Barycenter in Wasserstein Spaces: Existence and Consistency

Thibaut Le Gouic, Jean-Michel Loubes

Abstract: We study barycenters in the Wasserstein space $$\mathcal {P}_p(E)$$ of a locally compact geodesic space (E, d). In this framework, we define the barycenter of a measure $$\mathbb {P}$$ on $$\mathcal {P}_p(E)$$ as its Fréchet mean. The paper establishes its existence and states consistency with respect to $$\mathbb {P}$$ . We thus extends previous results on $$\mathbb {R}^d$$ , with conditions on $$\mathbb {P}$$ or on the sequence converging to $$\mathbb {P}$$ for consistency.