Existence and Consistency of Wasserstein Barycenters

TL;DR

This paper establishes the existence and consistency of Wasserstein barycenters for random distributions on geodesic spaces, extending the theoretical foundation for statistical analysis of probability measures.

Contribution

It proves the existence and consistency of Wasserstein barycenters in a general setting, including empirical and growing distribution scenarios.

Findings

Wasserstein barycenters exist for distributions on geodesic spaces.

Consistency of barycenters is proven in a broad, general framework.

The results extend the theoretical understanding of statistical means in measure spaces.

Abstract

In this paper, based on the Fr{\'e}chet mean, we define a notion of barycenter corresponding to a usual notion of statistical mean. We prove the existence of Wasserstein barycenters of random distributions defined on a geodesic space (E, d). We also prove the consistency of this barycenter in a general setting, that includes taking barycenters of empirical versions of the distributions or of a growing set of distributions.