Finding Hall blockers by matrix scaling

Abstract

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.

Shrunk subspaces via operator Sinkhorn iteration

“…[Aas14] showed that the accumulation points exhibit a block diagonal structure which can be characterized combinatorially. Recently, Hayashi and Hirai [HH22] proved that one can obtain a Hall blocker (dual certificate showing that a given matrix has no scaling) by the Sinkhorn iteration.…”

“…The authors thank Nikhil Srivastava for references to eigendecomposition algorithms. The authors also thank Koyo Hayashi and Hiroshi Hirai for references [Aas14,GR13] and sharing a draft of [HH22].…”