Regularization and convergence rates for Wasserstein barycenters

Abstract

The concept of barycenter in the Wasserstein space allows to define a notion of Frechet mean of a set of probability measures. The purpose of this talk is to discuss potential applications of Wasserstein barycenters for characterizing the mean of random histograms or random planar objects, and to discuss their statistical properties. However, depending on the data at hand, Wasserstein barycenters may be irregular. In this talk, we thus introduce a convex regularization of Wasserstein barycenters for random probability measures supported on a domain of any dimension. We prove the existence and uniqueness of such barycenters for a large class of regularizing functions. A stability result of regularized barycenters in terms of Bregman divergence associated to the convex regularization term is also given. This allows to study the case of data made of iid random probability measures. In particular, we prove the convergence in Bregman divergence of the regularized empirical barycenter of a set of n random probability measures towards its population counterpart, and we discuss its rate of convergence. This approach is shown to be appropriate for the study of discrete or absolutely continuous random measures. As illustrative examples, we focus on the analysis of probability measures supported on the real line, and we discuss the usefulness of this approach for the statistical analysis of random planar shapes.