On the geometry and dynamical formulation of the Sinkhorn algorithm for optimal transport

TL;DR

This paper explores the geometric and dynamical aspects of the Sinkhorn algorithm for optimal transport, framing it as a discretization of a nonlinear integral equation and connecting it to classical measure theory results.

Contribution

It provides a geometric perspective on the Sinkhorn algorithm and links it to historical measure theory, offering new insights into its foundational structure.

Findings

Highlights the geometric origin of the Sinkhorn algorithm

Connects the algorithm to classical results on product measures

Provides a new interpretation as a discretization of a nonlinear integral equation

Abstract

The Sinkhorn algorithm is a numerical method for the solution of optimal transport problems. Here, I give a brief survey of this algorithm, with a strong emphasis on its geometric origin: it is natural to view it as a discretization, by standard methods, of a non-linear integral equation. In the appendix, I also provide a short summary of an early result of Beurling on product measures, directly related to the Sinkhorn algorithm.