Functional Bayesian Filter

TL;DR

The paper introduces a novel nonlinear Bayesian filtering method using RKHS that learns system dynamics directly from data, outperforming traditional Kalman filters in high-dimensional, nonlinear, and noisy scenarios.

Contribution

It develops a fully data-driven Bayesian filter in RKHS, capable of handling nonlinear systems without predefined models, with demonstrated superior performance.

Findings

Outperforms existing Kalman-based algorithms in simulations

Effective in chaotic time-series estimation and inverse kinematics

Handles Gaussian and non-Gaussian noise robustly

Abstract

We present a general nonlinear Bayesian filter for high-dimensional state estimation using the theory of reproducing kernel Hilbert space (RKHS). Applying kernel method and the representer theorem to perform linear quadratic estimation in a functional space, we derive a Bayesian recursive state estimator for a general nonlinear dynamical system in the original input space. Unlike existing nonlinear extensions of Kalman filter where the system dynamics are assumed known, the state-space representation for the Functional Bayesian Filter (FBF) is completely learned from measurement data in the form of an infinite impulse response (IIR) filter or recurrent network in the RKHS, with universal approximation property. Using positive definite kernel function satisfying Mercer's conditions to compute and evolve information quantities, the FBF exploits both the statistical and time-domain information about the signal, extracts higher-order moments, and preserves the properties of covariances without the ill effects due to conventional arithmetic operations. This novel kernel adaptive filtering algorithm is applied to recurrent network training, chaotic time-series estimation and cooperative filtering using Gaussian and non-Gaussian noises, and inverse kinematics modeling. Simulation results show FBF outperforms existing Kalman-based algorithms.