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

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, each regularized by a particular entropy functional: (i) Gaussian distributions and (ii) q-Gaussian distributions. We propose an algorithm based on gradient projection method (GPM) in the space of matrices in order to compute these regularized barycenters. Finally, we numerically show the influence of parameters and stability of the algorithm under small perturbation of data and compare the gradient projection method with Riemannian gradient method.