Functional Bayesian Filter
This provides a novel method for state estimation in fields like robotics or signal processing, though it appears incremental as an extension of kernel methods to filtering.
The authors tackled the problem of high-dimensional state estimation in nonlinear dynamical systems by developing a Functional Bayesian Filter (FBF) that learns state-space representations from data using reproducing kernel Hilbert spaces, and simulation results show it outperforms existing Kalman-based algorithms.
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.