A decomposition method in the multivariate feedback particle filter via tensor product Hermite polynomials
Ruoyu Wang, Xue Luo
TL;DR
This work extends a scalar decomposition of the Poisson equation in the feedback particle filter (FPF) to the multivariate setting by leveraging tensor-product Hermite polynomials in a Galerkin framework and a weighted-radial construction for the auxiliary subproblem.By splitting the multivariate equation into two exactly solvable subproblems, the authors derive explicit gain functions $K$ with a provable structure (Theorem 3.5) and establish invertibility of critical coefficient matrices in several practical cases.The approach yields a polynomial-in-dimension computational complexity and demonstrates improved accuracy and efficiency over EKF, PF, and kernel/constant-gain FPF in ship-tracking and Lorenz-oscillator benchmarks, illustrating strong potential for high-dimensional nonlinear filtering.These results provide a practical, scalable alternative for gain computation in multivariate FPF, enabling accurate, resampling-free filtering with explicit, analyzable gain functions.
Abstract
The feedback particle filter (FPF), a resampling-free algorithm proposed over a decade ago, modifies the particle filter (PF) by incorporating a feedback structure. Each particle in FPF is regulated via a feedback gain function (lacking a closed-form expression), which solves a Poisson's equation with a probability-weighted Laplacian. While approximate solutions to this equation have been extensively studied in recent literature, no efficient multivariate algorithm exists. In this paper, we focus on the decomposition method for multivariate gain functions in FPF, which has been proven efficient for scalar FPF with polynomial observation functions. Its core is splitting the Poisson's equation into two exactly solvable sub-equations. Key challenges in extending it to multivariate FPF include ensuring the invertibility of the coefficient matrix in one sub-equation and constructing a weighted-radial solution in the other. The proposed method's computational complexity grows at most polynomially with the state dimension, a dramatic improvement over the exponential growth of most particle-based algorithms. Numerical experiments compare the decomposition method with traditional methods: the extended Kalman filter (EKF), PF, and FPF with constant-gain or kernel-based gain approximations. Results show it outperforms PF and FPF with other gain approximations in both accuracy and efficiency, achieving the shortest CPU time among methods with comparable performance.