Nonpositive curvature, the variance functional, and the Wasserstein barycenter

TL;DR

This paper explores the relationship between nonpositive curvature of manifolds and the convexity of the variance functional on probability measures, establishing new equivalences and inequalities involving Wasserstein barycenters.

Contribution

It establishes that nonpositive sectional curvature and trivial topology of a manifold are characterized by the displacement convexity of the variance functional on probability measures.

Findings

Variance functional is displacement convex iff the manifold has nonpositive curvature and trivial topology.

A Jensen type inequality for the variance functional with Wasserstein barycenters is proved.

Comparison results between Wasserstein and linear barycenters' variance are derived.

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 $\mathbf{R}^n$) 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 $L^2$ projection to the set of invariant measures.