Quantitative Stability and Numerical Resolution of the Moment Measure Problem

TL;DR

This paper develops a stability estimate for the nonlinear moment measure problem, introduces a numerical approach inspired by semi-discrete optimal transport, and demonstrates its effectiveness through experiments.

Contribution

It provides the first quantitative stability estimate for the moment measure problem and proposes a Newton-based numerical method for its approximation.

Findings

Established a stability estimate validating numerical approximation methods.

Developed a Newton method for solving the discretized moment measure problem.

Numerical experiments show convergence rates exceeding theoretical predictions.

Abstract

The moment measure problem consists in finding a convex function $\psi$ whose moment measure, i.e., the pushforward by $\nabla \psi$ of the measure with density $e^{-\psi(\,\cdot\,)}$, is prescribed. It is highly non-linear and less understood than the related optimal transport problem. We establish a quantitative stability estimate for this problem. This estimate validates, as well as leads us to introduce, an approach to the numerical resolution of the moment measure problem inspired by semi-discrete optimal transport, consisting in approximating the prescribed measure by a finitely supported one. We describe a Newton method for solving the discrete problem thus obtained, and perform numerical experiments, studying the experimental rates of convergence of the approximation beyond the predictions of the stability estimate.