# Newton-Flow Particle Filters based on Generalized Cram\'er Distance

## TL;DR

This paper introduces a novel recursive particle filter that uses a generalized Cramér distance and Newton flow to efficiently update particles in high-dimensional spaces without degeneracy, improving stability and simplicity.

## Contribution

The paper presents a new particle filtering method based on a differentiable generalized Cramér distance and Newton flow, avoiding density estimation and ensuring non-degeneracy in high dimensions.

## Key findings

- The filter is simple to implement and efficient.
- It never degenerates in high-dimensional problems.
- It can replace classic particle filters as a plugin.

## 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 Cram\'er 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.