Newton-Flow Particle Filters based on Generalized Cramér Distance

2025-08-29

AI Summary

In high-dimensional particle filtering, severe particle degeneracy and loss of sample diversity due to resampling severely degrade estimation performance. Method: We propose a non-degenerating recursive particle filter that introduces homotopy continuation to gradually embed the likelihood into an artificial time dimension, and employs Newton flow to continuously optimize particle positions along this path. A differentiable, rank-invariant closed-form expression for the generalized Cramér distance is derived to enable smooth particle migration from prior to posterior. The entire algorithm operates on a low-discrepancy deterministic particle set, eliminating the need for density estimation or resampling. Contribution/Results: The method requires only prior particles and the likelihood function—no tuning or auxiliary approximations—yielding an extremely lightweight, plug-and-play architecture. It achieves stable and efficient state estimation in high-dimensional settings, circumventing degeneracy entirely while preserving sample diversity throughout recursion.

Technology Category

• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Probabilistic Programming
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Developing a recursive particle filter for high-dimensional problems
• Addressing particle degeneracy through deterministic low-discrepancy sets
• Smoothly transitioning particles from prior to posterior using Newton flow

Innovation

• Newton flow minimizes Cramér distance between particle sets
• Homotopy continuation gradually introduces likelihood function
• Deterministic low-discrepancy particles prevent filter degeneration