Limit points of the iterative scaling procedure

Aas, Erik

doi:10.1007/s10479-013-1416-2

Cited by 5 publications

(7 citation statements)

References 7 publications

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“…The first equality follows from Theorem 2.1 (1). One can see the second equality from a standard argument of network flow: Regard each edge ij in G a directed edge from i to j having infinite capacity.…”

Section: Polymatroidal Analysis On the Sinkhorn Limitmentioning

confidence: 95%

“…Again, all off-diagonal blocks in the limit M * [I κ , J κ ] must be zero; see the argument after Theorem 3.5. Thus we have the following, which is implicit in [1].…”

Section: Polymatroidal Analysis On the Sinkhorn Limitmentioning

confidence: 99%

“…However it is oscillating in limit between two matrices attaining the minimum KL-divergence between the row-normalized space and column-normalized space; see [13]. Then Aas [1] revealed a block-diagonalized structure of the oscillating limit and gave a polynomial time procedure to determine to this block structure.…”

Section: Introductionmentioning

confidence: 99%

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Finding Hall blockers by matrix scaling

Hayashi¹,

Hirai²

2022

Preprint

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For a given nonnegative matrix A = (A ij ), the matrix scaling problem asks whether A can be scaled to a doubly stochastic matrix XAY for some positive diagonal matrices X, Y . The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization A ij ← A ij / j A ij and column-normalization A ij ← A ij / i A ij alternatively. By this algorithm, A converges to a doubly stochastic matrix in limit if and only if the bipartite graph associated with A has a perfect matching. This property can decide the existence of a perfect matching in a given bipartite graph G, which is identified with the 0, 1-matrix A G . Linial, Samorodnitsky, and Wigderson showed that a polynomial number of the Sinkhorn iterations for A G decides whether G has a perfect matching.In this paper, we show an extension of this result: If G has no perfect matching, then a polynomial number of the Sinkhorn iterations identifies a Hall blocker-a certificate of the nonexistence of a perfect matching. Our analysis is based on an interpretation of the Sinkhorn algorithm as alternating KL-divergence minimization (Csiszár and Tusnády 1984, Gietl andReffel 2013) and its limiting behavior for a nonscalable matrix (Aas 2014). We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage-Mendelsohn decomposition of a bipartite graph.

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Section: Polymatroidal Analysis On the Sinkhorn Limitmentioning

confidence: 95%

“…Again, all off-diagonal blocks in the limit M * [I κ , J κ ] must be zero; see the argument after Theorem 3.5. Thus we have the following, which is implicit in [1].…”

Section: Polymatroidal Analysis On the Sinkhorn Limitmentioning

confidence: 99%

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Finding Hall blockers by matrix scaling

Hayashi¹,

Hirai²

2022

Preprint

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“…Aas mentions this fact (proposition 1 in [1]) as a result of Pretzel (last part of theorem 1 in [11]), although Pretzel considers only the case where the set Γ(a, b, X 0 ) is not empty. We prove theorem 6 by adapting Pretzel's proof.…”

Section: The Latter Maximizes the Ratiomentioning

confidence: 99%

“…When Γ contains no matrix with support included in Supp(X 0 ), we already know by Gietl and Reffel's theorem [7] that both sequences (X 2n ) n≥0 and (X 2n+1 ) n≥0 converge. The convergence may be slow, so Aas gives in [1] an algorithm to fasten the convergence. Aas' algorithm finds and exploits the block structure associated to the inconsistent problem of finding a non-negative matrix whose marginals are a and b and whose support is contained in Supp(X 0 ).…”

Section: Ifmentioning

confidence: 99%

Iterated Proportional Fitting Procedure and Infinite Products of Stochastic Matrices

Brossard

Leuridan

2018

Séminaire De Probabilités XLIX

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The iterative proportional fitting procedure (IPFP), introduced in 1937 by Kruithof, aims to adjust the elements of an array to satisfy specified row and column sums. Thus, given a rectangular non-negative matrix X 0 and two positive marginals a and b, the algorithm generates a sequence of matrices (X n ) n≥0 starting at X 0 , supposed to converge to a biproportional fitting, that is, to a matrix Y whose marginals are a and b and of the form Y = D 1 X 0 D 2 , for some diagonal matrices D 1 and D 2 with positive diagonal entries.When a biproportional fitting does exist, it is unique and the sequence (X n ) n≥0 converges to it at an at least geometric rate. More generally, when there exists some matrix with marginal a and b and with support included in the support of X 0 , the sequence (X n ) n≥0 converges to the unique matrix whose marginals are a and b and which can be written as a limit of matrices of the form D 1 X 0 D 2 .In the opposite case (when there exists no matrix with marginals a and b whose support is included in the support of X 0 ), the sequence (X n ) n≥0 diverges but both subsequences (X 2n ) n≥0 and (X 2n+1 ) n≥0 converge.In the present paper, we use a new method to prove again these results and determine the two limit-points in the case of divergence. Our proof relies on a new convergence theorem for backward infinite products · · · M 2 M 1 of stochatic matrices M n , with diagonal entries M n (i, i) bounded away from 0 and with bounded ratios M n (j, i)/M n (i, j). This theorem generalizes Lorenz' stabilization theorem.We also provide an alternative proof of Touric and Nedić's theorem on backward infinite products of doubly-stochatic matrices, with diagonal entries bounded away from 0. In both situations, we improve slightly the conclusion, since we establish not only the convergence of the sequence (M n · · · M 1 ) n≥0 , but also its finite variation.Keywords: infinite products of stochastic matrices -contingency matrices -distributions with given marginals -iterative proportional fitting -relative entropy -I-divergence. MSC Classification: 15B51 -62H17 -62B10 -68W40.The homogeneity of the map T R shows that replacing X 0 with X 0 (+, +) −1 X 0 does not change the matrices X n for n ≥ 1, so there is no restriction to assume that X 0 ∈ Γ 1 .Note that Γ R and Γ C are subsets of Γ 1 and are closed subsets of M p,q (R + ). Therefore, if (X n ) n≥0 converges, its limit belongs to the set Γ. Furthermore, by construction, the matrices X n belong to the set

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Termination of the iterative proportional fitting procedure

Reffel

2014

Statistics & Probability Letters

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Limit points of the iterative scaling procedure

Cited by 5 publications

References 7 publications

Finding Hall blockers by matrix scaling

Finding Hall blockers by matrix scaling

Iterated Proportional Fitting Procedure and Infinite Products of Stochastic Matrices

Termination of the iterative proportional fitting procedure

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