Newton-Flow Particle Filters based on Generalized Cramér Distance

TL;DR

The paper addresses the degeneration-prone nature of high-dimensional particle filters by introducing a degeneration-free, fully differentiable filtering framework. It combines three core ideas: progressive likelihood introduction via homotopy continuation over an artificial time $\gamma$, deterministic low-discrepancy sampling (Dirac mixtures), and a Newton-flow update driven by a generalized Cramér-von Mises distance to move particles from the prior to the posterior. The methodology yields closed-form gradients and Hessians, enabling a mapping-free, recursive particle flow that preserves equal weights and can serve as a plugin replacement for traditional filter steps, with broad applicability to different density representations. Practically, the approach promises simpler implementation, improved robustness in high dimensions, and the potential for end-to-end learning by leveraging the differentiable nature of the flow.

Abstract

We propose a recursive particle filter for high-dimensional problems that inherently never degenerates. The state estimate is represented by deterministic low-discrepancy particle sets. We focus on the measurement update step, where a likelihood function is used for representing the measurement and its uncertainty. This likelihood is progressively introduced into the filtering procedure by homotopy continuation over an artificial time. A generalized Cramér distance between particle sets is derived in closed form that is differentiable and invariant to particle order. A Newton flow then continually minimizes this distance over artificial time and thus smoothly moves particles from prior to posterior density. The new filter is surprisingly simple to implement and very efficient. It just requires a prior particle set and a likelihood function, never estimates densities from samples, and can be used as a plugin replacement for classic approaches.