Wasserstein barycenters over Riemannian manifolds

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• 作者：Young-Heon Kim^a ; ^{yhkim@math.ubc.ca} ; Brendan Pass^b ; ^{pass@ualberta.ca}
• 关键词：Optimal Transport ; Riemannian manifolds ; Wasserstein barycenters ; Multi-marginal optimal transport ; Ricci curvature bounds ; Jensen's inequality ; Random Brunn&ndash ; Minkowski inequality
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：5 February 2017
• 年：2017
• 卷：307
• 期：Complete
• 页码：640-683
• 全文大小：708 K
• 卷排序：307

文摘

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 Ω∈P(P(M))Ω∈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 [1] to the Riemannian setting, and on the other hand, results in the Riemannian case of Cordero-Erausquin, McCann, Schmuckenschläger [12] for barycenters between two measures to the multi-marginal setting. Our work also extends these results to the case where Ω 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.