Wasserstein Barycenters over Riemannian manifolds

By Young-Heon Kim, Brendan Pass. 2014

96 citations recorded by the source corpus.

We study barycenters in the space of probability measures on a Riemannian manifold, equipped with the Wasserstein metric. Under reasonable assumptions, we establish absolute continuity of the barycenter of general measures $\Omega \in P(P(M))$ on Wasserstein space, extending on one hand, results in the Euclidean case (for barycenters between finitely many measures) of Agueh and Carlier \cite{ac} to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschl\"ager \cite{c-ems} for barycenters between two measures to the multi-marginal setting. Our work also extends these results to the case where $\Omega$ is not finitely supported. As applications, we prove versions of Jensen's inequality on Wasserstein space and a generalized Brunn-Minkowski inequality for a random measurable set on a Riemannian manifold.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/66aaada46aadd436d73d9a2a2f147ea45f174b97

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Consistent estimation of a population barycenter in the Wasserstein space (2013, background): “Some generalization of the results in [31] for probability measures defined on a Riemannian manifold, equipped with the Wasserstein metric, have been recently proposed in [30].”
Characterization of barycenters in the Wasserstein space by averaging optimal transport maps (2012, background): “Some generalization of these notions for probability measures defined on a Riemannian manifold, equipped with the Wasserstein metric, have been recently proposed in [34] and [33].”
Bayesian Learning with Wasserstein Barycenters (2018, background): “Let us now recall the Wasserstein barycenter, introduced in [1] and further studied in [35, 25, 28], among others.”