Fast convergence of empirical barycenters in Alexandrov spaces and the Wasserstein space

TL;DR

This paper proves that empirical barycenters in Alexandrov spaces, including Wasserstein space, converge rapidly under natural conditions, advancing understanding of convergence rates in non-Euclidean geometric spaces.

Contribution

It establishes fast convergence rates for empirical barycenters in a broad class of geodesic spaces with curvature bounds, including infinite-dimensional Wasserstein space.

Findings

Achieves parametric convergence rates under natural geometric conditions.

Extends results to infinite-dimensional spaces like Wasserstein space.

Links bi-extendibility of geodesics to regularity of Kantorovich potentials.

Abstract

This work establishes fast rates of convergence for empirical barycenters over a large class of geodesic spaces with curvature bounds in the sense of Alexandrov. More specifically, we show that parametric rates of convergence are achievable under natural conditions that characterize the bi-extendibility of geodesics emanating from a barycenter. These results largely advance the state-of-the-art on the subject both in terms of rates of convergence and the variety of spaces covered. In particular, our results apply to infinite-dimensional spaces such as the 2-Wasserstein space, where bi-extendibility of geodesics translates into regularity of Kantorovich potentials.