A fast variational approach for Bayesian compressive sensing with informative priors

The Sparse Bayesian learning (SBL) framework has been successfully adopted for sparse signal recovery. In SBL inference can be performed either via Type-II Maximum Likelihood or by following a Variational approach. When employing uninformative prior distributions, fast algorithms have been proposed for both renditions of SBL and it has been proven that they are equivalent. Unfortunately the use of such priors prohibits the incorporation of prior statistical information which can be beneficial in terms of convergence and accuracy. A modified variational approach is proposed, resulting in a fast variational algorithm for informative priors. A fixed point analysis is performed with the major challenge being the highly involved analytical expressions for the points in the fixed set. The given theoretical analysis demonstrates how this issue can be circumvented. Comprehensive empirical results are given to support the claims.