Sparsity constrained estimation via spike and slab priors

Abstract

We address sparse signal, i.e. image recovery in a Bayesian estimation framework where sparsity is enforced on reconstruction coefficients via probabilistic priors. In particular, we focus on the popular spike and slab prior which is considered the gold standard in the statistics literature. The optimization problem resulting from this model has broad applicability in recovery, regression and classification problems and is known to be a hard non-convex problem whose existing solutions involve simplifying assumptions and/or relaxations. We propose an approach called Iterative Convex Refinement (ICR) that aims to solve the aforementioned optimization problem directly allowing for greater generality in the sparse structure. Essentially, ICR solves a sequence of convex optimization problems such that sequence of solutions converges to a sub-optimal solution of the original hard optimization problem. Applications will be considered in image classification as well as image reconstruction.