Solving Monge problem by Hilbert space embeddings of probability measures

TL;DR

This paper introduces deep learning approaches for solving Monge's optimal transport problem using Hilbert space embeddings, with proven convergence and validation through numerical experiments, including large-scale applications.

Contribution

The paper develops a novel deep learning framework that employs Hilbert space embeddings and MMD penalties to solve Monge's problem, with theoretical convergence guarantees.

Findings

Methods converge to optimal transport maps in $L^2$ cost.

Validated on large-scale Monge problems.

Applicable to high-dimensional distributions.

Abstract

We propose deep learning methods for classical Monge's optimal mass transportation problems, where where the distribution constraint is treated as penalty terms defined by the maximum mean discrepancy in the theory of Hilbert space embeddings of probability measures. We prove that the transport maps given by the proposed methods converge to optimal transport maps in the problem with $L^2$ cost. Several numerical experiments validate our methods. In particular, we show that our methods are applicable to large-scale Monge problems. This is a corrected version of the ICORES 2025 proceedings paper.