The Riemannian geometry of Sinkhorn divergences

TL;DR

This paper introduces a Riemannian-like metric on probability measures based on Sinkhorn divergences, revealing geometric properties and limitations, and connecting optimal transport with RKHS structures.

Contribution

It defines a new Riemannian metric on probability measures derived from Sinkhorn divergences, linking optimal transport geometry with RKHS theory and analyzing its properties.

Findings

The metric is geodesic and metrizes the weak-star topology.

Translations are geodesics for quadratic cost on .

Sinkhorn divergence is not jointly convex and its square root is not a true distance.

Abstract

We propose a new metric between probability measures on a compact metric space that mirrors the Riemannian manifold-like structure of quadratic optimal transport but includes entropic regularization. Its metric tensor is given by the Hessian of the Sinkhorn divergence, a debiased variant of entropic optimal transport. We precisely identify the tangent space it induces, which turns out to be related to a Reproducing Kernel Hilbert Space (RKHS). As usual in Riemannian geometry, the distance is built by looking for shortest paths. We prove that our distance is geodesic, metrizes the weak-star topology, and is equivalent to a RKHS norm. Still it retains the geometric flavor of optimal transport: as a paradigmatic example, translations are geodesics for the quadratic cost on $\mathbb{R}^d$. We also show two negative results on the Sinkhorn divergence that may be of independent interest: that it is not jointly convex, and that its square root is not a distance because it fails to satisfy the triangle inequality.