ENTROPICAL OPTIMAL TRANSPORT, SCHRODINGER'S SYSTEM AND ALGORITHMS

In this exposition paper we present the optimal transport problem of Monge-Ampere-Kantorovitch (MAK in short) and its approximative entropical regularization. Contrary to the MAK optimal transport problem, the solution of the entropical optimal transport problem is always unique, and is characterized by the Schrodinger system. The relationship between the Schrodinger system, the associated Bernstein process and the optimal transport was developed by Leonard [32, 33] (and by Mikami [39] earlier via an h-process). We present Sinkhorn's algorithm for solving the Schrodinger system and the recent results on its convergence rate. We study the gradient descent algorithm based on the dual optimal question and prove its exponential convergence, whose rate might be independent of the regularization constant. This exposition is motivated by recent applications of optimal transport to different domains such as machine learning, image processing, econometrics, astrophysics etc..