OptiX
+✓
+X
+✓
+X
+X
+X
+X
+✓
+✓
+X
+✓
+✓
+X
+X
+P
+DoorGym [8]
+MuJoCo
+Unity
+X
+X
+✓
+X
+X
+X
+✓
+✓
+✓
+X
+X
+✓
+X
+X
+P, G
+DEDO [9]
+Bullet
+OpenGL
+✓
+X
+✓
+✓
+✓
+X
+X
+✓
+X
+X
+X
+✓
+X
+X
+P, G
+RLBench [10]
+Bullet/ODE
+OpenGL
+X
+X
+✓
+X
+X
+X
+X
+✓
+✓
+X
+✓
+✓
+X
+X
+P
+iGibson [11]
+Bullet
+MeshRenderer
+✓
+X
+✓
+X
+X
+X
+✓
+✓
+✓
+✓
+X
+X
+✓
+X
+M
+Habitat 2.0 [12]
+Bullet
+Magnum
+X
+X
+✓
+X
+X
+X
+✓
+✓
+✓
+X
+X
+✓
+✓
+✓
+P, M
+SoftGym [13]
+FleX
+OpenGL
+✓
+X
+✓
+✓
+✓
+✓
+X
+X
+X
+X
+X
+✓
+X
+X
+P
+ThreeDWorld [14]
+PhysX 4/FleX/Obi
+Unity3D
+X
+X
+✓∗
+✓∗
+✓∗
+✓∗
+✓
+✓
+✓
+X
+X
+X
+✓
+X
+P
+SAPIEN [15]
+PhysX 4
+OptiX, Kuafu
+✓
+X
+✓
+X
+X
+X
+✓
+✓
+✓
+X
+✓
+✓
+✓
+X
+P
+ManipulatorThor [16]
+PhysX 4
+Unity
+X
+X
+✓
+X
+X
+X
+✓
+✓
+✓
+X
+X
+X
+✓
+X
+P, G
+IsaacGymEnvs [17]
+PhysX 5
+Vulkan
+✓
+✓
+✓
+X
+X
+X
+X
+✓
+✓
+X
+✓
+✓
+X
+✓
+P
+ORBIT (ours)
+PhysX 5
+Omniverse RTX
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+✓
+P, M, G
+* ThreeDWorld supports simulation of rigid bodies and deformable bodies based on whether PhysX 4 or FleX/Obi is enabled respectively. Thus, it is limited in simulating interactions between rigid and deformable bodies.
+observations processing. This allows training and evaluation
+of a complete robotic system at scale, without abstracting
+out low-level details in robot-environment interactions.
+The release of ORBIT v1.0 features:
+1) models for three quadrupeds, seven robotic arms, four
+grippers, two hands, and four mobile manipulators;
+2) a selection of CPU and GPU-based motion generators
+implementations for each robot category, including pre-
+trained locomotion policies, inverse kinematics, opera-
+tional space control, and model predictive control;
+3) utilities for collecting human demonstrations using pe-
+ripherals (keyboard, gamepad or 3D mouse), replaying
+demonstration datasets, and utilizing them for learning;
+4) a suite of standardized tasks of varying complexity for
+benchmark purposes. These include eleven rigid object
+manipulation, thirteen deformable object manipulation,
+and two locomotion environments. Within each task, we
+allow switching robots, objects, and sensors easily.
+In the remaining of the paper, we describe the underlying
+simulation choices (Sec. II), the framework’s design deci-
+sions and abstractions (Sec. III), and its highlighted features
+(Sec. IV). We demonstrate the framework’s applicability for
+different workflows (Sec. V) – particularly RL using various
+libraries, LfD with robomimic [21], motion planning [22],
+[23], and connection to physical robots for deployment.
+II. RELATED WORK
+Recent years have seen several simulation frameworks,
+each specializing for particular robotic applications. In this
+section, we highlight the design choices crucial for building
+a unified simulation platform and how ORBIT compares to
+other frameworks (also summarized in Table I).
+a) Physics Engine: Increasing the complexity and re-
+alism of physically simulated environments is essential for
+advancing robotics research. This includes improving the
+contact dynamics, having better collision handling for non-
+convex geometries (such as threads), stable solvers for de-
+formable bodies, and high simulation throughput.
+Prior frameworks [7], [10] using MuJoCo [24] or Bul-
+let [25] focus mainly on rigid object manipulation tasks.
+Since their underlying physics engines are CPU-based, they
+need CPU clusters to achieve massive parallelization [17]. On
+the other hand, frameworks for deformable bodies [9], [13]
+mainly employ Bullet [25] or FleX [26], which use particle-
+based dynamics for soft bodies and cloth simulation. How-
+ever, limited tooling exists in these frameworks compared
+to those for rigid object tasks. ORBIT aims to bridge this
+gap by providing a robotics framework that supports rigid
+and deformable body simulation via PhysX SDK 5 [27]. In
+contrast to other engines, it features GPU-based hardware
+acceleration for high throughput, signed-distance field (SDF)
+collision checking [28], and more stable solvers based on
+finite elements for deformable body simulation.
+b) Sensor simulation: Various existing frameworks [7],
+[10], [17] use classic rasterization that limits the photo-
+realism in the generated images. Recent techniques [29], [30]
+simulate the interaction of rays with object’s textures in a
+physically correct manner. These methods helps capture fine
+visual properties such as transparency and reflection, thereby
+are promising for bridging sim-to-real visual domain gap.
+While recent frameworks [15], [12], [16] include physically-
+based renderers, they mainly support camera-based sen-
+sors (RGB, depth). This is insufficient for certain mobile
+robot applications that need range sensors, such as LiDARs.
+Leveraging the ray-tracing technology in NVIDIA Isaac Sim,
+ORBIT supports all these modalities and includes APIs to
+obtain additional information such as semantic annotations.
+c) Scene designing and asset handling: Frameworks
+support scene creation procedurally [6], [7], [15], via mesh
+scans [11], [12] or through game-engine style interfaces [31],
+[14]. While mesh scans simplify generating large amounts of
+scenes, they often suffer from geometric artifacts and lighting
+problems. On the other hand, procedural generation allows
+leveraging object datasets for diverse scenes. To not restrict
+to either possibility, we facilitate scene designing by using
+graphical interfaces and also providing tools for importing
+different datasets [32], [33], [34].
+Simulators are typically general-purpose and expose ac-
+cess to various internal properties, often alienating non-
+expert users due to a steep learning curve. ORBIT inherits
+many utilities from the NVIDIA Omniverse and Isaac Sim
+platforms, such as high-quality rendering, multi-format asset
+import, ROS support, and domain randomization (DR) tools.
+However, its contributions lie in the specialization of inter-
+faces for robot learning that simplify environment designing
+and facilitate transfer to a real robot. For instance, we provide
+unified abstractions for different robot and object types, allow
+
+Fig. 3: ORBIT’s abstractions comprise World, analogous to the real world, and Agent, the computation graph behind the embodied system.
+The nodes in the agent’s graph can perform observation-based or action-based processing. Through a graph-cut over this computation
+graph and specifying an extrinsic goal, it is feasible to design different tasks within the same World instance.
+injecting actuator models into the simulation to assist in
+sim-to-real transfer, and support various peripherals for data
+collection. Overall, it provides a highly featured state-of-the-
+art simulation framework (Table I) while preserving usability
+through intuitive abstractions.
+III. ORBIT: ABSTRACTIONS AND INTERFACES DESIGN
+At a high level, the framework design comprises a world
+and an agent, similar to the real world and the software
+stack running on the robot. The agent receives raw observa-
+tions from the world and computes the actions to apply on the
+embodiment (robot). Typically in learning, it is assumed
+that all the perception and motion generation occurs at the
+same frequency. However, in the real world, that is rarely the
+case: (1) different sensors tick at differing frequencies, (2)
+depending on the control architecture, actions are applied at
+different time-scales [35], and (3) various unmodeled sources
+cause delays and noises in the real system. In ORBIT, we
+carefully design the interfaces and abstractions to support
+(1) and (2), and for (3), we include implementation of
+different actuator and noise models as part of the robot
+and sensors respectively.
+a) World: Analogous to the real world, we define a
+world where robots, sensors, and objects (static
+or dynamic) exist on the same stage. The world can be de-
+signed procedurally (script-based), via scanned meshes [33],
+[32], through the game-based GUI of Isaac Sim, or a
+combination of them, such as importing scanned meshes
+and adding objects to it. This flexibility reaps the benefits
+of 3D reconstructed meshes, which capture various archi-
+tectural layouts, with game-based designing, that simplifies
+the experience of creating and verifying the scene physics
+properties by playing the simulation.
+Robots are a crucial component of the world since
+they serve as the embodiment for interaction. They consist
+of an articulated system, sensors, and low-level controllers.
+The robot class loads its model from USD files. It may
+DC Motor
+ Actuator Net
+(MLP/LSTM)
+Fig. 4: Illustration of actuator groups for a legged mobile manipu-
+lator. This allows decomposing a complex system into sub-groups
+and defining specific transmission models for each of them flexibly.
+have onboard sensors specified through the same USD file or
+configuration files. The low-level controller processes input
+actions through the configured actuator models and applies
+desired joint position, velocity, or torque commands to the
+simulator (as shown in Fig. 4). The actuator dynamics can be
+modeled using first-principle from physics or be learned as
+neural networks. This allows injection of real world actuator
+characteristics into simulation thereby facilitating sim-to-real
+transfer of control policies [36].
+Sensors may exist both on the articulation (as part of the
+robot) or externally (such as, third-person cameras). ORBIT
+interface unifies different physics-based (range, force, and
+contact sensor) and rendering-based (RGB, depth, normals)
+sensors under a common interface. To simulate asynchronous
+sensing and actuation, each sensor has an internal timer that
+governs its operating frequency. The sensor only reads the
+simulator buffers at the configured frequency. Between the
+timesteps, the sensor returns the previously obtained values.
+Objects are passive entities in the world. While several
+objects may exist in the scene, the user can define objects
+of interest for a specified task and retrieve data/properties
+only for them. Object properties mainly comprise visual and
+collision meshes, textures, and physics materials. For any
+given object, we support randomization of its textures and
+physics properties, such as friction and joint parameters.
+
+rt
+Learning
+Learning
+Task Logic
+Task
+Agent
+Rewards/Costs
+Oracle Reset
+World
+Agent
+Sensors
+Ot
+Node 1
+Passive
+Camera
+External Sensors
+Objects
+Node 2
+LiDAR
+Robot
+at
+....
+Node n
+Actuator Model
+Height Scan
+Visualization
+On-board
+Sensors
+Markers
+Contact Report
+Computation Nodes
+O NVIDIA Isaac Sim
+Motion Generation
+Perception
+Filtering
+Learning-based
+PhysX
+NVIDIA
+可
+USD
+Mapping
+Model-based
+Iray
+by NVIDIAGripper
+open/close
+joint velocity
+Mimic Group
+(1)
+(6)
+Actions
+joint position
+Arm
+joint torque
+DC Motor
+(6)
+(6)
+Base
+joint position
+Actuator Net
+joint torque
+(12)
+(MLP/LSTM)
+(12)Rigid
+Articulated
+Deformable
+Cloth
+IK
+OSC
+RMPFlow
+OCS2
+NN Policy
+Teleoperation
+End-Effector
+Arm
+Mobile Base
+Height Scan
+Camera
+Contact Reporter
+Proprioception
+Fig. 5: Overview of features included in ORBIT. We provide models of different sensors, robotic platforms, objects from different
+datasets, motion generators and teleoperation devices. Using RTX-accelerated ray-tracing, we can obtain high-fidelity images in real-time
+for different modalities such as RGB, depth, surface normal, instance and semantic segmentation (pixel-wise and bounding boxes).
+b) Agent: An agent refers to the decision-making
+process (“intelligence”) guiding the embodied system. While
+roboticists have embraced the modularity of ROS [19], most
+robot learning frameworks often focus only on the environ-
+ment definition. This practice requires code replication, &
+adds friction to switching between different implementations.
+Keeping modularity at its core, an agent in ORBIT com-
+prises various nodes that formulate a computation graph
+exchanging information between them. Broadly, we consider
+nodes are of two types: 1) perception-based i.e., they process
+inputs into another representation (such as RGB-D image to
+point-cloud/TSDF), or 2) action-based i.e., they process in-
+puts into action commands (such as task-level commands to
+joint commands). Currently, the flow of information between
+nodes happens synchronously via Python, which avoids the
+data exchange overhead of service-client protocols.
+c) Learning task and agent: Paradigms such as RL
+require specification of a task, a world and may include
+some computation nodes of the agent. The task logic
+helps specify the goal for the agent, compute metrics (re-
+wards/costs) to evaluate the agent’s performance, and manage
+the episodic resets. With this component as a separate mod-
+ule, it becomes feasible to use the same world definition
+for different tasks, similar to learning in the real world,
+where tasks are specified through extrinsic reward signals.
