From Knothe's transport to Brenier's map and a continuation method for optimal transport

TL;DR

This paper introduces a continuation method for solving optimal transport problems by connecting Knothe's rearrangement to Brenier's map through a limiting process involving quadratic cost problems with asymptotically dominating coordinate weights.

Contribution

It establishes a novel link between Knothe's rearrangement and Brenier's optimal transport map via a limiting process, enabling a new numerical continuation approach.

Findings

Knothe's rearrangement is the limit of certain Monge-Kantorovich solutions.

A continuation method for optimal transport is proposed based on this limit.

The method provides a new way to compute Brenier's map numerically.

Abstract

A simple procedure to map two probability measures in $\mathbb{R}^d$ is the so-called \emph{Knothe-Rosenblatt rearrangement}, which consists in rearranging monotonically the marginal distributions of the last coordinate, and then the conditional distributions, iteratively. We show that this mapping is the limit of solutions to a class of Monge-Kantorovich mass transportation problems with quadratic costs, with the weights of the coordinates asymptotically dominating one another. This enables us to design a continuation method for numerically solving the optimal transport problem.