Entropic-Wasserstein barycenters: PDE characterization, regularity and CLT

Guillaume Carlier (1, 2) , Katharina Eichinger (1) , Alexey Kroshnin (3, 4)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
3 UCBL - Université Claude Bernard Lyon 1
4 ICJ - Institut Camille Jordan

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.

Mots clés: Entropic-Wasserstein barycenters; Monge-Ampère equations and systems; Central limit theorem

Domaines: Optimisation et contrôle [math.OC]

Guillaume Carlier, Katharina Eichinger, Alexey Kroshnin. Entropic-Wasserstein barycenters: PDE characterization, regularity and CLT. SIAM Journal on Mathematical Analysis, 2021, ⟨10.1137/20M1387262⟩. ⟨hal-03084049⟩