A GPM-based algorithm for solving regularized Wasserstein barycenter problems in some spaces of probability measures

TL;DR

This paper introduces a gradient projection algorithm for computing regularized Wasserstein barycenters in spaces of probability measures, providing theoretical insights and numerical stability analysis for Gaussian and q-Gaussian distributions.

Contribution

It offers a novel algorithm for regularized Wasserstein barycenters and characterizes their uniqueness for Gaussian and q-Gaussian measures, expanding understanding in this area.

Findings

Algorithm effectively computes regularized barycenters.

Uniqueness established for Gaussian and q-Gaussian cases.

Numerical experiments demonstrate parameter influence and stability.

Abstract

In this paper, we focus on the analysis of the regularized Wasserstein barycenter problem. We provide uniqueness and a characterization of the barycenter for two important classes of probability measures: (i) Gaussian distributions and (ii) $q$-Gaussian distributions; each regularized by a particular entropy functional. We propose an algorithm based on gradient projection method in the space of matrices in order to compute these regularized barycenters. We also consider a general class of $\varphi$-exponential measures, for which only the non-regularized barycenter is studied. Finally, we numerically show the influence of parameters and stability of the algorithm under small perturbation of data.