Entropic-Wasserstein barycenters: PDE characterization, regularity and CLT

Abstract

In this paper, we investigate properties of entropy-penalized Wasserstein barycenters introduced in [5] as a regularization of Wasserstein barycenters [1]. After characterizing these barycenters in terms of a system of Monge-Ampère equations, we prove some global moment and Sobolev bounds as well as higher regularity properties. We finally establish a central limit theorem for entropic-Wasserstein barycenters.

Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm

“…The techniques involving the projective metric are less probabilistic in nature, which may be one reason why it is wide open how to relax the boundedness conditions. We remark that the initial result of [12] also covered the multimarginal problem which has recently become popular due to its role in the Wasserstein barycenter problem [1,11]. At least in the context of [10], it was observed that Hilbert-Birkhoff arguments may not be equally successful beyond two marginals.…”