Sinkhorn

Sinkhorn refers to a group of results in matrix analysis and optimization named after the mathematician Richard Sinkhorn. The most influential are the Sinkhorn–Knopp algorithm for matrix scaling, Sinkhorn’s theorem on diagonal scaling to doubly stochastic form, and the entropy-regularized distance used in optimal transport, commonly called the Sinkhorn distance.

Sinkhorn–Knopp algorithm: Given a nonnegative matrix A, the goal is to find diagonal matrices D_r and D_c with positive entries such that D_r A D_c has prescribed row sums and column sums (often all ones). The method alternates normalizing rows and then columns: scale each row to sum to one, then scale each column to sum to one, and repeat. If A has total support and strictly positive entries, the iterations converge to a matrix that is doubly stochastic. When A contains zeros, convergence becomes more delicate, and the limit may depend on the support pattern.

Sinkhorn’s theorem: For any square matrix with strictly positive entries, there exist positive diagonal matrices D_r and D_c, such that D_r A D_c is doubly stochastic (all row and column sums equal to one). This theorem underpins the convergence of the Sinkhorn–Knopp algorithm and has applications in matrix balancing and preconditioning.

Sinkhorn distance: In optimal transport theory, entropic regularization adds an entropy term to the transport problem, yielding a smooth objective that can be solved efficiently by Sinkhorn iterations. The resulting distance, the Sinkhorn distance, is computationally fast and widely used in machine learning and statistics. As the regularization parameter tends to zero, it approaches the classical Wasserstein distance.

Applications: the methods serve in data normalization and balancing, preconditioning, image and text processing, and transport-based comparisons in machine learning, economics, and statistical modeling. Related topics include entropy-regularized optimal transport and Kantorovich duality.