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Normalizing Flow-based Differentiable Particle Filters

Xiongjie Chen, Yunpeng Li

TL;DR

This work introduces NF-DPF, a normalizing-flow-based differentiable particle filter that learns flexible, density-backed dynamic, proposal, and measurement models for state-space estimation. By employing (conditional) normalizing flows and an entropy-regularized optimal transport resampler, NF-DPF yields tractable densities for all components and provable consistency as the number of particles grows. Theoretical results establish convergence rates and consistency, while extensive experiments on 1D and multivariate LGSSMs, disk localization, and robot localization tasks demonstrate improved posterior approximation, tracking accuracy, and sample efficiency over existing differentiable filters. The approach enables parameter learning and robust state estimation in complex, high-dimensional environments, with potential for broader deployment in navigation, robotics, and perception.

Abstract

Recently, there has been a surge of interest in incorporating neural networks into particle filters, e.g. differentiable particle filters, to perform joint sequential state estimation and model learning for non-linear non-Gaussian state-space models in complex environments. Existing differentiable particle filters are mostly constructed with vanilla neural networks that do not allow density estimation. As a result, they are either restricted to a bootstrap particle filtering framework or employ predefined distribution families (e.g. Gaussian distributions), limiting their performance in more complex real-world scenarios. In this paper we present a differentiable particle filtering framework that uses (conditional) normalizing flows to build its dynamic model, proposal distribution, and measurement model. This not only enables valid probability densities but also allows the proposed method to adaptively learn these modules in a flexible way, without being restricted to predefined distribution families. We derive the theoretical properties of the proposed filters and evaluate the proposed normalizing flow-based differentiable particle filters' performance through a series of numerical experiments.

Normalizing Flow-based Differentiable Particle Filters

TL;DR

This work introduces NF-DPF, a normalizing-flow-based differentiable particle filter that learns flexible, density-backed dynamic, proposal, and measurement models for state-space estimation. By employing (conditional) normalizing flows and an entropy-regularized optimal transport resampler, NF-DPF yields tractable densities for all components and provable consistency as the number of particles grows. Theoretical results establish convergence rates and consistency, while extensive experiments on 1D and multivariate LGSSMs, disk localization, and robot localization tasks demonstrate improved posterior approximation, tracking accuracy, and sample efficiency over existing differentiable filters. The approach enables parameter learning and robust state estimation in complex, high-dimensional environments, with potential for broader deployment in navigation, robotics, and perception.

Abstract

Recently, there has been a surge of interest in incorporating neural networks into particle filters, e.g. differentiable particle filters, to perform joint sequential state estimation and model learning for non-linear non-Gaussian state-space models in complex environments. Existing differentiable particle filters are mostly constructed with vanilla neural networks that do not allow density estimation. As a result, they are either restricted to a bootstrap particle filtering framework or employ predefined distribution families (e.g. Gaussian distributions), limiting their performance in more complex real-world scenarios. In this paper we present a differentiable particle filtering framework that uses (conditional) normalizing flows to build its dynamic model, proposal distribution, and measurement model. This not only enables valid probability densities but also allows the proposed method to adaptively learn these modules in a flexible way, without being restricted to predefined distribution families. We derive the theoretical properties of the proposed filters and evaluate the proposed normalizing flow-based differentiable particle filters' performance through a series of numerical experiments.
Paper Structure (37 sections, 7 theorems, 97 equations, 7 figures, 6 tables, 1 algorithm)

This paper contains 37 sections, 7 theorems, 97 equations, 7 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Statement
  4. Preliminaries
  5. Particle Filtering
  6. Differentiable particle filters
  7. Normalizing flows
  8. Normalizing flow-based differentiable particle filters
  9. Dynamic models with normalizing flows
  10. Proposal distributions with conditional normalizing flows
  11. Conditional normalizing flow measurement models
  12. Importance weights update
  13. Theoretical analysis
  14. Experiments
  15. One-dimensional Linear Gaussian State-Space Models
  16. ...and 22 more sections

Key Result

Proposition 5.1

For a bounded weight function $\omega_t(x_t)=p(y_t|x_t;\theta):\mathcal{X}\rightarrow\mathbb{R}$ and a measurable bounded $k$-Lipschitz function $\psi(\cdot):\mathcal{X}\rightarrow\mathbb{R}$, when the regularization coefficient in entropy-regularized optimal transport resampler $\epsilon_N=o(1/\log (replacing ${\beta}^{(t-1)}f$ by the initial distribution $\pi(x_0,\theta)$ at time $t=0$ defined i

Figures (7)

  • Figure 1: A diagram that shows the overall structure of the proposed NF-DPF, illustrating how to generate new particles and update particle weights in NF-DPFs. Blue circles refer to random variables. Green rectangles refer to operations such as drawing samples or evaluating certain functions.
  • Figure 2: Evaluation metrics of different methods evaluated on the validation set with 1000 sequences. (a) $L^2$-norm between the true parameter set and the estimated parameter sets. (b) $L^2$-norm of posterior mean error evaluated on validation set. (c) ELBO evaluated on validation set. (d) Effective sample size on validation set. Lower parameter estimation error, posterior mean error, higher effective sample size, and ELBO indicate better performance. The shaded area represents the standard deviation of the presented evaluation metrics among 50 random simulations.
  • Figure 3: Evaluation metrics of different methods evaluated on a test set with 1000 sequences. (a) $L^2$-norm between the true parameter set and the estimated parameter sets. (b) $L^2$-norm of posterior mean error evaluated on test set. (c) ELBO evaluated on test set. (d) Effective sample size on test set. Lower parameter estimation error, posterior mean error, higher effective sample size, and ELBO indicate better performance. The reported results are the mean of evaluation metrics computed over 50 random simulations.
  • Figure 4: (A) An example of observation images. (B) RMSE of different methods evaluated at selected time steps on test set. (C) RMSE of different differentiable particle filters on the validation set during training. Shaded areas represent the standard deviation of the presented evaluation metrics among 5 random simulations.
  • Figure 5: Ablation studies conducted in the disk localization experiment to investigate how each individual component of the NF-DPF affects its performance. (a) RMSE of variants of the NF-DPF evaluated at selected time steps on test set. (b) RMSE of different variants of the NF-DPF on the validation set during training. Shaded areas represent the standard deviation of the presented evaluation metrics among 5 random simulations.
  • ...and 2 more figures

Theorems & Definitions (13)

  • Proposition 5.1
  • Lemma B.1
  • proof
  • Lemma B.2
  • proof
  • Lemma B.3
  • Proposition B.1
  • proof
  • Lemma B.4
  • proof
  • ...and 3 more