How to avoid the curse of dimensionality: scalability of particle filters with and without importance weights

TL;DR

This paper investigates the scalability issues of particle filters in high-dimensional nonlinear filtering problems, highlighting the limitations of importance weights and demonstrating that a feedback particle filter can avoid the curse of dimensionality.

Contribution

The paper analyzes the impact of dimensionality on particle filter degeneracy and shows that a feedback particle filter based on optimal control can bypass the curse of dimensionality.

Findings

Standard particle filters require exponentially increasing samples with dimension.

Weight degeneracy occurs faster as dimension increases.

Feedback particle filter does not exhibit the curse of dimensionality.

Abstract

Particle filters are a popular and flexible class of numerical algorithms to solve a large class of nonlinear filtering problems. However, standard particle filters with importance weights have been shown to require a sample size that increases exponentially with the dimension D of the state space in order to achieve a certain performance, which precludes their use in very high-dimensional filtering problems. Here, we focus on the dynamic aspect of this curse of dimensionality (COD) in continuous time filtering, which is caused by the degeneracy of importance weights over time. We show that the degeneracy occurs on a time-scale that decreases with increasing D. In order to soften the effects of weight degeneracy, most particle filters use particle resampling and improved proposal functions for the particle motion. We explain why neither of the two can prevent the COD in general. In order to address this fundamental problem, we investigate an existing filtering algorithm based on optimal feedback control that sidesteps the use of importance weights. We use numerical experiments to show that this Feedback Particle Filter (FPF) by Yang et al. (2013) does not exhibit a COD.