+The task definition may also contain different nodes of the
+agent. An intuitive way to formalize this is by considering
+that learning for a particular node happens through a graph
+cut on the agent’s computation graph.
+To further concretize the design motivation, consider the
+example of learning over task space instead of low-level joint
+actions for lifting a cube [35]. In this case, the task-space
+controller, such as inverse kinematics (IK), would typically
+run at 50Hz, while the joint controller requires commands
+at 1000 Hz. Although the task-space controller is a part of
+the agent’s and not the world’s computation, it is possible
+to encapsulate that into the task design. This functionality
+easily allows switching between motion generators, such as
+IK, operational-space control (OSC), or reactive planners.
+IV. ORBIT: FEATURES
+While various robotic benchmarks have been proposed [9],
+[6], [10], the right choice of necessary and sufficient tasks to
+demonstrate “intelligent” behaviors remains an open ques-
+tion. Instead of being prescriptive about tasks, we provide
+ORBIT as a platform to easily design new tasks. To facilitate
+the same, we include a diverse set of supported robots,
+peripheral devices, and motion generators and a large set
+of tasks for rigid and soft object manipulation for essential
+skills such as folding cloth, opening the dishwasher, and
+screwing a nut into a bolt. Each task showcases aspects of
+physics and renderer that we believe will facilitate answering
+crucial research questions, such as building representations
+for deformable object manipulation and learning skills that
+generalize to different objects and robots.
+a) Robots: We support 4 mobile platforms (one om-
+nidirectional drive base and three quadrupeds), 7 robotic
+arms (two 6-DoF and five 7-DoF), and 6 end-effectors (four
+parallel-jaw grippers and two robotic hands). We provide
+tools to compose different combinations of these articulations
+into a complex robotic system such as a legged mobile
+manipulator. This provides a large set of robot platforms,
+each of which can be switched in the World.
+b) I/O Devices: Devices define the interface to periph-
+eral controllers that teleoperate the robot in real-time. The
+interface reads the input commands from an I/O device and
+parses them into control commands for subsequent nodes.
+This helps not only in collecting demonstrations [21] but also
+in debugging the task designs. Currently, we include support
+for Keyboard, Gamepad (Xbox controller), and Spacemouse
+from 3Dconnexion.
+c) Motion Generators: Motion generators transform
+high-level actions into lower-level commands by treating
+input actions as reference tracking signals. For instance,
+inverse kinematics (IK) [37] interprets commands as the
+desired end-effector poses and computes the desired joint
+positions. Employing these controllers, particularly in task
+space, has shown to help sim-to-real transferability of robot
+manipulation policies [7], [35].
+With ORBIT, we include GPU-based implementations for:
+differential IK [37], operational-space control [38] and joint-
+level control. Additionally, we provide CPU implementa-
+tion of state-of-the-art model-based planners such as RMP-
+Flow [22] for fixed-arm manipulators and OCS2 [23] for
+whole-body control of mobile manipulators. We also provide
+pre-trained policies for legged locomotion [39] to facilitate
+solving navigation tasks using base velocity commands.
+
+40%MORE
+French's
+YELLOL3DconnexionF1
+F4
+F6
+F7
+F8
+SYGR
+Lock
+Bresk
+2
+3
+6
+7
+8
+Q
+R
+T
+Home
+Pgup
+Cops Lock
+G
+H
+Enter
+Doier
+Booe
+Z
+X
+tsift
+B
+tshint
+Pon
+Ente
+Alt
+Alt
+Ctrt11GPU
+IK
+OSC
+NN PolicyCPU
+OCS2
+RMPF1owated
+Fluid
+ClotlTeleoperationFig. 6: Demonstration of the designed tasks using hand-crafted state machines and task-space controllers. Leveraging recent advances
+in physics engines, we support high-fidelity simulation of rigid and deformable objects. We include environments that allow switching
+between robots, objects, observations, and action spaces through configuration files (Task videos).
+d) Rigid-body Environments: For rigid-body environ-
+ments, it is vital to have accurate contact physics, fast
+collision checking, and articulate joints simulation. While
+some of these tasks exist in prior works [6], [10], [28],
+[39], we enhance them with our framework’s interfaces and
+provide more variability using DR tools. We also extend ma-
+nipulation tasks for fixed-arm robots to mobile manipulators.
+For brevity, we list the environments are as follows:
+1) Reach - Track desired pose of the end-effector.
+2) Lift - Take an object to a desired position.
+3) Beat the Buzz - Displace a key around a pole
+without touching the pole.
+4) Nut-Bolt - Tighten a nut on a given bolt.
+5) Cabinet - Open or close a cabinet (articulated object).
+6) Pyramid Stack - Stack blocks into pyramids.
+7) Hockey [10] - Shoot a puck into the net using a stick.
+8) Peg In Hole - Insert blocks into their holes.
+9) Jenga [10] - Remove and stack blocks into a tower.
+10) In-Hand Repose - Using dexterous robotic hands.
+11) Velocity Locomotion - Track a desired velocity
+command via a legged robot on various terrains.
+e) Deformable-body Environments:
+Deformable ob-
+jects have a high dimensional state and complex dynamics
+which are difficult to capture succinctly for robot learning.
+With ORBIT, we provide seventeen deformable objects assets
+(such as toys and garments) with valid physics configurations
+and methods to generate new assets (such as rectangular
+cloth) procedurally. A concise list of included environments
+are as follows:
+1) Cloth Lifting - Lift a cloth to a target position.
+2) Cloth Folding - Fold a cloth into a desired state.
+3) Cloth Spreading - Spread a cloth on a table.
+4) Cloth Dropping - Drop a cloth into a container.
+5) Flag Hoisting - Hoist a flag standing on a table.
+6) Soft Lifting - Lift a soft object to a target position.
+7) Soft Placing - Place a soft object on a shelf.
+8) Soft Stacking - Stack soft objects on each other.
+9) Soft Dropping - Drop soft objects into a container.
+10) Tower of Hanoi - Stack toruses around a pole.
+11) Rope Reshaping - Reshape a rope on a table.
+12) Fluid Pouring - Pour fluid into another container.
+13) Fluid Transport - Move a filled container without
+causing any spillages.
+It should be noted that the environments (1), (2), and
+(3) carry the same World definition. They only differ in
+their task logic module, i.e. the associated reward associ-
+ated, which is defined through configuration managers. This
+modularity allows code reusage and makes it easier to define
+a large set of tasks within the same World.
+V. EXEMPLAR WORKFLOWS WITH ORBIT
+ORBIT is a unified simulation infrastructure that provides
+both pre-built environments and easy-to-use interfaces that
+enables extendability and customization. Owing to high-
+quality physics, sensor simulation, and rendering, ORBIT us
+useful for multiple robotics challenges in both perception
+and decision-making. We outline a subset of such use cases
+through exemplar workflows.
+A. GPU-based Reinforcement Learning
+We provide wrappers to different RL frameworks (rl-
+games [40], RSL-rl [39], and stable-baselines-3 [41]). This
+allows users to test their environments on a larger set of RL
+algorithms and facilitate algorithmic developments in RL.
+In Fig. 7, we show the training of Franka-Reach with
+PPO [42] with different frameworks. Although we ensure
+same parameters settings for PPO in the frameworks, we
+notice a difference in their learning performance and training
+time. Since RSL-rl and rl-games are optimized for GPU, we
+observe a training speed of 50,000-75,000 frames per second
+(FPS) with 2048 environments on an NVIDIA RTX3090.
+With stable-baselines3, we receive 6,000-18,0000 FPS.
+We also demonstrate training results for different action
+spaces in the Franka-Cabinet-Opening task, and var-
+ious network architectures and domain randomizations (DR)
+in the ShadowHand-Reposing task. In our testing, we
+observed that simulation throughput for these environments
+are at par with the ones in IsaacGymEnvs [17].
+B. Teleoperation and Imitation Learning
+Many manipulation tasks are computationally expensive
+or beyond the reach of current RL algorithms. In these
+
+0.5
+1.0
+1.5
+Steps
+×107
+7
+8
+9
+10
+Average Return
+PPO on Franka-Reach
+Stable Baselines3
+RL Games
+RSL RL
+0.5
+1.0
+1.5
+2.0
+Steps
+×107
+20
+40
+60
+80
+100
+120
+Average Return
+RSLRL PPO on Franka-Cabinet-Opening
+Joint, position
+Joint, velocity
+0.5
+1.0
+1.5
+2.0
+Steps
+×107
+0
+10
+20
+30
+40
+Consecutive Successes
+RLGames PPO on ShadowHand-Repose
+Full-State Feed Forward (FF)
+Asymmetric actor-critic (AC) FF
+Asymmetric AC-FF with DR
+Asymmetric AC-LSTM
+Asymmetric AC-LSTM with DR
+Fig. 7: Franka-Reach is trained with joint position action space using PPO from Stable Baseline3, RL Games, and RSL RL.
+Franka-Cabinet-Opening is trained with PPO using different controllers. ShadowHand-Repose for in-hand manipulation of
+a cube is trained using variants of PPO with different randomizations, observations, and network types. We evaluate over five seeds and
+plot the mean and one standard deviation of the average reward.
+Fig. 8: Interactive grasp and motion planning demonstration using ORBIT. The World comprises of objects for table-top manipulation.
+The user can select an object from the GUI to grasp. This triggers an image-based grasp generator and allows previewing of the generated
+grasps and the robot motion sequence. The user can then choose the grasp and execute the motion on the robot.
+TABLE II: Evaluation of policies obtained from behavior cloning
+on Franka-Block-Lift environment in the same setting (No
+Change), changing initial states (I), goal states (G), and changing
+both initial and goal states (Both). We report the the success rate
+and trajectory lengths obtained over 100 trials.
+Algorithm
+Average Traj. Len
+Succ. Rate
+Eval. Setup
+BC
+234
+1.00
+No Change
+307
+0.89
+G
+321
+0.47
+I
+324
+0.43
+Both
+BC-RNN
+249
+1.00
+No Change
+251
+1.00
+G
+286
+0.88
+I
+293
+0.87
+Both
+scenarios, boostrapping from user demonstrations provides a
+viable path to skill learning. ORBIT provides a data collection
+interface that is useful for interacting with the provided
+environments using I/O devices and collect data similar to
+roboturk [43]. We also provide an interface robomimic [21]
+for training imitation learning models.
+As
+an
+example,
+we
+show
+LfD
+for
+the
+Franka-Block-Lift
+task.
+For
+each
+of
+the
+four
+settings of initial and desired object positions (fixed or
+random start and desired positions), we collect 2000
+trajectories. Using these demonstrations, we train policies
+using Behavior Cloning (BC) and BC with an RNN policy
+(BC-RNN). We show the performance at test time on 100
+trials in Table II.
+C. Motion planning
+Motion planning is one of the well-studied domains
+in robotics. The traditional Sense-Model-Plan-Act (SMPA)
+methodology decomposes the complex problem of reasoning
+and control into possible sub-components. ORBIT supports
+doing this both procedurally and interactively via the GUI.
+a) Hand-crafted policies: We create a state machine
+for a given task to perform sequential planning as a separate
+node in the agent. It provides the goal states for reaching a
+target object, closing the gripper, interacting with the object,
+and maneuvering to the next target position. We demonstrate
+this paradigm for several tasks in Fig. 6. These hand-crafted
+policies can also be utilized for collecting expert demonstra-
+tions for challenging tasks such as cloth manipulation.
+b) Interactive motion planning: We define a system of
+nodes for grasp generation, teleoperation, task-space control,
+and motion previewing (shown in Fig. 8). Through the GUI,
+the user can select an object to grasp and view the possible
+grasp poses and the robot motion sequences generated using
+the RMP controller . After confirming the grasp pose, the
+robot executes the motion and lifts the object. Following this,
+the user obtains teleoperation control of the robot.
+D. Deployment on real robot
+Deploying an agent on a real robot faces various chal-
+lenges, such as dealing with real-time control and safety con-
+straints. Different data transport layers, such as ROSTCP [19]
+or ZeroMQ (ZMQ) [44], exist for connecting a robotic stack
+
+FRANKA
+THORAFSFRANKA
+THOR ATSTHOR ATSTHOR ATSa.1
+a.2
+b.1
+b.2
+Fig. 9: Using simulator as a digital twin to compute and apply commands on the simulated and real robot via ZMQ connection. a) Franka
+Panda arm with Allegro hand lifting two objects at once (video). b) Franka Panda performing object avoidance using RMP (video).
+1
+2
+3
+Fig. 10: Deployment of an RL policy on ANYmal-D robot using
+ROS connection (video). The policy is trained in simulation and
+runs at 50 Hz while the actuator net functions at 200 Hz.
