Gradient systems and asymmetric relaxations in view of Riemannian geometry

TL;DR

This paper generalizes the relationship between gradient flows and geodesics from flat to general Riemannian manifolds, providing new criteria for comparing relaxation trajectories and illustrating applications to Gaussian chains.

Contribution

It extends the geometric framework of gradient flows to non-flat Riemannian manifolds without flatness or symmetry constraints, broadening the scope of information geometry.

Findings

Established a criterion for comparing relaxation along different gradient curves.

Applied the framework to Gaussian chains, confirming the universal asymmetry in relaxation.

Revealed new geometric insights for optimization and stochastic processes.

Abstract

In dually flat manifolds, there is a deep connection between gradient flows and pregeodesics. This was one of the many important contributions of Amari to information geometry. In this paper, we extend the study of this relationship to general Riemannian manifolds. Our result does not impose conditions of flatness on the connection or symmetry on its non-metricity tensor, thus broadening the geometric setting beyond Hessian manifolds. Within this framework, we provide a criterion for comparing relaxation along two different gradient descent curves of a function, formulated in terms of the non-metricity tensor of a connection for which the gradient curves are pregeodesics. We use it to study Gaussian chains, whose relaxation trajectories coincide with gradient descent curves in the space of Gaussian distributions.Thus, we recover a recent result that establishes a universal asymmetry: warming up is faster than cooling down. Our work illustrates how geometric insights rooted in Amari's legacy offer new perspectives for optimization problems and stochastic processes.