Newton-Flow Particle Filters based on Generalized Cramér Distance
This addresses the degeneracy issue in particle filters for high-dimensional problems, offering a potentially more robust alternative to classic approaches.
The authors tackled the problem of particle filter degeneracy in high-dimensional state estimation by developing a Newton-flow particle filter that uses deterministic particle sets and homotopy continuation to smoothly transition from prior to posterior densities. The result is a filter that never degenerates, is simple to implement, and requires only a prior particle set and likelihood function without density estimation.
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.