+to a real platform. We showcase how these mechanisms can
+be used with ORBIT to run policies on a real robot.
+a) Using
+ZMQ:
+To
+maintain
+a
+light-weight
+and
+effiecient communication between, we use ZMQ to send joint
+commands from ORBIT to a computer running the real-time
+kernel for Franka Emika robot. To abide by the real-time
+safety constraints, we use a quintic interpolator to upsample
+the 60 Hz joint commands from the simulator to 1000 Hz
+for execution on the robot (shown in Fig. 9).
+We run experiments on two configurations of the Franka
+robot: one with the Franka Emika hand and the other with
+an Allegro hand. For each configuration, we showcase three
+tasks: 1) teleoperation using a Spacemouse device, 2) de-
+ployment of a state machine, and 3) waypoint tracking with
+obstacle avoidance. The modular nature of the agent makes
+it easy to switch between different control architectures for
+each task while using the same interface for the real robot.
+b) Using ROS: A variety of existing robots come with
+their ROS software stack. In this demonstration, we focus on
+how policies trained using ORBIT can be exported and de-
+ployed on a robotic platform, particularly for the quadrupedal
+robot from ANYbotics, ANYmal-D.
+We train a locomotion policy entirely in simulation using
+an actuator network [36] for the legged base. To make the
+policy robust, we randomize the base mass (22 ± 5 kg) and
+add simulated random pushes. We use the contact reporter to
+obtain the contact forces and use them in reward design. The
+learned policy is deployed on the robot using the ANYmal
+ROS stack, (Fig. 10). This sim-to-real transfer indicates the
+viability of the simulated contact dynamics and its suitability
+for contact-rich tasks in ORBIT.
+VI. DISCUSSION
+In this paper, we proposed ORBIT: a framework to sim-
+plify environment design, enable easier task specifications
+and lower the barrier to entry into robotics and robot learn-
+ing. ORBIT builds on state-of-the-art physics and render-
+ing engines, and provides interfaces to easily design novel
+realistic environments comprising various robotic platforms
+interacting with rigid and deformable objects, physics-based
+sensor simulation and sensor noise models, and different
+actuator models. We readily support a broad set of robotic
+platforms, ranging from fixed-arm to legged mobile manip-
+ulators, CPU and GPU-based motion generators, and object
+datasets (such as YCB and Partnet-Mobility).
+The breadth of environments possible, as demonstrated
+in part in Sec. IV, makes ORBIT useful for broad set of
+research questions in robotics. Keeping modularity at its
+core, we demonstrated the framework’s extensibility to dif-
+ferent paradigms, including reinforcement learning, imitation
+learning, and motion planning. We also showcased the ability
+to interface the framework to the Franka Emika Panda robot
+via ZMQ-based message-passing and sim-to-real deployment
+of RL policies for quadrupedal locomotion.
+By open-sourcing this framework1, we aim to reduce the
+overhead for developing new applications and provide a
+unified platform for future robot learning research. While
+we continue improving and adding more features to the
+framework, we hope that researchers contribute to making
+it a one-stop solution for robotics research.
+VII. FUTURE WORK
+ORBIT can notably simulate physics at up to 125,000
+samples per second; however, camera rendering is currently
+bottlenecked to a total of 270 frames per second for ten
+cameras rendering 640×480 RGB images on an RTX 3090.
+While this number is comparable to other frameworks, we
+are actively improving it further by leveraging GPU-based
+acceleration for training for visuomotor policies.
+1NVIDIA Isaac Sim is free with an individual license. ORBIT will be
+open-sourced, and available at https://isaac-orbit.github.io.
+
+CHEEZLIT
+THORLABSCHEEZIT
+THORLABS
+.QHEEZIT
+THORLAIS
+THORAISCHEEZIT
+THORLATS
+THORLATSCHEEZ-T
+ORIGiNal
+THORLATS
+THORATSTHORLAIS
+THORATISCHEEZ-IT
+ORiGiNal
+THORLABSD
+OR
+THORLABSAdditionally, though our experiments showcase the fidelity
+of rigid-contact modeling, the accuracy of contacts in de-
+formable objects simulation is still unexplored. It is essential
+to note that until now, robot manipulation research in this
+domain has not relied on sim-to-real since existing solvers
+are typically fragile or slow. Using FEM-based solvers and
+physically-based rendering, we believe our framework will
+help answer these open questions in the future.
+ACKNOWLEDGMENT
+We thank Farbod Farshidian for helping with OCS2, Umid
+Targuliyev for assisting with imitation learning experiments,
+as well as Ossama Samir Ahmed, Lukasz Wawrzyniak, Avi
+Rudich, Bryan Peele, Nathan Ratliff, Milad Rakhsha, Vik-
+tor Makoviychuk, Jean-Francois Lafleche, Yashraj Narang,
+Miles Macklin, Liila Torabi, Philipp Reist, Adam Mora-
+vansky, and other members of the NVIDIA PhysX and
+Omniverse teams for their assistance with the simulator.
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+Astronomy & Astrophysics manuscript no. main
+©ESO 2023
+January 6, 2023
+Letter to the Editor
+Asteroids’ reflectance from Gaia DR3:
+Artificial reddening at near-UV wavelengths
+F. Tinaut-Ruano1, 2, E. Tatsumi1, 2, 3, P. Tanga4, J. de León1, 2, M. Delbo4, F. De Angeli5, D. Morate1, 2, J. Licandro1, 2,
+and L. Galluccio4
+1 Instituto de Astrofísica de Canarias (IAC), C/ Vía Láctea, s/n, E-38205, La Laguna, Spain
+e-mail: [email protected]
+2 Department of Astrophysics, University of La Laguna, Tenerife, Spain
+3 Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-0033 Tokyo, Japan
+4 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304
+Nice Cedex 4, France
+5 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
+Received 04/10/2022; accepted 02/01/2023
+ABSTRACT
+Context. Observational and instrumental difficulties observing small bodies below 0.5 µm make this wavelength range poorly studied
+compared with the visible and near-infrared. Furthermore, the suitability of many commonly used solar analogues, essential in the
+computation of asteroid reflectances, is usually assessed only in visible wavelengths, while some of these objects show spectra that
+are quite different from the spectrum of the Sun at wavelengths below 0.55 µm. Stars HD 28099 (Hyades 64) and HD 186427 (16
+Cyg B) are two well-studied solar analogues that instead present spectra that are also very similar to the spectrum of the Sun in the
+wavelength region between 0.36 and 0.55 µm.
+Aims. We aim to assess the suitability in the near-ultraviolet (NUV) region of the solar analogues selected by the team responsible for
+the asteroid reflectance included in Gaia Data Release 3 (DR3) and to suggest a correction (in the form of multiplicative factors) to
+be applied to the Gaia DR3 asteroid reflectance spectra to account for the differences with respect to the solar analogue Hyades 64.
+Methods. To compute the multiplicative factors, we calculated the ratio between the solar analogues used by Gaia DR3 and Hyades
+64, and then we averaged and binned this ratio in the same way as the asteroid spectra in Gaia DR3. We also compared both the
+original and corrected Gaia asteroid spectra with observational data from the Eight Color Asteroid Survey (ECAS), one UV spectrum
+obtained with the Hubble Space Telescope (HST) and a set of blue-visible spectra obtained with the 3.6m Telescopio Nazionale
+Galileo (TNG). By means of this comparison, we quantified the goodness of the obtained correction.
+Results. We find that the solar analogues selected for Gaia DR3 to compute the reflectance spectra of the asteroids of this data release
+have a systematically redder spectral slope at wavelengths shorter than 0.55 µm than Hyades 64. We find that no correction is needed
+in the red photometer (RP, between 0.7 and 1 µm), but a correction should be applied at wavelengths below 0.55 µm, that is in the
+blue photometer (BP). After applying the correction, we find a better agreement between Gaia DR3 spectra, ECAS, HST, and our set
+of ground-based observations with the TNG.
+Conclusions. Correcting the near-UV part of the asteroid reflectance spectra is very important for proper comparisons with laboratory
+spectra (minerals, meteorite samples, etc.) or to analyse quantitatively the UV absorption (which is particularly important to study
+hydration in primitive asteroids). The spectral behaviour at wavelengths below 0.5 µm of the selected solar analogues should be fully
+studied and taken into account for Gaia DR4.
+Key words. Gaia – asteroids – Solar analogues – UV – spectra
+1. Introduction
+Asteroid reflectance spectra and/or spectrophotometry pro-
+vide(s) information on their surfaces’ composition and the pro-
+cesses that modify their properties such as space weathering
+(Reddy et al. 2015). Historically, the use of photoelectric detec-
+tors (or photometers), which are more sensitive at bluer wave-
+lengths (e.g. < 0.5 µm), and the development of the standard
+UBV photometric system (Johnson & Morgan 1951) led to the
+appearance of the first asteroid taxonomies in the 1970s (Zell-
+ner 1973; Chapman et al. 1975), which contained information
+at blue-visible wavelengths or what we call near-UV (NUV).
+The introduction of the charge-coupled-devices (CCDs) in as-
+tronomy in the 1990s and later on contributed to the ’loss’ of
+NUV information, as CCDs were much less sensitive at those
+wavelengths. Therefore, the large majority of the modern spec-
+troscopic and spectrophotometric surveys cover the wavelength
+range from ∼0.5 µm up to 2.5 µm. Nevertheless, there are some
+exceptions. One of the first large surveys with information in the
+NUV is the Eight Asteroid Survey (ECAS, Zellner et al. 1985).
+In this survey, we can find the photometry in eight broad-band
+filters between 0.34 to 1.04 µm for 589 minor planets, includ-
+ing two filters below 0.45 µm. These observations were used to
+develop a new taxonomy (see Tholen 1984). Other recent cat-
+alogues, such as the Sloan Digital Sky Survey (SDSS) Moving
+Objects catalogue (Ivezi´c et al. 2002), the Moving Objects Ob-
+served from Javalambre (MOOJa) catalogue from the J-PLUS
+survey (Morate et al. 2021), or the Solar System objects obser-
+vations from the SkyMapper Southern Survey (Sergeyev et al.
+Article number, page 1 of 13
+arXiv:2301.02157v1  [astro-ph.EP]  5 Jan 2023
+
+A&A proofs: manuscript no. main
+2022), also include photometry in five, 12, and six filters be-
+tween 0.3 and 1.1 µm with 104,449, 3122, and 205,515 objects
+observed, respectively. The new Gaia data release 3 (DR3 here-
+after) catalogue, which was released in June 2022, offers 60,518
+objects binned in 16 wavelengths between 0.352 and 1.056 µm
+to mean reflectance spectra.
+Even though some laboratory measurements suggest the po-
+tential of the NUV absorption as a diagnostic region of hydrated
+and ferric material (Gaffey & McCord 1979; Feierberg 1981;
+Feierberg et al. 1985; Hiroi et al. 1996; Cloutis et al. 2011a,b;
+Hendrix et al. 2016; Hiroi et al. 2021), a quantitative distribu-
+tion of the NUV absorption among asteroids has not been dis-
+cussed before (Tatsumi et al. 2022). The small sensitivity of
+CCDs and the lower Sun’s emission in NUV wavelengths make
+observations difficult. Moreover, the Rayleigh scattering by the
+atmosphere is stronger on shorter wavelengths, decreasing the
+signal-to-noise ratio (S/N) for the NUV region observed from
+the ground. To compute the reflectance spectra, we needed to di-
+vide wavelength by wavelength of the measured spectra by the
+spectra of the Sun. As it is unpractical to observe the Sun with
+the same instrument used to observe asteroids, we used solar
+analogues (SAs), that is stars selected by their known similar
+spectra to that of the Sun. As the large majority of the spec-
+troscopic and spectrophotometric surveys cover the wavelength
+range that goes from the visible to near-infrared (NIR), the most
+commonly used SAs are well characterised at those wavelengths
+but they can behave very differently in the NUV. This flux dif-
+ference at bluer wavelengths can introduce systematic errors in
+the asteroid reflectance spectra. A good example is the work by
+de León et al. (2016), where they searched for the presence of
+F-type asteroids in the Polana collisional family since the parent
+body of the family, asteroid (142) Polana, was classified as an
+F type. The authors obtained reflectance spectra in the NUV of
+the members of the family, finding that the large majority were
+classified as B types. As most of the observers, they used SAs
+that were widely used by the community. Interestingly, after ob-
+taining the asteroid reflectances again using only Hyades 64 as
+the SA, Tatsumi et al. (2022) found that the large majority of
+the observed members of the Polana family were indeed F types
+and not B types. This evidences the importance of using ade-
+quate SAs when observing in the NUV, and it has been the main
+motivation for this work.
