Nonpositive curvature, the variance functional, and the Wasserstein barycenter

2015/03/22 by Kim, Young-Heon, Pass, Brendan · 2 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1503.06460

Abstract

This paper connects nonpositive sectional curvature of a Riemannian manifold with the displacement convexity of the variance functional on the space P(M) of probability measures over M. We show that M has nonpositive sectional curvature and has trivial topology (i.e, is homeomorphic to Rⁿ) if and only if the variance functional on P(M) is displacement convex. This is followed by a Jensen type inequality for the variance functional with respect to Wasserstein barycenters, as well as by a result comparing the variance of the Wasserstein and linear barycenters of a probability measure on P(M) (that is, an element of P(P(M))). These results are applied to invariant measures under isometry group actions, giving a comparison for the variance functional between the Wasserstein projection and the L² projection to the set of invariant measures.

Cited by

• Concavity of spacetimes
• Sectional curvature and matrix displacement convexity

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