Nonpositive curvature, the variance functional, and the Wasserstein barycenter

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 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.

Optimal transport and barycenters for dendritic measures

“…• The vertical variance R d ×R ≥0 |y− ȳ| 2 dµ(x, y) = R ≥0 |y− ȳ| 2 dµ V (y), a measure the spread of the mass in the vertical direction [13]. • The vertical internal energy R ≥0 (f V (y)) r dy for r ≥ 1.…”