+In this Letter, we present a comparison between the SAs se-
+lected to compute the reflectance spectra in the frame of the data
+processing of Gaia DR3 (Gaia Collaboration et al. 2022) and
+Hyades 64. We analyse the results from this comparison and
+propose a multiplicative correction that can be applied to the
+archived asteroids’ reflectance spectra. We finally tested it by
+comparing corrected Gaia reflectance spectra with ground-based
+observations that have also been corrected against the same SA
+(ECAS survey, TNG spectra) and with one observation with the
+Hubble Space Telescope (HST).
+2. Sample
+2.1. Solar analogues in Gaia DR3
+The Gaia DR3 catalogue (Gaia Collaboration et al. 2022) gives
+access to internally and externally calibrated mean spectra for a
+large subset of sources. Internally calibrated spectra refer to an
+internal reference system that is homogeneous across all differ-
+ent instrumental configurations, while externally calibrated spec-
+tra are given in an absolute wavelength and flux scale (see De
+Angeli et al. 2022; Montegriffo et al. 2022, for more details).
+Epoch spectra (spectra derived from a single observation rather
+than averaging many observations of the same source) are not
+included in this release. For this Letter, we relied on internally
+calibrated data when computing the correction for the Gaia re-
+flectances to ensure consistency and to avoid artefacts that could
+appear when dividing two externally calibrated spectra, as they
+are polynomial fits.
+To select the SAs, the Gaia team did a bibliographic search
+and selected a list of stars that are widely used as solar analogues
+for asteroid spectroscopy (Bus & Binzel 2002; Lazzaro et al.
+2004; Soubiran & Triaud 2004; Fornasier et al. 2007; Popescu
+et al. 2014; Perna et al. 2018; Popescu et al. 2019; Lucas et al.
+2019). First of all, we note that the star identified as 16 Cygnus
+B in Gaia Collaboration et al. (2022) is in fact 16 Cygnus A and
+that the parameters in their Table C.1. correspond to those of 16
+Cygnus A. Luckily enough, the spectrum of 16 Cygnus B was
+also available in DR3. Among the referenced works, only Soubi-
+ran & Triaud (2004) carried out a search for SAs by comparing
+their spectra to that of the Sun down to 0.385 µm. The rest sim-
+ply used G2V stars or cited previous works that presented SAs,
+as in Hardorp (1978). In this later work, Hardorp selected SAs
+by comparing their spectra with the spectrum of the Sun using
+wavelengths down to 0.36 µm. He highlighted the variations that
+can exist at NUV wavelengths even between stars of the same
+spectral class.
+2.2. Asteroids in Gaia DR3
+Among the Gaia DR3 products for Solar System objects (SSOs),
+neither the internally nor the externally calibrated spectra are
+available to the community, as is the case for the stars. This is due
+to a specific choice of the Data Processing and Analysis Consor-
+tium (DPAC) caused by the difficulty of calculating those quanti-
+ties owing to the intrinsic variability and proper motion of SSOs.
+Instead, for each SSO and each epoch, the nominal, pre-launch
+dispersion function was used to convert pseudo-wavelengths to
+physical wavelengths. The reflectance spectra were calculated by
+dividing each epoch spectrum by the mean of the SAs selected
+and then averaging over the set of epochs. After that, a set of
+fixed wavelengths every 44 nm in the range between 374 and
+1034 nm was defined, with a set of bins centred at those wave-
+lengths and with a size of 44 nm. For each bin (a total of 16 are
+provided), a σ-clipping filter was applied and a weighted aver-
+age using the inverse of the standard deviation as weight was
+obtained. Finally, the reflectances were normalised to unity us-
+ing the value at 550±25 nm. This final product is the only one
+available in DR3.
+2.3. Hyades 64 & 16 Cyg B
+As mentioned in Sect. 2.1, Hardorp (1978) concluded that
+Hyades 64 and 16 Cyg B are two of the four stars that exhibit
+’almost indistinguishable’ NUV spectra (quoting the author’s
+words) from the spectrum of the Sun. This was confirmed in
+subsequent papers from the same author (Hardorp 1980a,b) and
+from other researchers (Cayrel de Strobel 1996; Porto de Mello
+& da Silva 1997; Farnham et al. 2000; Soubiran & Triaud 2004).
+We used these two stars as a ’reference’ to compute the correc-
+tion factor to be applied to the Gaia DR3 asteroids spectra, as
+they are in the list of SAs selected by Gaia Collaboration et al.
+(2022). The methodology is described in the following section.
+We note that the obtained correction factor using Hyades 64 as
+opposed to 16 Cygnus B differs less than 0.5%. We, therefore,
+Article number, page 2 of 13
+
+F. Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths
+Table 1. Multiplicative correction factors for Gaia asteroid binned spec-
+tra. We include the wavelengths below 0.55 µm.
+Wavelength (µm)
+Correction factor
+0.374
+1.07
+0.418
+1.05
+0.462
+1.02
+0.506
+1.01
+0.550
+1.00
+decided to use Hyades 64, as it was the star that was used for
+both the ECAS survey and our ground-based observations.
+3. Methodology
+3.1. Computing the correction factor: Internally calibrated
+data
+In order to compute a correction applicable to the Gaia DR3
+reflectances, we proceeded as follows: first, using the internally
+calibrated data, we computed the ratio between the Gaia sample
+of SAs, as well as the mean spectrum of these SAs, and Hyades
+64 (Fig. 1). As we can observe in the right panel of Fig. 1, which
+corresponds to the red photometer (RP), the deviation from the
+unity of the ratio between Gaia’s mean SA and Hyades 64 (black
+line) is always below 1%. Therefore, this mean spectrum can
+confidently be used to obtain the reflectance spectra of asteroids
+above 0.55 µm.
+However, the situation in the the blue photometer (BP) is
+quite different. We can see in the left panel of Fig. 1 that the de-
+viation from the unity of the above defined ratio can reach values
+of up to 10%, indicating that the mean spectrum of the SAs used
+in Gaia DR3 differs significantly from Hyades 64 at wavelengths
+below 0.55 µm. The biggest effect when using this mean spec-
+trum to obtain asteroids’ reflectance spectra is the introduction
+of a systematic (and not real) positive slope, in particular in the
+range between 0.4 and 0.55 µm, mimicking a drop in reflectance
+below 0.55 µm. Furthermore, the division by this mean spec-
+trum can also introduce a ’fake’ absorption around 0.38 µm. We
+have quantified this spectral slope in two separate wavelength
+ranges, trying to reproduce the observed behaviour of the ratio:
+one slope between 0.4 to 0.55 µm, which we named S Blue, and
+another one for wavelengths below 0.4 µm, named µm S UV. The
+obtained values for the individual SAs used in Gaia DR3 (blue
+stars), as well as for the mean spectrum (blue cross) are shown
+in Fig. 2. For the mean spectrum of the SAs used in Gaia DR3,
+we found that the introduced slopes are S Blue = -0.38 µm−1 and
+S UV = 0.69 µm−1.
+From this analysis, we conclude that a correction is needed
+in the NUV wavelengths, that is below 0.55 µm. To arrive at
+the multiplicative correction factors, we binned the ratio between
+the mean spectra of SAs selected by the DPAC and Hyades 64,
+using the same wavelengths and bin size as the ones adopted for
+the asteroid reflectance spectra in the Gaia DR3 (see Sect. 2). In
+this way, the users can easily correct the asteroid spectra at NUV
+wavelengths. The obtained values are shown in Table 1.
+3.2. Comparison of corrected reflectances with existing data
+To correct the artificial slopes introduced by the use of the mean
+Gaia SAs, we multiplied the binned asteroid reflectance spec-
+tra below 0.55 µm by the corresponding correction factors. We
+compared the corrected Gaia spectra with spectra or spectropho-
+tometry of the same asteroids obtained using other facilities. As
+a first step, we selected only those Gaia asteroid spectra with
+a S/N > 160, as we detected a systematic decrease in spectral
+slope values at blue wavelengths with decreasing S/N for objects
+with a smaller S/N than 150. We then selected spectrophotomet-
+ric data from the ECAS survey for asteroids that have more than
+one observation, and NUV spectra obtained with the Telesco-
+pio Nazionale Galileo (TNG) and previously published by Tat-
+sumi et al. (2022). The resulting comparison dataset is shown in
+Fig. A.1, where the red lines correspond to the original Gaia re-
+flectances, black lines are the corrected ones, dark blue lines cor-
+respond to ECAS data, and TNG spectra are shown in light blue.
+As can be seen, the corrected reflectances are in better agree-
+ment with the ECAS and TNG data than the original ones. We
+also included the UV spectrum of asteroid (624) Hector down-
+loaded from the ESA archive using the python package astro-
+query.esa.hubble1. It was obtained with STIS at HST (Wong
+et al. 2019). We converted the flux to reflectance using the spec-
+trum of the Sun provided for the STIS instrument2. We note
+that even after the correction, some asteroids show discrepancies
+with the reference data. This is discussed in the next section.
+4. Results and discussion
+We have shown that the artificial slope introduced at blue
+wavelengths in the Gaia DR3 asteroid data due to the selected
+SAs is -0.38 µm−1 in the range between 0.4 to 0.55 µm and 0.69
+µm−1 below 0.4 µm. Following Zellner et al. (1985), the b and v
+filters of the ECAS survey have central effective wavelengths of
+0.437 and 0.550 µm, respectively. According to Tholen (1984),
+the (b-v) colours of the mean F and B taxonomical classes are
+-0.049 and -0.015 magnitudes, respectively. Transforming these
+colours to relative reflectances results in 1.046 and 1.014, which
+gives slopes of -0.407 and -0.124 µm−1 between 0.437 and 0.55
+µm. Therefore, the difference between these computed slopes
+for F and B types (-0.283 µm−1) is smaller than the artificial
+slope introduced by the use of the mean SA of Gaia, implying
+that unless we apply the correction proposed in this Letter,
+asteroids can be easily misclassified as B types when actually
+being F types (see the described example in the Introduction for
+the case of members of the Polana family).
+To test and quantify the goodness of our proposed correction,
+we computed the spectral slope between 0.437 and 0.55 µm for
+the ECAS comparison dataset, and between 0.418 and 0.55 for
+Gaia original and corrected spectra. In Fig. 3 we plotted the
+difference between those slopes. After applying our correction
+factor, we could see that the large majority (148 out of 152) of
+the asteroids have more similar slopes to those of ECAS.
+Nevertheless, our correction has limitations. First, we were
+testing its goodness over space-based observations using
+ground-based observations. For wavelengths down to 0.3 µm,
+ground-based observations present some difficulties, mainly due
+to the atmospheric absorption and the lower sensitivity of the
+detectors. Furthermore, Gaia observations at those wavelengths
+also have other artifacts that we do not fully understand, such
+as the detected strong decrease in the spectral slope below S/N
+150. Another point to consider when comparing asteroid spectra
+observed in different epochs is the effect of the different viewing
+geometries. This difference in the viewing geometry, and thus, in
+1 https://astroquery.readthedocs.io/en/latest/esa/
+hubble/hubble.html
+2 https://archive.stsci.edu/hlsps/reference-atlases/
+cdbs/current_calspec/sun_reference_stis_002.fits
+Article number, page 3 of 13
+
+A&A proofs: manuscript no. main
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+Wavelength [ m]
+0.95
+1.00
+1.05
+1.10
+1.15
+Counts relative to Hyades 64
+BP
+0.7
+0.8
+0.9
+1.0
+Wavelength [ m]
+RP
+HD060234
+HD123758
+16 Cyg B(A)1
+HD6400
+HD220022
+HD220764
+HD016640
+HD292561
+HD100044
+HD155415
+SA110-361
+HD182081
+HD144585
+HD146233
+HD138159
+HD139287
+HD020926
+HD154424
+HD202282
+16 Cyg B
+mean
+Fig. 1. Ratio between the internally calibrated spectra of each of the Gaia SAs and Hyades 64 in the blue photometer (BP, left panel) and the red
+photometer (RP, right panel). We also plotted the ratio of the mean Gaia SA and Hyades 64 (black solid line) and the binned version of this ratio
+at the wavelengths provided for SSO in Gaia DR3 (black dots).
+1 We note that the star identified as 16 Cygnus B in Gaia Collaboration et al. (2022) is in fact 16 Cyg A (see the main text for more details).
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+SUV [ m
+1]
+0.8
+0.6
+0.4
+0.2
+0.0
+SBlue [ m
+1]
+Gaia SAs
+Gaia mean
+Fig. 2. Slopes introduced by each of the SAs in the Gaia sample (blue
+stars) and their mean (blue cross), compared to Hyades 64. We note that
+S Blue was computed in the 0.4–0.55 µm range, while S UV was computed
+using wavelengths below 0.4 µm.
