Large Deviations Principle for Bures-Wasserstein Barycenters

TL;DR

This paper establishes the large deviations principle for empirical Bures-Wasserstein barycenters, revealing new insights into measure concentration and exponential tilting in non-linear metric spaces, with broad applicability to various geometric settings.

Contribution

It introduces the first large deviations principles for Fréchet means in non-linear metric spaces, including Bures-Wasserstein barycenters, and develops a novel exponential tilting method.

Findings

Proves large deviations principle for Bures-Wasserstein barycenters.

Demonstrates dimension-free concentration of measure phenomena.

Extends large deviations results to Riemannian and Wasserstein spaces.

Abstract

We prove the large deviations principle for empirical Bures-Wasserstein barycenters of independent, identically-distributed samples of covariance matrices and covariance operators. As an application, we explore some consequences of our results for the phenomenon of dimension-free concentration of measure for Bures-Wasserstein barycenters. Our theory reveals a novel notion of exponential tilting in the Bures-Wasserstein space, which, in analogy with Cr\'amer's theorem in the Euclidean case, solves the relative entropy projection problem under a constraint on the barycenter. Notably, this method of proof is easy to adapt to other geometric settings of interest; with the same method, we obtain large deviations principles for empirical barycenters in Riemannian manifolds and the univariate Wasserstein space, and we obtain large deviations upper bounds for empirical barycenters in the general multivariate Wasserstein space. In fact, our results are the first known large deviations principles for Fr\'echet means in any non-linear metric space.