Learning Distributional Parameters for Adaptive Bayesian Sparse Signal Recovery

Power Exponential Scale Mixture (PESM), a generalized scale mixture family of distributions, has been recently proposed to model the sparsity inducing prior distributions currently in use for Sparse Signal Recovery (SSR). In this paper, we review this generalized scale mixture family and establish the necessary and sufficient condition for a distribution (symmetric with respect to origin) to have a PESM representation, which is a generalization of the results previously known for the Gaussian Scale Mixture (GSM) family. On the algorithmic front, we propose an adaptive Bayesian Sparse Signal Recovery (B-SSR) framework by learning the distributional parameters of a Generalized t-distribution (GT) which belongs to the PESM family. For specific choice of distributional parameters of GT our proposed framework corresponds to popular sparse recovery algorithms such as, LASSO, Reweighted ,1 norm minimization, Reweighted ,2 norm minimization etc. The tail nature of GT distribution family is extensively studied in this paper and an adaptive algorithm has been proposed where the tail nature of the prior is adapted over iterations based on the observation. Extensive experimental results based on traditional SSR setup have also been presented to show the efficacy of this adaptive approach.