Gradient Flows in Filtering and Fisher-Rao Geometry

TL;DR

This paper explores the geometric interpretation of filtering as gradient flows on probability density manifolds, focusing on Fisher-Rao geometry, and extends previous work from Wasserstein to Fisher-Rao metrics.

Contribution

It introduces a new formulation of filtering as proximal operators with respect to Fisher-Rao metric, providing deeper geometric insights and extending prior Wasserstein-based approaches.

Findings

Derived evolution equations as proximal operators in Fisher-Rao geometry

Detailed development of linear Gaussian case

Clarified implications of different metric choices in filtering

Abstract

Uncertainty propagation and filtering can be interpreted as gradient flows with respect to suitable metrics in the infinite dimensional manifold of probability density functions. Such a viewpoint has been put forth in recent literature, and a systematic way to formulate and solve the same for linear Gaussian systems has appeared in our previous work where the gradient flows were realized via proximal operators with respect to Wasserstein metric arising in optimal mass transport. In this paper, we derive the evolution equations as proximal operators with respect to Fisher-Rao metric arising in information geometry. We develop the linear Gaussian case in detail and show that a template two step optimization procedure proposed earlier by the authors still applies. Our objective is to provide new geometric interpretations of known equations in filtering, and to clarify the implication of different choices of metric.