Quantitative Stability of Barycenters in the Wasserstein Space

TL;DR

This paper investigates how approximations of probability measures affect Wasserstein barycenters, demonstrating their stability under mild conditions using advanced optimal transport theory.

Contribution

It provides a quantitative analysis of the stability of Wasserstein barycenters with respect to their input measures, introducing new bounds based on strong convexity and push-forward continuity.

Findings

Wasserstein barycenters depend Hölder-continuously on their marginals.

The stability results rely on recent estimates of strong convexity in optimal transport.

A new control on the modulus of continuity of push-forward maps is established.

Abstract

Wasserstein barycenters define averages of probability measures in a geometrically meaningful way. Their use is increasingly popular in applied fields, such as image, geometry or language processing. In these fields however, the probability measures of interest are often not accessible in their entirety and the practitioner may have to deal with statistical or computational approximations instead. In this article, we quantify the effect of such approximations on the corresponding barycenters. We show that Wasserstein barycenters depend in a H{\"o}lder-continuous way on their marginals under relatively mild assumptions. Our proof relies on recent estimates that quantify the strong convexity of the dual quadratic optimal transport problem and a new result that allows to control the modulus of continuity of the push-forward operation under a (not necessarily smooth) optimal transport map.