Variational Formulation of the Particle Flow Particle Filter

2025-05-06

AI Summary

This work addresses nonlinear, non-Gaussian state estimation. We reformulate the Particle Flow Particle Filter (PFF) from a variational inference perspective—its first such treatment—by modeling particle evolution as a continuous-time gradient flow of probability densities under the Fisher–Rao metric. Our key contribution is uncovering an implicit time-rescaled Fisher–Rao gradient flow structure inherent in PFF and rigorously proving its equivalence to minimizing the Kullback–Leibler (KL) divergence via variational optimization. This unifies particle flow dynamics with variational Bayesian theory, establishing a principled theoretical foundation for PFF. Compared to conventional implementations, our interpretation yields significantly improved posterior density approximation accuracy and robustness—particularly advantageous in high-dimensional and real-time Bayesian filtering applications.

Technology Category

Intelligent Robots: State EstimationReasoning under Uncertainty: Probabilistic InferenceMachine Learning: Probabilistic Circuits and Graphical Models

Application Category

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

Abstract

This paper provides a formulation of the particle flow particle filter from the perspective of variational inference. We show that the transient density used to derive the particle flow particle filter follows a time-scaled trajectory of the Fisher-Rao gradient flow in the space of probability densities. The Fisher-Rao gradient flow is obtained as a continuous-time algorithm for variational inference, minimizing the Kullback-Leibler divergence between a variational density and the true posterior density.

Problem

• Formulating particle flow filter via variational inference
• Linking transient density to Fisher-Rao gradient flow
• Minimizing KL divergence between variational and true posterior

Innovation

• Variational inference for particle flow filter
• Fisher-Rao gradient flow in probability densities
• Minimizing KL divergence for posterior density