Nonpositive curvature, the variance functional, and the Wasserstein barycenter

Kim, Young-Heon; Pass, Brendan

doi:10.1090/proc/14840

Nonpositive curvature, the variance functional, and the Wasserstein barycenter
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Proc. Amer. Math. Soc. 148 (2020), 1745-1756 Request permission

Abstract:

We show that a Riemannian manifold $M$ has nonpositive sectional curvature and is simply connected if and only if the variance functional on the space $P(M)$ of probability measures over $M$ is displacement convex. We then establish convexity over Wasserstein barycenters of the variance, and derive an inequality between the variance of the Wasserstein and linear barycenters of a probability measure on $P(M)$. These results are applied to invariant measures under isometry group actions, implying that the variance of the Wasserstein projection to the set of invariant measures is less than that of the $L^2$ projection to the same set.

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