+the phase angle, causes a change in the spectral slope known as
+phase reddening or phase coloring Alvarez-Candal et al. (2022).
+This effect has not been well studied at blue wavelengths. Still,
+even in the event that we were able to correct it, Gaia’s spectra
+are, on average, over different epochs and the information on
+the phase angle values is not provided.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ECAS slope - original Gaia slope (1/ )
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+ECAS slope - corrected Gaia slope (1/ )
+Fig. 3. Difference between the blue slope for ECAS and for Gaia origi-
+nal data (x-axis) and corrected data (y-axis) in the comparison sample.
+5. Conclusions
+We have found that the use of the SAs selected to compute the
+reflectance spectra of the asteroids in Gaia DR3 introduces an
+artificial reddening in the spectral slope below 0.5 µm, that is
+an artificial drop in reflectance. By comparing those SAs with
+Hyades 64, one of the best characterised SAs at NUV wave-
+lengths, we obtain multiplicative correction factors for each of
+the reflectance wavelengths below 0.55 µm (a total of four) that
+can be applied to the asteroids’ reflectance spectra in Gaia DR3.
+By applying this correction, we found a better agreement be-
+tween the Gaia spectra and other data sources such as ECAS.
+Article number, page 4 of 13
+
+F. Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths
+The behaviour of the SAs in the red wavelengths is in agree-
+ment with Hyades 64 within 1%. This was somehow expected,
+as the majority of the SAs used by the Gaia team were previ-
+ously tested and widely used by the community to obtain visible
+reflectance spectra of asteroids, typically beyond 0.45–0.5 µm.
+Correcting the NUV part of the asteroid reflectance spectra is
+fundamental to study the presence of the UV absorption, which
+has been associated with hydration in primitive asteroids, or to
+discriminate between B and F types, which are two taxonom-
+ical classes that have proven to have very distinct polarimetric
+properties. The NUV region has not yet been fully exploited for
+asteroids and, in this way, Gaia spectra constitute a major step
+forward in our understanding of these wavelengths.
+Acknowledgements. FTR, JdL, ET, DM, and JL acknowledge support from the
+Agencia Estatal de Investigación del Ministerio de Ciencia e Innovación (AEI-
+MCINN) under the grant ’Hydrated Minerals and Organic Compounds in Prim-
+itive Asteroids’ with reference PID2020-120464GB-100.
+FTR also acknowledges the support from the COST Action and the ESA
+Archival Visitor Programme.
+DM acknowledges support from the ESA P3NEOI programme (AO/1-
+9591/18/D/MRP).
+This work has made use of data from the European Space Agency (ESA)
+mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia
+Data Processing and Analysis Consortium (DPAC, https://www.cosmos.
+esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been pro-
+vided by national institutions, in particular, the institutions participating in the
+Gaia Multilateral Agreement.
+The work of MD is supported by the CNES and by the project Origins of the
+French National Research Agency (ANR-18-CE31-0014).
+F. De Angeli is supported by the United Kingdom Space Agency (UKSA)
+through the grants ST/X00158X/1 and ST/W002469/1.
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+Article number, page 5 of 13
+
+A&A proofs: manuscript no. main
+Appendix A: Comparison figures
+In this appendix, we are showing the spectra of asteroids that
+have at least two observations from the ground from ECAS (dark
+blue) or from TNG (light blue) and also have available spectra in
+Gaia DR3. We plotted the original (red) and the corrected (black)
+version of the Gaia spectra together. For asteroid 624 (Hector),
+we also added an observation from HST (further information can
+be found in the main text).
+Article number, page 6 of 13
+
+F. Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths
+1.0
+0.8
+1
+ECAS
+original Gaia
+corrected Gaia
+2
+3
+1.0
+0.8
+4
+6
+7
+1.0
+0.8
+8
+9
+10
+1.0
+0.8
+Relative reflectance
+12
+14
+16
+1.0
+0.8
+18
+21
+23
+1.0
+0.8
+24
+27
+29
+1.0
+0.8
+37
+38
+39
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+1.0
+0.8
+42
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+44
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+45
+Fig. A.1. Comparison between ground-based observations from the Eight Asteroid Survey (ECAS, dark blue line), TNG observations (light blue
+line) original Gaia data (red line), and corrected data (black line). We also included a UV spectrum of asteroid (624) downloaded from ESA
+archive and obtained with the instrument STIS, on board the Hubble Space Telescope (HST).
+Article number, page 7 of 13
+
+A&A proofs: manuscript no. main
+1.0
+0.8
+46
+47
+TNG
+ECAS
+original Gaia
+corrected Gaia
+51
+1.0
+0.8
+62
+64
+65
+1.0
+0.8
+71
+80
+82
+1.0
+0.8
+Relative reflectance
+83
+85
+86
+1.0
+0.8
+87
+88
+90
+1.0
+0.8
+93
+94
+95
+1.0
+0.8
+97
+101
+103
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+1.0
+0.8
+105
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+106
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+107
+Article number, page 8 of 13
+
+F. Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths
+1.0
+0.8
+109
+111
+114
+1.0
+0.8
+117
+124
+132
+1.0
+0.8
+134
+135
+137
+1.0
+0.8
+Relative reflectance
+142
+153
+158
+1.0
+0.8
+168
+171
+179
+1.0
+0.8
+187
+190
+198
+1.0
+0.8
+211
+213
+216
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+1.0
+0.8
+221
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+225
+TNG
+ECAS
+original Gaia
+corrected Gaia
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+229
+Article number, page 9 of 13
+
+A&A proofs: manuscript no. main
+1.0
+0.8
+233
+236
+261
+TNG
+ECAS
+original Gaia
+corrected Gaia
+1.0
+0.8
+268
+275
+279
+1.0
+0.8
+287
+306
+308
+1.0
+0.8
+Relative reflectance
+322
+323
+326
+1.0
+0.8
+334
+339
+349
+1.0
+0.8
+354
+361
+368
+1.0
+0.8
+369
+374
+379
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+1.0
+0.8
+380
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+383
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+389
+Article number, page 10 of 13
+
+F. Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths
+1.0
+0.8
+394
+406
+407
+1.0
+0.8
+419
+TNG
+ECAS
+original Gaia
+corrected Gaia
+420
+433
+1.0
+0.8
+434
+442
+443
+1.0
+0.8
+Relative reflectance
+470
+471
+480
+1.0
+0.8
+483
+509
+512
+1.0
+0.8
+522
+529
+532
+1.0
+0.8
+558
+566
+570
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+1.0
+0.8
+579
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+602
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+616
+Article number, page 11 of 13
+
+A&A proofs: manuscript no. main
+1.0
+0.8
+624
+HST
+TNG
+ECAS
+original Gaia
+corrected Gaia
+635
+639
+1.0
+0.8
+654
+664
+686
+1.0
+0.8
+699
+702
+704
+1.0
+0.8
+Relative reflectance
+712
+714
+721
+1.0
+0.8
+733
+739
+748
+1.0
+0.8
+773
+778
+785
+1.0
+0.8
+786
+849
+863
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+1.0
+0.8
+897
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+914
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+931
+Article number, page 12 of 13
+
+F. Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths
+1.0
+0.8
+980
+ECAS
+original Gaia
+corrected Gaia
+1001
+1021
+1.0
+0.8
+1105
+1144
+1172
+1.0
+0.8
+Relative reflectance
+1180
+1266
+1268
+1.0
+0.8
+1275
+1509
+1604
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+1.0
+0.8
+1606
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+1650
+0.35
+0.40
+0.45
+0.50
+0.55
+Wavelength [ m]
+1754
+Article number, page 13 of 13
+

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+arXiv:2301.13257v1  [math.RA]  30 Jan 2023
+CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES
+MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP
+Abstract. The Fiedler matrices are a large class of companion matrices that include the
+well-known Frobenius companion matrix. The Fiedler matrices are part of a larger class
+of companion matrices that can be characterized with a Hessenberg form. In this paper,
+we demonstrate that the Hessenberg form of the Fiedler companion matrices provides a
+straight-forward way to compare the condition numbers of these matrices. We also show
+that there are other companion matrices which can provide a much smaller condition num-
+ber than any Fiedler companion matrix. We finish by exploring the condition number of a
+class of matrices obtained from perturbing a Frobenius companion matrix while preserving
+the characteristic polynomial.
+1. Introduction
+The Frobenius companion matrix is a template that provides a matrix with a prescribed
+characteristic polynomial. More recently, it was discovered that the Frobenious companion
+matrix belongs to a larger class of Fiedler companion matrices [5], which in turn is a
+subset of the intercyclic companion matrices [4]. Other recent templates include nonsparse
+companion matrices [2] and generalized companion matrices [6].
+The Frobenius companion matrix is employed in algorithms that use matrix methods to
+determine roots of polynomials, but this matrix is not always well-conditioned [3]. Recent
+work [3] has explored under what circumstances other Fielder companion matrices can have
+a better condition number than the Frobenius matrix, with respect to the Frobenius norm.
+After covering background details in Section 2, we use a Hessenberg characterization of the
+Fiedler companion matrices in Section 3 to provide a concise argument for the condition
+number of a Fielder companion matrix. The characterization allows us to avoid dealing
+with the particular permutation in Fiedler’s construction of companion matrices [5], as well
+as associated concepts around consecutions and inversions developed in [3]. In Section 4,
+we provide some examples of non-Fiedler companion matrices that demonstrate that there
+are intercyclic companion matrices that have a smaller condition number than any Fielder
+companion matrix for some specific polynomials. In Section 5, we provide a method for con-
+structing a generalized companion matrix that, in some cases, can improve on the condition
+number of any Fiedler companion matrix.
+Date: February 1, 2023.
+2010 Mathematics Subject Classification. 15A12, 15B99.
+Key words and phrases. companion matrix, Fiedler companion matrix, condition number, generalized
+companion matrix.
+Research of Vander Meulen was supported in part by NSERC Discovery Grant 2022-05137.
+Research of Van Tuyl was supported in part by NSERC Discovery Grant 2019-05412.
+Research of Voskamp was supported in part by NSERC USRA 504279.
+1
+
+2
+MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP
+
+
+0
+1
+0
+0
+0
+0
+1
+0
+0
+0
+0
+1
+−c0
+−c1
+−c2
+−c3
+
+ ,
+
+
+0
+1
+0
+0
+0
+−c3
+1
+0
+0
+−c2
+0
+1
+−c0
+−c1
+0
+0
+
+ ,
+
+
+0
+1
+0
+0
+−c2
+−c3
+1
+0
+0
+0
+0
+1
+−c0
+−c1
+0
+0
+
+ .
+Figure 1. Some 4 × 4 unit sparse companion matrices.
+
+
+0
+1
+0
+0
+−c2
+0
+1
+0
+−c1 + c3c2
+0
+−c3
+1
+−c0
+0
+0
+0
+
+ ,
+
+
+−c3
+1
+0
+0
+0
+0
+1
+0
+−c1 + c3c2
+−c2
+0
+1
+−c0
+0
+0
+0
+
+ ,
+
+
+−c3
+1
+0
+0
+−c2 + a
+0
+1
+0
+−c1 + ac3
+−a
+0
+1
+−c0
+0
+0
+0
+
+ .
+Figure 2. Some 4 × 4 companion matrices.
+2. Technical definitions and background
+In this section we recall the relevant background on companion matrices and condition
+numbers that will be required throughout the paper.
+Let n ≥ 2 be an integer and p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c0. A compan-
+ion matrix to p(x) is an n × n matrix A over R[c0, . . . , cn−1] such that the characteristic
+polynomial of A is p(x). A unit sparse companion matrix to p(x) is a companion matrix A
+that has n − 1 entries equal to one, n variable entries −c0, . . . , −cn−1, and the remaining
+n2 − 2n + 1 entries equal to zero. The unit sparse companion matrix of the form
+
+
+0
+1
+0
+· · ·
+0
+0
+0
+0
+1
+· · ·
+0
+0
+0
+0
+0
+· · ·
+1
+0
+...
+...
+...
+· · ·
+0
+1
+−c0
+−c1
+−c2
+· · ·
+−cn−2
+−cn−1
+
+
+is called the Frobenius companion matrix of p(x). Sparse companion matrices have also
+been called intercyclic companion matrices due to the structure of the digraph associated
+with the matrix (see [7] and [4] for details).
+The matrices in Figure 1 are examples of unit sparse companion matrices to p(x) =
+x4 + c3x3 + c2x2 + c1x + c0. The first matrix in Figure 1 is a Frobenius companion matrix.
+The matrices in Figure 2 are also companion matrices to p(x), but they are not unit sparse
+since not every nonzero variable entry is the negative of a single coefficient of p(x). Note
+that in the last matrix, the value of a can be any real number; when a = 0, then this matrix
+becomes a unit sparse companion matrix.
