Consistent estimation of a population barycenter in the Wasserstein space

Jérémie Bigot (1) , Thierry Klein (2)

1 DMIA - Département de Mathématiques, Informatique, Automatique
2 IMT - Institut de Mathématiques de Toulouse UMR5219

We define a notion of barycenter for random probability measures in the Wasserstein space. We give a characterization of the population barycenter in terms of existence and uniqueness for compactly supported measures. Then, the problem of estimating this barycenter from n independent and identically distributed random probability measures is considered. We study the convergence of the empirical barycenter proposed in Agueh and Carlier (2011) to its population counterpart as the number of measures n tends to infinity. To illustrate the benefits of this approach for data analysis and statistics, we finally discuss the usefulness of barycenters in the Wasserstein space for curve and image warping.

Mots clés: Wasserstein space; Empirical and population barycenters; Fréchet mean; Convergence of random variables; Optimal transport; Curve and image warping.; Curve and image warping

Domaines: Théorie [stat.TH]

Jérémie Bigot, Thierry Klein. Consistent estimation of a population barycenter in the Wasserstein space. 2012. ⟨hal-00763668v1⟩