Hessian transport gradient flows

TL;DR

This paper introduces a novel class of gradient flows in probability spaces based on Hessian-transported Riemannian metrics, generalizing Fokker-Planck equations and linking to stochastic differential equations for various divergence measures.

Contribution

It develops a new framework for gradient flows using Hessian-transported metrics, extending the theory of divergence-based flows and their stochastic counterparts.

Findings

Derived new gradient flows for divergence functions

Connected gradient flows to stochastic differential equations

Presented examples for multiple divergence measures

Abstract

We derive new gradient flows of divergence functions in the probability space embedded with a class of Riemannian metrics. The Riemannian metric tensor is built from the transported Hessian operator of an entropy function. The new gradient flow is a generalized Fokker-Planck equation and is associated with a stochastic differential equation that depends on the reference measure. Several examples of Hessian transport gradient flows and the associated stochastic differential equations are presented, including the ones for the reverse Kullback--Leibler divergence, alpha-divergence, Hellinger distance, Pearson divergence, and Jenson--Shannon divergence.