Multivariable Feedback Particle Filter

TL;DR

This paper advances the feedback particle filter by developing new algorithms and theoretical results for multivariable systems, improving gain computation, and connecting to MCMC and reinforcement learning methods.

Contribution

It introduces new representations and algorithms for computing the filter gain in multivariable systems, with theoretical guarantees and practical finite-element methods.

Findings

Established multivariate consistency of the FPF.

Expressed the gain as a gradient solving Poisson's equation.

Demonstrated the finite-element algorithm's effectiveness in numerical experiments.

Abstract

In recent work it is shown that importance sampling can be avoided in the particle filter through an innovation structure inspired by traditional nonlinear filtering combined with Mean-Field Game formalisms. The resulting feedback particle filter (FPF) offers significant variance improvements; in particular, the algorithm can be applied to systems that are not stable. The filter comes with an up-front computational cost to obtain the filter gain. This paper describes new representations and algorithms to compute the gain in the general multivariable setting. The main contributions are, (i) Theory surrounding the FPF is improved: Consistency is established in the multivariate setting, as well as well-posedness of the associated PDE to obtain the filter gain. (ii) The gain can be expressed as the gradient of a function, which is precisely the solution to Poisson's equation for a related MCMC diffusion (the Smoluchowski equation). This provides a bridge to MCMC as well as to approximate optimal filtering approaches such as TD-learning, which can in turn be used to approximate the gain. (iii) Motivated by a weak formulation of Poisson's equation, a Galerkin finite-element algorithm is proposed for approximation of the gain. Its performance is illustrated in numerical experiments.