+Since matrix transposition and permutation similarity does not affect the characteristic
+polynomial, nor the set of nonzero entries in a matrix, we call two companion matrices
+equivalent if one can be obtained from the other via transposition and/or permutation
+similarity.
+The matrices in Figure 3 are equivalent to the 4 × 4 Frobenius companion
+matrix. Note that if A and B are equivalent matrices, then the multiset of entries in any
+
+CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES
+3
+
+
+−c3
+1
+0
+0
+−c2
+0
+1
+0
+−c1
+0
+0
+1
+−c0
+0
+0
+0
+
+ ,
+
+
+−c3
+−c2
+−c1
+−c0
+1
+0
+0
+0
+0
+1
+0
+0
+0
+0
+1
+0
+
+ ,
+
+
+0
+0
+0
+−c0
+1
+0
+0
+−c1
+0
+1
+0
+−c2
+0
+0
+1
+−c3
+
+ .
+Figure 3. Some companion matrices equivalent to the 4×4 Frobenius com-
+panion matrix.
+row of A is exactly the multiset of entries of some row or column of B. No two matrices
+from Figures 1 and 2 are equivalent (assuming a ̸= 0).
+Fielder [5] introduced a class of companion matrices that are constructed as a product
+of certain block diagonal matrices. In particular, let F0 be a diagonal matrix with diagonal
+entries (1, . . . , 1, −c0) and for k = 1, . . . , n − 1, let
+Fk =
+
+
+In−k−1
+O
+O
+O
+Tk
+O
+O
+O
+Ik−1
+
+ with Tk =
+�
+−ck
+1
+1
+0
+�
+.
+Fiedler showed (see [5, Theorem 2.3]) that the product of these n matrices, in any or-
+der, will produce a companion matrix of p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c0.
+Consequently, given any permutation σ = (σ0, σ2, . . . , σn−1) of {0, 1, 2, . . . , n − 1}, we say
+that Fσ = Fσ0Fσ1 · · · Fσn−1 is a Fiedler companion matrix. The Frobenius companion ma-
+trix is a Fiedler companion matrix since the Frobenius companion matrix is equivalent to
+F0F1 · · · Fn−1, as noted in [5].
+In [4] it was demonstrated that every unit sparse companion matrix is equivalent to a
+unit lower Hessenberg matrix, as summarized in Theorem 2.1. Note that, for 0 ≤ k ≤ n−1,
+the k-th subdiagonal of a matrix A = [aij] consists of the entries {ak+1,1, ak+2,2, . . . , an,n−k}.
+The 0-th subdiagonal is usually called the main diagonal of a matrix.
+Theorem 2.1. [4, Corollary 4.3] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be
+a polynomial over R with n ≥ 2. Then A is an n × n unit sparse companion matrix to p(x)
+if and only if A is equivalent to a unit lower Hessenberg matrix
+(1)
+C =
+
+
+0
+Im
+O
+R
+In−m−1
+0T
+
+
+for some (n − m) × (m + 1) matrix R with m(n − 1 − m) zero entries, such that C has
+−cn−1−k on its k-th subdiagonal, for 0 ≤ k ≤ n − 1.
+Note that in (1), the unit lower Hessenberg matrix C always has Cn,1 = −c0 and R1,m+1 =
+−cn−1. Given this Hessenberg characterization of the unit sparse companion matrices, one
+can deduce the corresponding inverse matrix if c0 ̸= 0.
+Lemma 2.2. [7, Section 7] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a
+polynomial over R with n ≥ 2.
+Suppose that C is a unit lower Hessenberg companion
+
+4
+MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP
+matrix to p(x) as in (1). Assuming c0 ̸= 0, if
+C =
+
+
+0
+Im
+O
+u
+H
+In−m−1
+−c0
+yT
+0T
+
+ , for some u, y, H, then C−1 =
+
+
+1
+c0yT
+0T
+− 1
+c0
+Im
+O
+0
+− 1
+c0uyT − H
+In−m−1
+1
+c0 u
+
+ .
+Throughout this paper, we use the Frobenius norm of an n × n matrix A = [ai,j] given
+by
+||A|| =
+��
+i,j
+a2
+i,j.
+Remark 2.3. If A and B are both unit sparse companion matrices to the same polyno-
+mial p(x), then it follows that ||A|| = ||B|| since A and B have exactly the same entries.
+Furthermore, if A = PBP T for some permutation matrix P, then A−1 and B−1 also have
+the same entries, and hence ||A−1|| = ||B−1||.
+The condition number of A, denoted κ(A), is defined to be
+κ(A) = ||A|| · ||A−1||.
+Remark 2.3 implies the following lemma.
+Lemma 2.4. If A and B are equivalent companion matrices, then κ(A) = κ(B).
+3. Condition numbers of Fiedler matrices via the Hessenberg
+characterization
+The condition numbers of Fiedler companion matrices were first calculated by de Ter´an,
+Dopico, and P´erez [3, Theorem 4.1]. In this section we demonstrate how a characterization
+of Fielder companion matrices via unit lower Hessenberg matrices, as given by Eastman,
+et al. [4], provides an efficient way to obtain the condition numbers for Fiedler companion
+matrices.
+Our approach avoids the use of the consecution-inversion structure sequence,
+described in [3, Definition 2.3], which was used in the original computation of these numbers.
+The following theorem gives a characterization of the Fielder companion matrices in
+terms of unit lower Hesenberg matrices.
+Theorem 3.1. [4, Corollary 4.4] If p(x) = xn + cn−1xn−1 + · · · + c1x + c0 is a polynomial
+over R with n ≥ 2, then F is an n × n Fiedler companion matrix to p(x) if and only if F is
+equivalent to a unit lower Hessenberg matrix as in (1) with the additional property that if
+−ck is in position (i, j) then −ck+1 is in position (i − 1, j) or (i, j + 1) for 1 ≤ k ≤ n − 1.
+An alternative way to describe the unit lower Hesenberg matrix in Theorem 3.1 is to say
+that the variable entries of R in (1) form a lattice-path from the bottom-left corner to the
+top-right corner of R. The first two matrices in Figure 1 are examples of Fiedler companion
+matrices since the variable entries of R form a lattice-path. The last matrix in Figure 1 is
+not a Fiedler companion matrix.
+If F is a Fiedler companion matrix, the initial step size of F is the number of coefficients
+other than c0 in the row or column containing both c0 and c1. The first matrix in Figure 1
+has initial step size three and the second matrix in Figure 1 has initial step size one.
+
+CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES
+5
+Remark 3.2. Note that equivalent matrices have the same initial step size since transpo-
+sitions and permutation equivalence does not change the number of coefficients in the row
+or column containing c0 and c1.
+Using Theorem 3.1 and Lemma 2.2, one can describe the nonzero entries of the inverse
+of a Fiedler companion matrix:
+Lemma 3.3. [3, Theorem 3.2] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be
+a polynomial over R with n ≥ 2 and c0 ̸= 0. Let F be a Fiedler companion matrix to p(x)
+with an initial step size t. Then
+(1) F −1 has t + 1 entries equal to − 1
+c0, − c1
+c0, . . . , − ct
+c0,
+(2) F −1 has n − 1 − t entries equal to ct+1, ct+2, . . . , cn−1,
+(3) F −1 has n − 1 entries equal to 1, and
+(4) the remaining entries of F −1 are 0.
+Proof. Since F is a companion matrix to p(x), by Theorem 2.1, the matrix F is equivalent
+to a lower Hessenberg matrix C of the form (1). Since F and C are equivalent, it follows
+that the matrices F −1 and C−1 are equivalent, so it suffices to show that the matrix C−1
+satisfies conditions (1) − (4).
+Since F is a Fielder companion matrix, Theorem 3.1 implies that c1 is either directly
+above c0 in C or directly to the right of c1. If c1 is to right of c0 in C, then all other entries
+in the column containing c0 is zero. Alternatively, if c1 is above c0, all entries to the right
+of c0 in C are zero.
+Lemma 2.2, which gives us the inverse of a unit lower Hessenberg matrix, applies to the
+matrix C. By our above observation, the vector u or the vector y must be the zero vector.
+Without loss of generality, let yT be zero, which means that − 1
+c0uyT − H = −H. If the
+initial step size of A is t, then there will be t nonzero elements in u, and it will have the
+form
+u =
+
+
+0
+...
+0
+−ct
+...
+−c1
+
+
+.
+By Lemma 2.2 the inverse of the matrix C then has the form
+(2)
+C−1 =
+
+
+0T
+0T
+− 1
+c0
+Im
+O
+0
+−H
+In−m−1
+1
+c0u
+
+
+.
+From (2), we can describe the entries of C−1: m + n − m − 1 = n − 1 entries are 1 (coming
+from the submatrices Im and In−m−1); ct+1, . . . , cn−1, which all belong to the submatrix
+−H; the entry − 1
+c0 from the top-right corner; and the entries − c1
+c0 , . . . , − ct
+c0 from the term
+
+6
+MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP
+1
+c0u. Moreover, the rest of the entries of C−1 are zero. We have now shown that C−1, and
+hence F −1, has the desired properties.
+□
+Remark 3.4. Lemma 3.3 mimics [3, Theorem 3.2]. As observed in [7], the initial step size
+of a Fiedler companion matrix is equal to the number of initial consecutions or inversions
+of the permuation associated with the Fielder companion matrix, as defined in [3].
+We can now compute the condition number for any Fiedler companion matrix. This
+result first appeared in [3], but we can avoid the formal analysis of the permutation that
+was used to construct the Fiedler companion matrix, as well as the associated concepts of
+consecution and inversion of a permutation.
+Theorem 3.5. [3, Theorem 4.1] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be
+a polynomial over R with n ≥ 2 and c0 ̸= 0. Let F be a Fiedler companion matrix to p(x)
+with an initial step size t. Then
+κ(F)2 = ||F||2 ·
+�
+(n − 1) + 1 + |c1|2 + · · · + |ct|2
+|c0|2
++ |ct+1|2 + · · · + |cn−1|2
+�
+,
+with
+||F||2 = (n − 1) + |c0|2 + |c1|2 + · · · + |cn−1|2.
+Proof. This result follows from the fact that F is a unit sparse companion matrix (so it
+contains n − 1 entries equal to 1 and the entries −c0, . . . , −cn−1), and Lemma 3.3, which
+describes the entries of F −1.
+□
+Because the condition number κ(F) of a Fiedler companion matrix F depends only upon
+the initial step size and not the permutation σ, we can derive the following corollary.
+Corollary 3.6. [3, Corollary 4.3] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be
+a polynomial over R with n ≥ 2 and c0 ̸= 0. Let A and B be Fiedler companion matrices
+to the polynomial p(x). If the initial step size of both A and B is t, then κ(A) = κ(B).
+Since condition numbers of Fiedler companion matrices depend on the initial step size,
+let
+St = {F | F is a Fiedler companion matrix to p(x) with initial step size t},
+and define κ(t) = κ(F) for F ∈ St. We can now recover a result of [3] that allows us to
+compare the condition numbers of Fielder matrices while again avoiding any reference to
+the permutation σ used to define a Fiedler matrix.
+Corollary 3.7. [3, Corollary 4.5] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be
+a polynomial over R with n ≥ 2 and c0 ̸= 0. Then
+(1) if |c0| < 1, then κ(1) ≤ κ(2) ≤ · · · ≤ κ(n − 1);
+(2) if |c0| = 1, then κ(1) = κ(2) = · · · = κ(n − 1); and
+(3) if |c0| > 1, then κ(1) ≥ κ(2) ≥ · · · ≥ κ(n − 1).
+Proof. Note that by Corollary 3.6, κ(A) is the same for all A ∈ St, so κ(t) is well-defined.
+The conclusions follow from Theorem 3.5.
+□
+
+CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES
+7
+One of our new results is to compare the condition number of a Fielder companion
+matrix of p(x) to the condition number of other companion matrices of p(x). In particular,
+if a Fiedler companion matrix F has a smaller condition number than another companion
+matrix C to the same polynomial p(x), then the ratio κ(C)
+κ(F ) can be bounded. This result is
+similar in spirit to [3, Theorem 4.12].
+Theorem 3.8. Let p(x) = xn + cn−1xn−1 + · · · + c1x + c0 be a polynomial over R with
+n ≥ 2, and c0 ̸= 0. Let F be a Fielder companion matrix to p(x). Further, suppose C is
+any companion matrix to p(x) whose lower Hessenberg form is
+C =
+
+
+0
+Im
+O
+uC
+HC
+In−m−1
+−c0
+yT
+C
+0T
+
+
+such that either uC or yT
+C is the zero vector. If κ(F) ≤ κ(C), then
+1 ≤ κ(C)
+κ(F) ≤ κ(F).
+Proof. The conclusion that 1 ≤ κ(C)
+κ(F ) is immediate from the hypothesis that κ(F) ≤ κ(C).
