Barycenter in Wasserstein Spaces: Existence and Consistency

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.