Optimal transport via a Monge-Amp\`ere optimization problem

TL;DR

This paper reformulates Monge's optimal transportation problem as a convex optimization problem using a Monge-Ampère equation, providing a discretization method that converges to the true solution and demonstrating its practical effectiveness.

Contribution

It introduces a novel convex optimization framework for OT via Monge-Ampère equations and proves convergence of discrete solutions to the true OT map under regularity conditions.

Findings

Discrete problems admit solutions via convex optimization

Solutions converge to the true OT map under regularity conditions

Visualizations demonstrate practical applicability

Abstract

We rephrase Monge's optimal transportation (OT) problem with quadratic cost--via a Monge-Amp\`ere equation--as an infinite-dimensional optimization problem, which is in fact a convex problem when the target is a log-concave measure with convex support. We define a natural finite-dimensional discretization to the problem and associate a piecewise affine convex function to the solution of this discrete problem. The discrete problems always admit a solution, which can be obtained by standard convex optimization algorithms whenever the target is a log-concave measure with convex support. We show that under suitable regularity conditions the convex functions retrieved from the discrete problems converge to the convex solution of the original OT problem furnished by Brenier's theorem. Also, we put forward an interpretation of our convergence result that suggests applicability to the convergence of a wider range of numerical methods for OT. Finally, we demonstrate the practicality of our convergence result by providing visualizations of OT maps as well as of the dynamic OT problem obtained by solving the discrete problem numerically.