Existence and Consistency of Wasserstein Barycenters

Thibaut Le Gouic (1) , Jean-Michel Loubes (2)

1 I2M - Institut de Mathématiques de Marseille
2 IMT - Institut de Mathématiques de Toulouse UMR5219

In this paper, based on the Fréchet mean, we define a notion of barycenter corresponding to a usual notion of statistical mean. We prove the existence of Wasserstein barycenters of random distributions defined on a geodesic space (E, d). We also prove the consistency of this barycenter in a general setting, that includes taking barycenters of empirical versions of the distributions or of a growing set of distributions.

Domaines: Statistiques [math.ST]

Thibaut Le Gouic, Jean-Michel Loubes. Existence and Consistency of Wasserstein Barycenters. Probability Theory and Related Fields, 2017, 168 (3-4), pp.901-917. ⟨10.1007/s00440-016-0727-z⟩. ⟨hal-01163262v2⟩