Calculus of Variations and Geometric Measure Theory

H. Lavenant - J. Luckhardt - G. Mordant - B. Schmitzer - L. Tamanini

The Riemannian geometry of Sinkhorn divergences

created by tamanini1 on 09 May 2024



Inserted: 9 may 2024

Year: 2024

ArXiv: 2405.04987 PDF


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.