Sharp failure rates for the bootstrap particle filter in high dimensions

TL;DR

This paper rigorously demonstrates that in high-dimensional Gaussian particle filters, importance weights tend to collapse unless the ensemble size grows exponentially with the system dimension, highlighting fundamental limitations of the bootstrap particle filter.

Contribution

It weakens previous assumptions on eigenvalues and proves the conjecture that weight collapse occurs unless ensemble size grows exponentially with system dimension.

Findings

Maximum importance weight converges to unity without exponential ensemble growth

Weight collapse occurs in high dimensions unless ensemble size is exponentially large

Allows for a system dominated by a single mode under certain eigenvalue growth restrictions

Abstract

We prove that the maximum of the sample importance weights in a high-dimensional Gaussian particle filter converges to unity unless the ensemble size grows exponentially in the system dimension. Our work is motivated by and parallels the derivations of Bengtsson, Bickel and Li (2007); however, we weaken their assumptions on the eigenvalues of the covariance matrix of the prior distribution and establish rigorously their strong conjecture on when weight collapse occurs. Specifically, we remove the assumption that the nonzero eigenvalues are bounded away from zero, which, although the dimension of the involved vectors grow to infinity, essentially permits the effective system dimension to be bounded. Moreover, with some restrictions on the rate of growth of the maximum eigenvalue, we relax their assumption that the eigenvalues are bounded from above, allowing the system to be dominated by a single mode.