+By Theorem 3.1 and Lemma 2.4, we can assume F is in unit lower Hessenberg form. As
+such, let
+F =
+
+
+0
+Il
+O
+uF
+HF
+In−l−1
+−c0
+yT
+F
+0T
+
+
+.
+and let t be the initial step size of F. We want to show that
+||C|| · ||C−1||
+||F|| · ||F −1|| ≤ ||F|| · ||F −1||.
+Since C and F are unit sparse companion matrices, ||C|| = ||F||. It suffices to show that
+||C−1|| ≤ ||F|| · ||F −1||2.
+Using equivalence, we may assume without loss of generality that uC = 0. By Lemma 2.2,
+C−1 =
+
+
+1
+c0yT
+C
+0T
+− 1
+c0
+Im
+O
+0
+−HC
+In−m−1
+0
+
+
+.
+since uC = 0. Then
+(3)
+||C−1||2 = (n − 1) +
+� 1
+c0
+�2
++
+�
+ci∈yT
+C
+����
+ci
+c0
+����
+2
++
+�
+ck∈HC
+|ck|2.
+
+8
+MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP
+where c ∈ H (resp. c ∈ y) means −c is an entry in H (resp. y). On the other hand, using
+Lemma 3.3,
+(4) ||F||2 · ||F −1||4 =
+�
+(n − 1) +
+n−1
+�
+i=0
+|ci|2
+� 
+(n − 1) +
+� 1
+c0
+�2
++
+t
+�
+i=1
+����
+ci
+c0
+����
+2
++
+n−1
+�
+j=t+1
+|cj|2
+
+
+2
+.
+We want to show that ||C−1|| ≤ ||F||·||F −1||2 which is equivalent to showing that ||C−1||2 ≤
+||F||2 · ||F −1||4. To do this, for each of the four different summands in (3), we show that
+there exists distinct terms in ||F||2 ·||F −1||4 that are greater than or equal to the summand.
+Here we rely on the fact that there are no negative summands in (4).
+Partially expanding out (4), we have
+||F||2 · ||F −1||4 = (n − 1)3 + (n − 1)2
+� 1
+c0
+�2
++ (n − 1)
+�n−1
+�
+i=0
+|ci|2
+� � 1
+c0
+�2
++ (n − 1)2
+n−1
+�
+j=0
+|cj|2 + other non-negative terms.
+Consequently,
+||C−1||2
+=
+(n − 1) +
+� 1
+c0
+�2
++
+�
+ci∈yT
+C
+����
+ci
+c0
+����
+2
++
+�
+ck∈HC
+|ck|2
+≤
+(n − 1)3 + (n − 1)2
+� 1
+c0
+�2
++ (n − 1)
+�n−1
+�
+i=0
+|ci|2
+� � 1
+c0
+�2
++ (n − 1)2
+n−1
+�
+j=0
+|cj|2
+≤
+||F||2 · ||F −1||4.
+□
+4. Striped Companion Matrices
+In this section we explore a particular class of companion matrices known as striped
+companion matrices, which were introduced in [4].
+A striped companion matrix to a
+polynomial p(x) = xn + cn−1xn−1 + · · · + c1x + c0 has the property that the coefficients
+−c0, −c1, . . . , −cn−1 form horizontal stripes in the matrix. In particular, if t = (t1, t2, . . . , tr)
+is an ordered r-tuple of positive integers with t1 +t2 +· · ·+tr = n, and t1 ≥ ti for 2 ≤ i ≤ n,
+then we define the striped companion matrix Sn(t) to be the companion matrix of unit Hes-
+senberg form
+Sn(t) =
+
+
+0
+It1−1
+O
+R
+In−t1
+0T
+
+
+(5)
+with the (n − t1 + 1) × t1 matrix R having r nonzero rows and with the ith nonzero row of
+R having ti variables in the first ti positions and ti − 1 zero rows immediately above it in
+
+CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES
+9
+R, for 1 < i ≤ r. Note that this implies the first row of R is a nonzero row with t1 leading
+nonzero entries. For example,
+S7(3, 2, 2) =
+�
+����������
+0
+1
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+−c4
+−c5
+−c6
+1
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+−c2
+−c3
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+1
+−c0
+−c1
+0
+0
+0
+0
+0
+�
+����������
+, and S8(3, 3, 2) =
+�
+������������
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+−c5
+−c6
+−c7
+1
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+−c2
+−c3
+−c4
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+1
+−c0
+−c1
+0
+0
+0
+0
+0
+0
+�
+������������
+.
+As the next theorem shows, in some cases the stripped companion matrices can have a
+better condition number than a Fielder companion matrix.
+Theorem 4.1. Suppose n = k(m + 1) for some positive k, m ∈ Z and p(x) = xn +
+cn−1xn−1 + · · · + c1x + c0 with c0 = 1, c1, . . . , cn−1 ∈ R. There exists a striped companion
+matrix S = Sn(k, . . . , k) for p(x) such that κ(S) ≤ κ(F) for every Fiedler companion matrix
+F if and only if
+(6)
+m
+�
+j=1
+�k−1
+�
+i=1
+|cicjk − cjk+i|2
+�
+≤
+m
+�
+j=1
+�k−1
+�
+i=1
+|cjk+i|2
+�
+.
+Proof. Let S = Sk(m+1)(k, . . . , k), and let F be a Fiedler companion matrix. Since ||S|| =
+||F|| as noted in Remark 2.3, it suffices to show that ||S−1|| ≤ ||F −1|| if and only if equation
+(6) holds. By Lemma 2.2,
+S−1 =
+
+
+−c1
+−c2
+. . .
+−ck−1
+0T
+−1
+Ik−1
+O
+0
+−c1cmk + cmk+1
+−c2cmk + cmk+2
+. . .
+−ck−1cmk + c(m+1)k−1
+0
+0
+. . .
+0
+...
+...
+. . .
+...
+...
+...
+. . .
+...
+0
+0
+. . .
+0
+−c1c2k + c2k+1
+−c2c2k + c2k+2
+. . .
+−ck−1c2k + c3k−1
+0
+0
+. . .
+0
+...
+...
+. . .
+...
+0
+0
+. . .
+0
+−c1ck + ck+1
+−c2ck + ck+2
+. . .
+−ck−1ck + c2k−1
+0
+0
+. . .
+0
+...
+...
+. . .
+...
+0
+0
+. . .
+0
+Imk
+−cmk
+0
+...
+...
+0
+−c2k
+0
+...
+0
+−ck
+0
+...
+0
+
+
+.
+Thus
+||S−1||2 = n +
+k−1
+�
+j=1
+|cj|2 +
+m
+�
+j=1
+|cjk|2 +
+m
+�
+j=1
+�k−1
+�
+i=1
+|cicjk − cjk+i|2
+�
+.
+
+10
+MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP
+By Theorem 3.5,
+||F −1||2 = n +
+k−1
+�
+j=1
+|cj|2 +
+m
+�
+j=1
+|cjk|2 +
+m
+�
+j=1
+�k−1
+�
+i=1
+|cjk+i|2
+�
+.
+Therefore κ(S) ≤ κ(F) if and only if
+m
+�
+j=1
+�k−1
+�
+i=1
+|cicjk − cjk+i|2
+�
+≤
+m
+�
+j=1
+�k−1
+�
+i=1
+|cjk+i|2
+�
+.
+□
+We can deduce the following corollary.
+Corollary 4.2. Suppose n = k(m + 1) for some m, k ∈ Z and p(x) = xn + cn−1xn−1 +
+· · · + c1x + c0 with c0 = 1, c1, . . . , cn−1 ∈ R. Suppose F is any Fiedler companion matrix
+for p(x). If
+|cicjk − cjk+i| ≤ |cjk+i|, for 1 ≤ j ≤ m and 1 ≤ i ≤ k − 1,
+then there exists a striped companion matrix S = Sn(k, . . . , k), such that κ(S) ≤ κ(F).
+Example 4.3. Let
+p(x) = x9 + 8x8 + 6x7 + 2x6 + 5x5 + 8x4 + 3x3 + 3x2 + 2x + 1.
+Note that the inequalities in Corollary 4.2 hold. Let F be any Fiedler companion to p(x)
+and consider the striped companion matrix S = S9(3, 3, 3), i.e.,
+S9(3, 3, 3) =
+
+
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+−2
+−6
+−8
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+−3
+−8
+−5
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+−1
+−2
+−3
+0
+0
+0
+0
+0
+0
+
+
+.
+Then ||S|| = ||F|| =
+√
+224, but κ(S) =
+√
+224
+√
+63 < κ(F) =
+√
+224
+√
+224.
+One extreme example of how the inequalities in Corollary 4.2 can be met is if c0 = 1 and
+the striped companion matrix in line (5) has rank(R) = 1. In this case, the inequalities are
+trivially met as described in the following corollary. A more general result can be developed
+for striped companion matrices with differing stripe lengths; e.g., see [1, Section 4.2].
+Corollary 4.4. Given p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 with c0 = 1, and
+c1, . . . , cn−1 ∈ R, let S be a striped companion matrix to the polynomial p(x). If
+S =
+
+
+0 Im
+O
+R
+In−m−1
+0T
+
+
+with rank(R) = 1, then κ(S) ≤ κ(F) for any Fiedler companion matrix F.
+
+CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES
+11
+Proof. This result follows from Corollary 4.2 by observing that |cicjk − cjk+i| = 0 for all
+1 ≤ j ≤ m and 1 ≤ i ≤ k − 1, if and only if rank(R) = 1. In particular, rank(R) = 1
+if and only every 2 × 2 submatrix of R has determinant zero, which is true if and only if
+|cicjk − cjk+i| = 0 for 1 ≤ j ≤ m and 1 ≤ i ≤ k − 1. Note that we are using the fact that
+�
+−cjk
+−cjk+i
+−c0
+−ci
+�
+is a 2 × 2 submatrix of R and c0 = 1.
+□
+Example 4.5. Let b, k ∈ R and consider the polynomial p(x) = x6 + (bk3)x5 + (bk2)x4 +
+(bk2)x3 + (bk)x2 + kx + 1. If
+S = S6(2, 2, 2) =
+
+
+0
+1
+0
+0
+0
+0
+−bk2
+−bk3
+1
+0
+0
+0
+0
+0
+0
+1
+0
+0
+−bk
+−bk2
+0
+0
+1
+0
+0
+0
+0
+0
+0
+1
+−1
+−k
+0
+0
+0
+0
+
+
+and F is any Fiedler companion matrix for p(x), then
+�κ(F)
+κ(S)
+�2
+= b2k6 + b2k4 + b2k4 + b2k2 + k2 + 6
+b2k4 + b2k2 + k2 + 6
+.
+In this case, for sufficiently large k,
+κ(F)
+κ(S) ≈ k
+demonstrating a significantly better condition number for S compared to any Fiedler com-
+panion matrix.
+As shown in Corollary 4.4, if the rank of the submatrix R in the striped companion matrix
+S has rank(R) = 1, then the inequality κ(S) ≤ κ(F) holds for any Fiedler companion matrix
+F. Note that in the striped companion matrix given in Example 4.5, the corresponding
+submatrix R has rank one. Observe also that we can write p(x) has
+p(x) = q(x) + (bk)x2q(x) + (bk2)x4q(x) + x6 with q(x) = 1 + kx.
+This generalizes: if the matrix S in Corollary 4.4 has rank(R) = 1, then p(x) = xn+q(x)f(x)
+for some polynomial q(x) with deg(q(x)) = m and deg(f(x)) = n − m − 1. Moreover,
+Corollary 4.4 can be improved by giving an estimate on κ(F )
+κ(S) for any Fiedler companion
+matrix F.
+Theorem 4.6. Suppose n = k(m + 1) and p(x) = q(x) + b1xkq(x) + b2x2kq(x) + · · · +
+bmxmkq(x) + x(m+1)k with
+q(x) = ak−1xk−1 + ak−2xk−2 + · · · + a1x + 1.
+Let S = Sn(k, k, . . . , k) and F be any Fiedler companion matrix to p(x). If (b2
+1 + · · · + b2
+m)
+is sufficiently large, then
+�κ(F)
+κ(S)
+�2
+≈ (a2
+1 + · · · + a2
+k−1 + 1),
+
+12
+MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP
+or if (a2
+1 + · · · + a2
+k−1) is sufficiently large, then
+�κ(F)
+κ(S)
+�2
+≈ (1 + b2
+1 + · · · + b2
+m).
+Proof. By Remark 2.3, κ(F )
+κ(S) = ||F −1||
+||S−1||. By Lemma 2.2,
+||S−1||2 = a2
+1 + · · · + a2
+k−1 + b2
+1 + · · · + b2
+m + n.
