Gradient Flows of Potential Energies in the Geometry of Sinkhorn Divergences

TL;DR

This paper studies the gradient flow of potential energies in probability measures using Sinkhorn divergences, proving well-posedness, stability, and convergence, with numerical illustrations of the flow's properties.

Contribution

It introduces a novel gradient flow framework based on Sinkhorn divergences, extending optimal transport geometry and analyzing its long-term behavior and numerical aspects.

Findings

Proved well-posedness and stability of the Sinkhorn-based gradient flow.

Showed convergence of the modified JKO scheme to the flow as time step vanishes.

Numerical experiments demonstrate unique properties of the new gradient flow.

Abstract

We analyze the gradient flow of a potential energy in the space of probability measures when we substitute the optimal transport geometry with a geometry based on Sinkhorn divergences, a debiased version of entropic optimal transport. This gradient flow appears formally as the limit of the minimizing movement scheme, a.k.a. JKO scheme, when the squared Wasserstein distance is substituted by the Sinkhorn divergence. We prove well-posedness and stability of the flow, and that, in the long term, the energy always converges to its minimal value. The analysis is based on a change of variable to study the flow in a Reproducing Kernel Hilbert Space, in which the evolution is no longer a gradient flow but described by a monotone operator. Under a restrictive assumption we prove the convergence of our modified JKO scheme towards this flow as the time step vanishes. We also provide numerical illustrations of the intriguing properties of this newly defined gradient flow.