Gaussian variational Bayes Kalman filtering for dynamic sparse Bayesian learning

Sparse Baysesian Learning (SBL) provides sophisticated (state) model order selection with unknown support distribution. This allows to handle problems with big state dimensions and relatively limited data. The techniques proposed in this paper allow to handle the extension of SBL to time-varying states, modeled as diagonal first-order vector autoregressive (VAR(1)) processes with unknown parameters. Adding the parameters to the state leads to an augmented state and a non-linear (at least bilinear) state-space model. The proposed approach, which applies also to more general non-linear models, uses Variational Bayes (VB) techniques to approximate the posterior distribution by a factored form, with Gaussian or exponential factors. The granularity of the factorization can take on various levels. In one extreme instance, called Gausian Space Alternating Variational Estimation Kalman Filtering (GSAVE-KF), all state components are treated individually, leading to low complexity filtering. Simulations illustrate the performance of the proposed GVB-KF techniques, which represent an alternative to Linear MMSE (LMMSE) filtering.