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A Decomposition Approach for the Gain Function in the Feedback Particle Filter

Ruoyu Wang, Huimin Miao, Xue Luo

TL;DR

This work addresses the gain function computation in the Feedback Particle Filter by introducing a decomposition of the Poisson equation into two exactly solvable subproblems under the assumption that the observation is a polynomial. By approximating the posterior with a Gaussian mixture and exploiting Hermite-spectral methods in 1D, the authors derive an explicit, near-linear-time solution for the gain, with a free parameter per particle chosen to ensure solvability. The approach demonstrates improved accuracy over particle-filter variants and kernel-based methods while achieving substantially lower CPU times in 1D tests and benchmark tracking scenarios. Although primarily validated in one dimension, the method shows promise for higher-dimensional problems via Galerkin-based determination of constants, suggesting a path to mitigating the curse of dimensionality in nonlinear filtering.

Abstract

The feedback particle filter (FPF) is an innovative, control-oriented and resampling-free adaptation of the traditional particle filter (PF). In the FPF, individual particles are regulated via a feedback gain, and the corresponding gain function serves as the solution to the Poisson's equation equipped with a probability-weighted Laplacian. Owing to the fact that closed-form expressions can only be computed under specific circumstances, approximate solutions are typically indispensable. This paper is centered around the development of a novel algorithm for approximating the gain function in the FPF. The fundamental concept lies in decomposing the Poisson's equation into two equations that can be precisely solved, provided that the observation function is a polynomial. A free parameter is astutely incorporated to guarantee exact solvability. The computational complexity of the proposed decomposition method shows a linear correlation with the number of particles and the polynomial degree of the observation function. We perform comprehensive numerical comparisons between our method, the PF, and the FPF using the constant-gain approximation and the kernel-based approach. Our decomposition method outperforms the PF and the FPF with constant-gain approximation in terms of accuracy. Additionally, it has the shortest CPU time among all the compared methods with comparable performance.

A Decomposition Approach for the Gain Function in the Feedback Particle Filter

TL;DR

This work addresses the gain function computation in the Feedback Particle Filter by introducing a decomposition of the Poisson equation into two exactly solvable subproblems under the assumption that the observation is a polynomial. By approximating the posterior with a Gaussian mixture and exploiting Hermite-spectral methods in 1D, the authors derive an explicit, near-linear-time solution for the gain, with a free parameter per particle chosen to ensure solvability. The approach demonstrates improved accuracy over particle-filter variants and kernel-based methods while achieving substantially lower CPU times in 1D tests and benchmark tracking scenarios. Although primarily validated in one dimension, the method shows promise for higher-dimensional problems via Galerkin-based determination of constants, suggesting a path to mitigating the curse of dimensionality in nonlinear filtering.

Abstract

The feedback particle filter (FPF) is an innovative, control-oriented and resampling-free adaptation of the traditional particle filter (PF). In the FPF, individual particles are regulated via a feedback gain, and the corresponding gain function serves as the solution to the Poisson's equation equipped with a probability-weighted Laplacian. Owing to the fact that closed-form expressions can only be computed under specific circumstances, approximate solutions are typically indispensable. This paper is centered around the development of a novel algorithm for approximating the gain function in the FPF. The fundamental concept lies in decomposing the Poisson's equation into two equations that can be precisely solved, provided that the observation function is a polynomial. A free parameter is astutely incorporated to guarantee exact solvability. The computational complexity of the proposed decomposition method shows a linear correlation with the number of particles and the polynomial degree of the observation function. We perform comprehensive numerical comparisons between our method, the PF, and the FPF using the constant-gain approximation and the kernel-based approach. Our decomposition method outperforms the PF and the FPF with constant-gain approximation in terms of accuracy. Additionally, it has the shortest CPU time among all the compared methods with comparable performance.

Paper Structure

This paper contains 14 sections, 3 theorems, 34 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Decomposition Method
  4. The decomposition of \ref{['eqn-62']}
  5. The decomposition method in the scalar case
  6. For \ref{['eqn-1']}
  7. For \ref{['eqn-71']}
  8. Boundary condition \ref{['eqn-41']}
  9. Discussions on the decomposition method in high dimensional scenario
  10. Algorithm
  11. Numeric
  12. The comparison of the gain functions
  13. A benchmark example
  14. CONCLUSIONS

Key Result

Proposition III.1

For each $j = 1,\cdots,m$, the gain function $\nabla\varphi_j(\mathbf{x})$ in eqn-pN_phi is given by where $\vec{\lambda}=(\lambda_1,\cdots,\lambda_d)$ represents the eigenvalues of $\Sigma^{-1}$, and $\nabla_{\vec{\lambda}}\psi_j^i(\mathbf{x}):=\left(\frac{1}{\lambda_l}\frac{\partial\psi_j^i(\mathbf{x})}{\partial x_l}\right)_{l = 1}^{d}$. The functions $\varphi_j^i(\mathbf{x})$ and $\psi_j^i(\ma

Figures (5)

  • Figure 1: The coefficients $\hat{K}_k^i$ in \ref{['eqn-92']}, $k=p-1,\cdots,1,0$, are calculated backwardly with the initial values $\hat{K}_p^i=\hat{K}_{p+1}^i\equiv0$ and the convention that $\hat{K}_k^i\equiv0$ when $k<0$.
  • Figure 2: Comparison of the exact gain function and those obtained by the decomposition method with $\epsilon=0.05$, $\epsilon=0.2$, and $\epsilon=1$, respectively.
  • Figure 3: The upper subplot: the relation between CPU times and the degrees of $h(x)$. The lower subplot: the relation between CPU times and the numbers of particles $N_p$.
  • Figure 4: The estimations of the true state (black) obtained by the FPF with the decomposition method (red), the kernel-based approach (yellow), the constant-gain approximation (blue), and the PF (green), respectively.
  • Figure 5: The MSEs averaged over $100$ MC simulations of the FPF with the decomposition method (red), the kernel-based approach (yellow) and the constant-gain approximation (blue), and the PF approximation (green), respectively.

Theorems & Definitions (7)

  • Proposition III.1
  • proof
  • Corollary III.2
  • Theorem III.3: Decomposition method for $d=1$
  • Example IV.1: Section V, TMM:17
  • Example IV.2
  • Example IV.3