An Optimal Transport Formulation of the Linear Feedback Particle Filter

TL;DR

This paper formulates the feedback particle filter as an optimal transport problem to derive a unique control law, reducing simulation variance and improving the approximation of the posterior distribution in nonlinear filtering.

Contribution

It introduces an optimal transport-based formulation for the feedback particle filter, providing a new method to obtain a unique control law with reduced variance.

Findings

The optimal control law replaces the noise term with a deterministic term.

The new formulation decreases simulation variance.

Numerical examples demonstrate improved performance.

Abstract

Feedback particle filter (FPF) is an algorithm to numerically approximate the solution of the nonlinear filtering problem in continuous time. The algorithm implements a feedback control law for a system of particles such that the empirical distribution of particles approximates the posterior distribution. However, it has been noted in the literature that the feedback control law is not unique. To find a unique control law, the filtering task is formulated here as an optimal transportation problem between the prior and the posterior distributions. Based on this formulation, a time stepping optimization procedure is proposed for the optimal control design. A key difference between the optimal control law and the one in the original FPF, is the replacement of noise term with a deterministic term. This difference serves to decreases the simulation variance, as illustrated with a simple numerical example.