A Unified Approach for Computing Wasserstein Barycenters of Discrete and Continuous Measures

Computing the unregularized Wasserstein barycenter for measure-valued data is a challenging optimization task. Recent algorithms have been tailored to either discrete measures as point clouds or continuous measures discretized on regular grids. In this work, we propose a primal mirror descent algorithm for computing the exact Wasserstein barycenter in the Fisher-Rao geometry. Our algorithm is a unified approach that is flexible enough to simultaneously cover discrete and absolutely continuous input measures, with convergence guarantees in both settings. In particular, when all input measures are discrete, our algorithm, initialized from any probability density, solves a sequence of semi-discrete optimal transport subproblems and produces absolutely continuous iterates that converge to the discrete barycenter. We use synthetic and real data examples to demonstrate the promising result in terms of accuracy and computational cost.