Wasserstein barycenters : statistics and optimization

Abstract

We study a geometric notion of average, the barycenter, over 2-Wasserstein space. We significantly advance the state of the art by introducing extendible geodesics, a simple synthetic geometric condition which implies non-asymptotic convergence of the empirical barycenter in non-negatively curved spaces such as Wasserstein space. We further establish convergence of first-order methods in the Gaussian case, overcoming the nonconvexity of the barycenter functional. These results are accomplished by various novel geometrically inspired estimates for the barycenter functional including a variance inequality, new so-called quantitative stability estimates, and a Polyak-Łojasiewicz (PL) inequality. These inequalities may be of independent interest.