The Advective Fisher-Rao Geometry of Deterministic Measure Transport

2026-08-12

AI Summary

This work addresses the absence of a natural geometric structure on paths of probability measures—constrained by the continuity equation—that is amenable to optimization. The authors propose a novel convection Fisher–Rao metric and, for the first time, establish its connection to measure transport optimization. They reveal its geometric essence through three complementary perspectives: its emergence as a zero-noise limit, its characterization as the expected form of the second variation of a large deviation rate functional, and its identification with the Hessian of the dynamical optimal transport action functional. Integrating tools from information geometry, large deviation theory, and variational analysis, the theoretical development—supported by numerical experiments—demonstrates that this metric effectively enables optimal fitting of probability densities, with the Gauss–Newton method proving particularly well-suited for optimizing the associated velocity fields.

Abstract

A novel advective Fisher-Rao metric is introduced for optimization tasks on paths of probability measures governed by the continuity equation. This metric is shown to lead to optimal descent directions. It is then shown that this metric arises naturally from three different perspectives: As the rescaled zero-noise limit of the Fisher-Rao metric on path measures, as the expected value of the second variation of the Freidlin--Wentzell large deviation rate functional, and as the Hessian of the Benamou--Brenier action functional from dynamic optimal transport. We supplement this geometric construction with computational experiments. Here, we demonstrate empirically that the advective Fisher-Rao metric yields the desired optimal fitting of probability densities, whereas the Gauss--Newton method yields optimal fitting of velocity fields.

Problem

• Advective Fisher-Rao metric
• probability measures
• continuity equation
• optimal transport
• optimization

Innovation

• advective Fisher-Rao metric
• continuity equation
• dynamic optimal transport
• large deviation theory
• geometric optimization