+By Theorem 3.5 we can determine that
+||F −1||2 = (1 + b2
+1 + · · · + b2
+m)(a2
+1 + · · · + a2
+k−1) + (b2
+1 + · · · + b2
+m) + n.
+Therefore,
+�κ(F)
+κ(S)
+�2
+= (1 + b2
+1 + · · · + b2
+m)(a2
+1 + · · · + a2
+k−1) + (b2
+1 + · · · + b2
+m) + n
+(a2
+1 + · · · + a2
+k−1) + (b2
+1 + · · · + b2m) + n
+and the result follows.
+□
+5. Generalized companion matrices: a case study
+In the previous sections, we focused on the condition numbers of unit sparse companion
+matrices. In this section, we initiate an investigation into the condition numbers of a family
+of matrices that are not companion matrices, but have properties similar to companion
+matrices. To date, there appears to be little work done on this approach, so the work in
+this section can be seen as providing a proof-of-concept for future projects. These results can
+also be viewed in the broader context of developing the properties of generalized companion
+matrices (e.g., see [4, 6]). Roughly speaking, given a polynomial p(x) = xn + cn−1xn−1 +
+· · ·+c1x1 +c0, a generalized companion matrix A is a matrix whose entries are polynomials
+in the c0, . . . , cn and whose characteristic polynomial is p(x). See [6] for more explicit detail.
+Instead of considering the general case, we focus on a particular family of matrices and
+their condition numbers. This case study shows that the condition numbers can improve
+on those of Frobenius (or Fiedler) companion matrices under some extra hypotheses.
+We now define our special family. Let p(x) = xn+cn−1xn−1+· · ·+c1x+c0 be a polynomial
+over R with n ≥ 2 and let a ∈ R be any real number. Fix an integer ℓ ∈ {3, . . . , n − 2} and
+let
+aT
+=
+(−cn−1, −cn−2, . . . , −cℓ+1) and bT = (−cℓ−2, −cℓ−3, . . . , −c1).
+Then let
+(7)
+Mn(a, ℓ) =
+
+
+a
+In−ℓ−1
+O
+O
+−cℓ + a
+W
+I2
+O
+−cℓ−1 + acn−1
+b
+O
+O
+Iℓ−2
+−c0
+O
+O
+O
+
+
+.
+
+CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES
+13
+
+
+−c6
+1
+0
+0
+0
+0
+0
+−c5
+0
+1
+0
+0
+0
+0
+−c4 + a
+0
+0
+1
+0
+0
+0
+−c3 + ac6
+−a
+0
+0
+1
+0
+0
+−c2
+0
+0
+0
+0
+1
+0
+−c1
+0
+0
+0
+0
+0
+1
+−c0
+0
+0
+0
+0
+0
+0
+
+
+Figure 4. The matrix M7(a, 4)
+where W is a 2 × (n − ℓ − 1) matrix having W2,1 = −a and zeroes in every other entry.
+Informally, the matrix Mn(a, ℓ) is constructed by starting with the Frobenius companion
+matrix which has all the coefficents of p(x) in the first column. Then we fix a row that is
+neither the top row nor one of the bottom two rows (this corresponds to picking the ℓ), and
+then adding a to cℓ in the (n − ℓ)-th row, and −a in the column to the right and one below.
+We then also add acn−1 to the first entry in the (n − ℓ + 1)-th row. Note that when a = 0,
+Mn(0, ℓ) is equivalent to the Frobenius companion matrix. We can thus view Mn(a, ℓ) as a
+perturbation of the Frobenius companion matrix when a ̸= 0. As an example, the matrix
+M7(a, 4) is given in Figure 4.
+We wish to compare the condition number of Mn(a, ℓ) with the Frobenius (and Fiedler)
+companion matrices. In some cases our new matrix Mn(a, ℓ) can provide us with a smaller
+condition number. The next lemma gives the inverse of Mn(a, ℓ) and shows that the char-
+acteristic polynomial of Mn(a, ℓ) is p(x).
+Lemma 5.1. Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over
+R, with n ≥ 2 and c0 ̸= 0. Let a ∈ R and ℓ ∈ {3, . . . , n − 2}, and let M = Mn(a, ℓ) be
+constructed from p(x) as above. Then
+(i) the characteristic polynomial of M is p(x), and
+(ii) if c0 ̸= 0, then
+M−1 = 1
+c0
+
+
+0T
+0T
+0T
+−1
+c0In−ℓ
+O
+O
+a
+−c0W
+c0I2
+O
+−cℓ + a
+−cℓ−1
+O
+O
+c0Iℓ−2
+b
+
+
+.
+Proof. (i) We employ the fact that the determinant of a matrix is a linear function of its
+rows. In particular, if M = Mn(a, ℓ), we observe that row n − ℓ of xIn − M can be written
+
+14
+MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP
+as u + av for some vectors u and v such that u is not a function of a. Row n − ℓ + 1 of
+xIn − M can also be written in a similar manner. Let k = n − ℓ. Thus applying linearity to
+row k gives us
+det(xIn − M) = det
+
+
+
+
+
+
+
+xIn −
+
+
+a
+In−ℓ−1
+O
+O
+−cℓ
+W
+I2
+O
+−cℓ−1 + acn−1
+b
+O
+O
+Iℓ−2
+−c0
+O
+O
+O
+
+
+
+
+
+
+
+
+
++ a(−1)xℓ.
+(8)
+Note that the term a(−1)xℓ in (8) comes from computing the determinant of the matrix A′
+formed by replacing the k-th row of the matrix xIn−M with the row
+�
+−a
+0
+· · ·
+0
+�
+. Do-
+ing a row expansion along the k-th row of A′, the determinant of A′ is (−1)k+1(−a)det(A′′)
+where A′′ is a block lower diagonal matrix with diagonal blocks D1 and D2. Furthermore,
+D1 is a (k−1)×(k−1) lower triangular matrix with −1 on all the diagonal entries, and D2 is
+a ℓ × ℓ upper triangular matrix with x on all the diagonal entries. So det(A′′) = (−1)k−1xℓ,
+and hence det(A′) = (−1)k+1(−a)(−1)k−1xℓ = (−a)xℓ, as desired.
+We now apply linearity to row k + 1 in the matrix that appears on the right-hand side
+of (8); in particular, a similar argument shows that the right-hand side (8) is equal to
+(9)
+det
+
+
+
+
+
+
+
+xI −
+
+
+a
+Ik−1
+O
+O
+−cℓ
+O
+I2
+O
+−cℓ−1
+b
+O
+O
+Iℓ−2
+−c0
+O
+O
+O
+
+
+
+
+
+
+
+
+
++a(−1)xℓ +acn−1(−1)xℓ−1 +a(x+cn−1)xℓ−1.
+Note that the first summand in (9) is the characteristic polynomial of a Frobenius companion
+matrix of p(x), and hence is p(x). Thus, (9) reduces to
+p(x) + a(−1)xℓ + acn−1(−1)xℓ−1 + a(x + cn−1)xℓ−1 = p(x).
+(ii) A direct multiplication will show that the given matrix is the inverse M.
+□
+Because both Mn(a, ℓ) and its inverse are known, we are able to compute its condition
+number. In the next lemma, instead of providing the general formula, we compute the
+condition number under the extra assumption that c0 = 1 in the polynomial p(x).
+Lemma 5.2. Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over
+R, with n ≥ 2, and suppose that c0 = 1.
+Let a ∈ R and ℓ ∈ {3, . . . , n − 2}, and let
+M = Mn(a, ℓ). Then
+κ(M)2 =
+�
+v + a2 + (a − cℓ)2 + (acn−1 − cℓ−1)2� �
+v + a2 + (a − cℓ)2 + c2
+ℓ−1 + 1
+�
+with
+v = n − c2
+ℓ−1 − c2
+ℓ +
+n−1
+�
+i=1
+c2
+i .
+
+CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES
+15
+The next result illustrates the desired proof-of-concept. In particular, the result shows
+that in special cases, the condition number of the matrix Mn(a, ℓ), which has properties
+similar to a companion matrix, has a condition number smaller than any Fielder compan-
+ion matrix. Although the scope of this result is limited, it does suggest that generalized
+companion matrices, and in particular perturbations of the Frobenius companion matrix,
+can provide better condition numbers in some cases.
+Theorem 5.3. Let n ≥ 2, and fix ℓ ∈ {3, . . . , n − 2} and t ∈ R. Set
+p(x) = xn + txn−1 + txℓ + t2xℓ−1 + 1.
+Let M = Mn(t, ℓ). Then, for any Fieldler companion matrix F of p(x),
+κ(F)2
+κ(M)2 =
+(n + 2t2 + t4)2
+(n + 2t2)(n + 1 + 2t2 + t4).
+In particular, for t for sufficiently large,
+κ(F )
+κ(M) ≈
+1
+√
+2t.
+Proof. By Lemma 5.2,
+κ(M)2 =
+�
+1 + t2 + (n − 1) + a2 + (a − t)2 + (at − t2)2� �
+1 + t2 + (n − 1) + a2 + (a − t)2 + t4 + 1
+�
+.
+Setting a = t gives κ(M)2 = (n + 2t2)(n + 1 + 2t2 + t4). We use Theorem 3.5 to compute
+κ(F)2. Note that since c0 = 1, κ(F) is independent of the initial step size of F. Hence
+κ(F)
+=
+((n − 1) + 1 + t4 + t2 + t2) = (n + 2t2 + t4).
+Thus we have
+κ(F)2
+κ(M)2 =
+(n + 2t2 + t4)2
+(n + 2t2)(n + 1 + 2t2 + t4).
+The limit of the right hand side is t2
+2 as t → ∞, which implies the final statement.
+□
+The following result gives another case where we can make a matrix with smaller condi-
+tion number than any other Fielder companion matrix, providing additional evidence that
+generalized companion matrices may be of interest.
+Theorem 5.4. Let n ≥ 2, and fix ℓ ∈ {3, . . . , n−2}. Let p(x) = xn+cn−1xn−1+· · ·+c1x+c0
+with c0 = 1, and (cℓcn−1)2 < 2cℓ−1cℓcn−1 − 1. Let M = Mn(cℓ, ℓ). Then κ(M) < κ(F) for
+every Fieldler companion matrix F of p(x).
+Proof. Let v = n − c2
+ℓ − c2
+ℓ−1 + �n−1
+i=1 c2
+i .
+Because c0 = 1, by Theorem 3.5 all Fielder
+companion matrices F have condition number
+κ(F) = (v + c2
+ℓ + c2
+ℓ−1).
+By Lemma 5.2, with a = cℓ,
+κ(M)2
+=
+(v + c2
+ℓ + (cℓcn−1 − cℓ−1)2)(v + c2
+ℓ + c2
+ℓ−1 + 1)
+=
+(v + c2
+ℓ + c2
+ℓ−1 + ((cℓcn−1)2 − 2cℓ−1cℓcn−1))(v + c2
+ℓ + c2
+ℓ−1 + 1).
+If we set w = (v +c2
+ℓ +c2
+ℓ−1), then κ(M)2 = (w−y)(w+1) with y = 2cℓ−1cℓcn−1 −(cℓcn−1)2.
+But y > 1 by hypothesis, thus κ(M)2 < w2 = κ(F)2.
+□
+
+16
+MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP
+References
+[1] M. Cox, On conditions numbers of companion matrices, M.Sc. Thesis, McMaster University, 2018.
+[2] L. Deaett, J. Fischer, C. Garnett, K.N. Vander Meulen, Non-sparse companion matrices, Electron. J.
+Linear Algebra 35 (2019) 223–247.
+[3] F. de Ter´an, F.M. Dopico, J. P´erez, Condition numbers for inversion of Fiedler companion matrices,
+Linear Algebra Appl. 439 (2013) 944–981.
+[4] B. Eastman, I.J. Kim, B.L. Shader, K.N. Vander Meulen, Companion matrix patterns, Linear Algebra
+Appl. 463 (2014) 255–272.
+[5] M. Fiedler, A note on companion matrices, Linear Algebra Appl. 372 (2003) 325–331.
+[6] C. Garnett, B.L. Shader, C.L. Shader, P. van den Driessche, Characterization of a family of generalized
+companion matrices, Linear Algebra Appl. 498 (2016) 360–365.
+[7] K.N. Vander Meulen, T. Vanderwoerd, Bounds on polynomial roots using intercyclic companion matri-
+ces, Linear Algebra Appl. 539 (2018) 94–116.
+Unit 202 - 133 Herkimer Street, Hamilton, ON, L8P 2H3, Canada
+Email address: [email protected]
+Department of Mathematics, Redeemer University College, Ancaster, ON, L9K 1J4, Canada
+Email address: [email protected]
+Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4L8,
+Canada
+Email address: [email protected]
+Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4L8,
+Canada
+Email address: [email protected]
+

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