# Approximation of Optimal Transport problems with marginal moments constraints

## TL;DR

This paper introduces a method to approximate optimal transport problems using moment constraints, resulting in finite discrete solutions that scale well with problem dimension and converge rapidly as the number of moments increases.

## Contribution

It demonstrates that moment-constrained OT problems can be solved with finite discrete measures, providing convergence rates and practical algorithms for approximation.

## Key findings

- Finite discrete measures solve MCOT problems.
- Number of points scales linearly with marginal laws.
- Convergence rates of O(1/n) and O(1/n^2) for moments.

## Abstract

Optimal Transport (OT) problems arise in a wide range of applications, from physics to economics. Getting numerical approximate solution of these problems is a challenging issue of practical importance. In this work, we investigate the relaxation of the OT problem when the marginal constraints are replaced by some moment constraints. Using Tchakaloff's theorem, we show that the Moment Constrained Optimal Transport problem (MCOT) is achieved by a finite discrete measure. Interestingly, for multimarginal OT problems, the number of points weighted by this measure scales linearly with the number of marginal laws, which is encouraging to bypass the curse of dimension. This approximation method is also relevant for Martingale OT problems. We show the convergence of the MCOT problem toward the corresponding OT problem. In some fundamental cases, we obtain rates of convergence in $O(1/n)$ or $O(1/n^2)$ where $n$ is the number of moments, which illustrates the role of the moment functions. Last, we present algorithms exploiting the fact that the MCOT is reached by a finite discrete measure and provide numerical examples of approximations.