Quantitative Stability and Numerical Resolution of the Moment Measure Problem
Provides a theoretical foundation and practical algorithm for solving the moment measure problem, which is important for applications in convex geometry and optimal transport.
The authors establish a quantitative stability estimate for the moment measure problem and introduce a numerical method inspired by semi-discrete optimal transport, using a Newton method to solve the discrete approximation. Numerical experiments show convergence rates beyond theoretical predictions.
The moment measure problem consists in finding a convex function $ψ$ whose moment measure, i.e., the pushforward by $\nabla ψ$ of the measure with density $e^{-ψ(\,\cdot\,)}$, 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.