Variational Formulation of the Particle Flow Particle Filter

TL;DR

The paper addresses nonlinear Bayesian filtering by linking particle flow methods with variational inference via the Fisher-Rao gradient flow on the space of probability densities.It derives a Gaussian Fisher-Rao flow that reduces to the Exact Daum–Huang flow under linear-Gaussian assumptions and extends to an approximated Gaussian mixture FR flow to capture multimodality.Derivative- and inverse-free formulations with Stein's lemma enable efficient computation and particle preservation, and empirical results on Gaussian mixtures, nonlinear likelihoods, and Bayesian logistic regression show improved posterior approximation and robustness relative to Wasserstein gradient flow and Gaussian-sum PF.This work offers practical variational tools for flexible priors and likelihoods and points toward applications in robotics and Lie-group state estimation.

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.

Figures (8)

• Figure 1: Comparison of the EDH flow and the Gaussian Fisher-Rao flow under linear Gaussian assumptions. We propagate 10 randomly selected particles through both flows. The trajectories of the particles are identical, verifying the results stated in Theorem \ref{['theorem: linear_gaussian_connection']}.
• Figure 2: Comparison of the Gaussian Fisher-Rao particle flow \ref{['gaussian_approx: natural_para_liouville']} with the Gaussian particle flow zhang2024multisensor and the Wasserstein gradient flow lambert2022wass for the Gaussian mixture prior case. For each method, we use a single Gaussian to approximate the posterior density. The covariance contour corresponding to one Mahalanobis distance is overlaid on the reference contour. The top two figures display results generated by Gaussian particle flow with different initializations. The top left figure shows the result with particles generated from ${\cal N}(\hat{\mathbf{x}}^{(1)}, P)$, while the top right figure shows the result with particles generated from ${\cal N}(\hat{\mathbf{x}}^{(2)}, P)$. This method is sensitive to initialization, and as a result, its accuracy is highly dependent on the initial condition. The bottom left figure shows the result generated by Wasserstein gradient flow, which converges to the mode with the largest component weight. The bottom right figure shows the result generated by our Gaussian Fisher-Rao particle flow, which achieves the lowest approximated KL divergence.
• Figure 3: Comparison of the approximated Gaussian mixture Fisher-Rao particle flow \ref{['gaussian_mixture_approx: natural_para_liouville']} with the Gaussian sum particle flow zhang2024multisensor and the Wasserstein gradient flow lambert2022wass for the Gaussian mixture prior case. The approximated posterior contour is shown for each method. The top left figure shows the reference contour. The top right figure shows the contour generated by Gaussian sum particle flow, which does not capture the four components of the reference contour and achieves the highest KL divergence. The bottom left figure shows the contour generated by Wasserstein gradient flow, which captures the locations of the four components of the reference contour but fails to capture the component weights. The bottom right figure shows the contour generated by the approximated Gaussian mixture Fisher-Rao particle flow, which captures both the locations and the weights of the four components of the reference contour, and as a result, achieves the lowest KL divergence.
• Figure 4: Comparison of expectation evaluations of the $V(\mathbf{x})$ function \ref{['fisher_rao: kl_kernel']} for the Gaussian mixture prior case. In each figure, the results obtained by Stein's method \ref{['dif: derivative_free']} with Gauss-Hermite particles of degree $32$ and resampled particles are represented by solid blue and dashed orange lines. The results obtained by analytical calculations with Gauss-Hermite particles of degree $4$, Gauss-Hermite particles of degree $32$, and resampled particles are represented by dashed green, dotted red, and solid purple lines, respectively. The top figure shows the difference in the expected $V(\mathbf{x})$ function, the middle plot shows the difference in the expected gradient of $V(\mathbf{x})$ function $\mathbb{E}\left[ \partial V(\mathbf{x}) / \partial \mathbf{x}^{\top} \right]$, and the bottom plot depicts the difference in the expected Hessian of $V(\mathbf{x})$ function $\mathbb{E}\left[ \partial^2 V(\mathbf{x}) / \partial \mathbf{x}^{\top} \partial \mathbf{x} \right]$. The maximum difference between the compared and reference methods across all Gaussian mixture components is reported. The reference method uses Stein's gradient and Hessian with Gauss-Hermite particles of degree $4$.
• Figure 5: Comparison of the Gaussian Fisher-Rao particle flow \ref{['gaussian_approx: natural_para_liouville']} with the Gaussian particle flow zhang2024multisensor and the Wasserstein gradient flow lambert2022wass for the nonlinear observation model case. For each method, we use a single Gaussian to approximate the posterior density. The covariance contour corresponding to one Mahalanobis distance is overlaid on the reference contour. The left figure shows the result obtained by the Gaussian particle flow method, which produces an approximation that is slightly misaligned with the region of highest posterior probability, leading to a higher approximated KL divergence. The middle figure shows the result obtained by the Wasserstein gradient flow method, which fails to accurately capture the region of the highest posterior probability, resulting in the highest approximated KL divergence. The right figure shows the result obtained by the Gaussian Fisher-Rao particle flow, which provides the most accurate Gaussian approximation of the posterior density, resulting in the lowest approximated KL divergence.

Theorems & Definitions (11)

• Lemma 1: EDH Flow is Exact
• Proposition 2: Gradient of KL Divergence
• Theorem 3: Transient Density as a Solution to Fisher-Rao Gradient Flow
• Lemma 4: Gaussian Fisher-Rao Particle Flow
• Theorem 5: EDH Flow as Fisher-Rao Particle Flow
• Proposition 6: Approximated Gaussian Mixture Fisher-Rao Parameter Flow
• Proposition 7: Approximated Gaussian Mixture Fisher-Rao Particle Flow
• Remark 8
• Theorem 9: Mahalanobis Distance is Invariant
• Theorem 10: Fisher-Rao Particle Flow as Gauss-Hermite Transform