Analysis of the feedback particle filter with diffusion map based approximation of the gain

TL;DR

This paper analyzes the feedback particle filter with a diffusion map-based gain approximation, providing insights into its form, well-posedness, and convergence, advancing understanding of nonlinear Bayesian filtering methods.

Contribution

It offers an analytic characterization of the approximate gain in feedback particle filters using diffusion maps, and establishes conditions for their well-posedness and convergence.

Findings

Analytic form of the approximate gain derived

Conditions for well-posedness established

Convergence of finite N system to mean field limit proven

Abstract

Control-type particle filters have been receiving increasing attention over the last decade as a means of obtaining sample based approximations to the sequential Bayesian filtering problem in the nonlinear setting. Here we analyse one such type, namely the feedback particle filter and a recently proposed approximation of the associated gain function based on diffusion maps. The key purpose is to provide analytic insights on the form of the approximate gain, which are of interest in their own right. These are then used to establish a roadmap to obtaining well-posedness and convergence of the finite $N$ system to its mean field limit. A number of possible future research directions are also discussed.