Variational Formulation of Particle Flow

TL;DR

This paper introduces a variational inference perspective on particle flow, deriving a Fisher-Rao gradient flow that unifies and extends existing particle filtering methods with Gaussian and mixture approximations.

Contribution

It formulates particle flow as a Fisher-Rao gradient flow in variational inference, deriving Gaussian and mixture approximations that generalize existing particle filtering techniques.

Findings

Fisher-Rao gradient flow aligns with variational inference principles.

Gaussian approximation reduces to classical particle flow under linear Gaussian assumptions.

Mixture approximation enhances model expressiveness for multi-modal distributions.

Abstract

This paper provides a formulation of the log-homotopy particle flow from the perspective of variational inference. We show that the transient density used to derive the particle flow 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. When considering a parametric family of variational densities, the function space Fisher-Rao gradient flow simplifies to the natural gradient flow of the variational density parameters. By adopting a Gaussian variational density, we derive a Gaussian approximated Fisher-Rao particle flow and show that, under linear Gaussian assumptions, it reduces to the Exact Daum and Huang particle flow. Additionally, we introduce a Gaussian mixture approximated Fisher-Rao particle flow to enhance the expressive power of our model through a multi-modal variational density. Simulations on low- and high-dimensional estimation problems illustrate our results.