Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (6): 2183-2197.doi: 10.1007/s10473-021-0623-1

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Liming WU   

  1. Institute for Advanced Study in Mathematics, Harbin's Institute of Technology, Harbin 150001, China;Laboratoire de Mathématiques Blaise Pascal, CNRS-UMR 6620, Université Clermont-Auvergne(UCA), 63000 Clermont-Ferrand, France
  • Received:2021-05-26 Revised:2021-10-13 Online:2021-12-25 Published:2021-12-27

Abstract: In this exposition paper we present the optimal transport problem of MongeAmpère-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 Schrödinger system. The relationship between the Schrödinger system, the associated Bernstein process and the optimal transport was developed by Léonard[32, 33] (and by Mikami[39] earlier via an h-process). We present Sinkhorn's algorithm for solving the Schrödinger 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..

Key words: entropical optimal transport, Schrödinger system, Sinkhorn's algorithm, gradient descent

CLC Number: 

  • 46N10