Approximating the zero-norm penalized sparse signal recovery using a hierarchical Bayesian framework

Abstract

Sparse Bayesian learning (SBL) with its self-regulation and uncertainty estimation features, has become a popular topic in the field of sparse signal recovery. However, it is a challenging problem to employ zero-norm penalty on signals in the hierarchical framework of SBL. In this paper, an SBL approach that approximates the zero-norm penalized spare signal recovery is proposed based on the idea that signals can be modeled using generalized Gaussian distribution (GGDs). It should be noted that the mean and variance of GGDs are related to its shape parameters, making the direct use of GGDs in Bayesian inference impractical. To solve this problem, we derives a proposition and further build a new hierarchical Bayesian framework (HrBayFw). The proposed HrBayFw is equivalent to assign ℓp-norm penalty to signals, and the parameter p can be designed to approximate 0. Thus, the zero-norm penalized spare signal recovery can be realized using SBL, and the use of the proposition enables tractable marginalization over all parameters. The main advantage of the proposed approach is that it achieves a lower reconstruction error than current Bayesian methods. Numerical simulations evidence that the proposed SBL approach achieves better accuracy performance in terms of normalized-mean-square-error comparing with other contemporary algorithms.