Can local particle filters beat the curse of dimensionality?

TL;DR

This paper explores the potential of local particle filtering algorithms to overcome the curse of dimensionality by leveraging decay of correlations, providing error bounds that are uniform in time and model dimension.

Contribution

It introduces a framework for developing dimension-free local particle filters based on decay of correlations, with theoretical error bounds.

Findings

Error bounds are uniform in time and dimension under certain assumptions.

Local particle filters can potentially mitigate the curse of dimensionality.

The decay of correlations is key to the stability of high-dimensional filtering.

Abstract

The discovery of particle filtering methods has enabled the use of nonlinear filtering in a wide array of applications. Unfortunately, the approximation error of particle filters typically grows exponentially in the dimension of the underlying model. This phenomenon has rendered particle filters of limited use in complex data assimilation problems. In this paper, we argue that it is often possible, at least in principle, to develop local particle filtering algorithms whose approximation error is dimension-free. The key to such developments is the decay of correlations property, which is a spatial counterpart of the much better understood stability property of nonlinear filters. For the simplest possible algorithm of this type, our results provide under suitable assumptions an approximation error bound that is uniform both in time and in the model dimension. More broadly, our results provide a framework for the investigation of filtering problems and algorithms in high dimension.