通过矩阵缩放找到大厅阻塞物

IF 1.4 3区数学 Q2 MATHEMATICS, APPLIED

Mathematics of Operations Research Pub Date : 2023-10-18 DOI:10.1287/moor.2022.0198

Koyo Hayashi, Hiroshi Hirai, Keiya Sakabe

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引用次数: 0

摘要

给定一个非负矩阵[公式:见文]，矩阵缩放问题是问对于一些正对角线矩阵d1, d2, a是否可以缩放成一个双随机矩阵[公式:见文]。Sinkhorn算法是一种简单的迭代算法，交替重复行归一化[公式:见文本]和列归一化[公式:见文本]。通过该算法，当且仅当与A相关的二部图具有完美匹配时，A收敛到极限的双随机矩阵。这个性质可以决定给定的二部图G中存在一个完全匹配，该二部图用矩阵a G来标识。Linial et al.(2000)表明[公式:见文]A G的迭代决定了G是否具有完美匹配。在这里，n是g的一个颜色类的顶点数。在本文中，我们给出了这个结果的一个扩展。如果G没有完美匹配，那么Sinkhorn迭代的一个多项式数可以识别一个Hall block——一个顶点子集X与[公式:见文]相邻，这是一个不存在完美匹配的证明。具体来说，我们证明了[公式:见文]迭代可以识别一个霍尔阻滞剂，进一步的多项式迭代也可以识别最大化[公式:见文]的所有参数霍尔阻滞剂X。前者的结果是基于对Sinkhorn算法的解释为几何规划的交替最小化。后者是对Kullback-Leibler (KL)散度的交替极小化解释及其对不可伸缩矩阵的极限行为。我们还将Sinkhorn极限与参数网络流、多拟阵的主划分以及二部图的Dulmage-Mendelsohn分解联系起来。资助:K. Hayashi由日本科学促进会资助[Grant JP19J22605]。H. Hirai得到了胚胎科学与技术前期研究的支持[Grant JPMJPR192A]。

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Finding Hall Blockers by Matrix Scaling

Given a nonnegative matrix [Formula: see text], the matrix scaling problem asks whether A can be scaled to a doubly stochastic matrix [Formula: see text] for some positive diagonal matrices D 1 , D 2 . The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization [Formula: see text] and column-normalization [Formula: see text] 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 et al. (2000) showed that [Formula: see text] iterations for A G decide whether G has a perfect matching. Here, n is the number of vertices in one of the color classes of G. 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 vertex subset X having neighbors [Formula: see text] with [Formula: see text], which is a certificate of the nonexistence of a perfect matching. Specifically, we show that [Formula: see text] iterations can identify one Hall blocker and that further polynomial iterations can also identify all parametric Hall blockers X of maximizing [Formula: see text] for [Formula: see text]. The former result is based on an interpretation of the Sinkhorn algorithm as alternating minimization for geometric programming. The latter is on an interpretation as alternating minimization for Kullback–Leibler (KL) divergence and on its limiting behavior for a nonscalable matrix. We also relate the Sinkhorn limit with parametric network flow, principal partition of polymatroids, and the Dulmage–Mendelsohn decomposition of a bipartite graph. Funding: K. Hayashi was supported by the Japan Society for the Promotion of Science [Grant JP19J22605]. H. Hirai was supported by Precursory Research for Embryonic Science and Technology [Grant JPMJPR192A].

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来源期刊

Mathematics of Operations Research 管理科学-应用数学

CiteScore

3.40

自引率

5.90%

发文量

178

审稿时长

15.0 months

期刊介绍： Mathematics of Operations Research is an international journal of the Institute for Operations Research and the Management Sciences (INFORMS). The journal invites articles concerned with the mathematical and computational foundations in the areas of continuous, discrete, and stochastic optimization; mathematical programming; dynamic programming; stochastic processes; stochastic models; simulation methodology; control and adaptation; networks; game theory; and decision theory. Also sought are contributions to learning theory and machine learning that have special relevance to decision making, operations research, and management science. The emphasis is on originality, quality, and importance; correctness alone is not sufficient. Significant developments in operations research and management science not having substantial mathematical interest should be directed to other journals such as Management Science or Operations Research.