Aspects of a Generalized Theory of Sparsity based Inference in Linear Inverse Problems

Abstract: Linear inverse problems are ubiquitous in various science and engineering disciplines. Of particular importance in the past few decades, is the incorporation of sparsity based priors, in particular $\ell_1$ priors, into linear inverse problems, which led to the flowering of fields of compressive sensing (CS) and sparsity based signal processing. More recently, methods based on a Compound Gaussian (CG) prior have been investigated and demonstrate improved results over CS in practice. This paper is the first attempt to identify and elucidate the fundamental structures underlying the success of CG methods by studying CG in the context of a broader framework of generalized-sparsity-based-inference. After defining our notion of generalized sparsity we introduce a weak null space property and proceed to generalize two well-known methods in CS, basis pursuit and iteratively reweighted least squares (IRLS). We show how a subset of CG-induced regularizers fits into this framework.