Huygens' equations and the gradient-flow equations in information geometry

TL;DR

This paper explores the connection between gradient-flow equations and Hamilton's equations within information geometry, revealing their geometric interpretation and relation to geodesic flows and replicator equations.

Contribution

It introduces a geometric framework linking gradient flows to Hamiltonian and geodesic flows in information geometry, using Huygens' equations and Jacobi-Maupertuis transformation.

Findings

Gradient flows are related to geodesic flows via Huygens' equations.

The evolution parameter in gradient flows corresponds to arc-length in Riemannian geometry.

A new relation between gradient-flow equations and replicator equations is established.

Abstract

We revisit the relation between the gradient-flow equations and Hamilton's equations in information geometry. By regarding the gradient-flow equations as Huygens' equations in geometric optics, we have related the gradient flows to the geodesic flows induced by the geodesic Hamiltonian in an appropriate Riemannian geometry. The original evolution parameter $\textit{t}$ in the gradient-flow equations is related to the arc-length parameter in the associated Riemannian manifold by Jacobi-Maupertuis transformation. As a by-product, it is found the relation between the gradient-flow equation and replicator equations.