For a given nonnegative matrix $A=(A_{ij})$, the matrix scaling problem asks whether $A$ can be scaled to a doubly stochastic matrix $D_1AD_2$ for some positive diagonal matrices $D_1,D_2$.The Sinkhorn algorithm is a simple iterative algorithm, which repeats row-normalization $A_{ij} \leftarrow A_{ij}/\sum_{j}A_{ij}$ and column-normalization $A_{ij} \leftarrow A_{ij}/\sum_{i}A_{ij}$ 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 $O(n^2 \log n)$ 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 $\Gamma(X)$ with $|X| > |\Gamma(X)|$. Specifically, we show that $O(n^2 \log n)$ iterations can identify one Hall blocker, and that further polynomial iterations can also identify all parametric Hall blockers $X$ of maximizing $(1-\lambda) |X| - \lambda |\Gamma(X)|$ for $\lambda \in [0,1]$.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 KL-divergence (Csisz\'{a}r and Tusn\'{a}dy 1984, Gietl and Reffel 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.

翻译：对于给定的非负矩阵 $A=(A_{ij})$，矩阵缩放问题考察是否存在正对角矩阵 $D_1,D_2$，使得 $A$ 可被缩放为双随机矩阵 $D_1AD_2$。Sinkhorn算法是一种简单的迭代算法，交替执行行归一化 $A_{ij} \leftarrow A_{ij}/\sum_{j}A_{ij}$ 与列归一化 $A_{ij} \leftarrow A_{ij}/\sum_{i}A_{ij}$。通过此算法，当且仅当 $A$ 关联的二部图具有完美匹配时，$A$ 在极限下收敛至双随机矩阵。该性质可用于判定给定二部图 $G$（其对应 $0,1$ 矩阵记为 $A_G$）是否存在完美匹配。Linial、Samorodnitsky与Wigderson证明，对 $A_G$ 执行 $O(n^2 \log n)$ 次迭代即可判定 $G$ 是否存在完美匹配，其中 $n$ 为 $G$ 某一色部顶点数。本文证明该结果的推广：若 $G$ 无完美匹配，则多项式次Sinkhorn迭代可识别出一个霍尔阻碍——即满足 $|X| > |\Gamma(X)|$ 的顶点子集 $X$（$\Gamma(X)$ 为其邻域）。具体而言，我们证明 $O(n^2 \log n)$ 次迭代可识别一个霍尔阻碍，进一步的多项式迭代可识别所有参数化霍尔阻碍 $X$，即最大化 $(1-\lambda) |X| - \lambda |\Gamma(X)|$（$\lambda \in [0,1]$）的子集。前者结果基于将Sinkhorn算法解释为几何规划的交替最小化，后者则基于将其解释为KL散度的交替最小化（Csiszár与Tusnády 1984，Gietl与Reffel 2013）及其对不可缩放矩阵的极限行为（Aas 2014）。我们还将Sinkhorn极限与参数化网络流、多拟阵主划分及二部图的Dulmage-Mendelsohn分解建立了联系。