Generalized range space property for group sparsity of linear underdetermined systems

Abstract

Group sparse vectors are the generalization of unstructured sparse vectors, with zero and non-zero elements occurring in groups. In group sparse signal recovery problems, we are interested in finding the signal with the smallest number of active groups that satisfy our observations. Because this problem has exponential complexity, a convex relaxation is typically used, which minimizes the sum of the groups' second norm that satisfies the observations. In this paper, we provide a set of deterministic necessary and sufficient conditions that the sensing matrix should satisfy for equivalence between the ℓ0 solution of a structured group sparse problem and its convex relaxation. These conditions are generalization of the Range Space Property that has been previously proposed for equivalence between the ℓ0- and the ℓ1- norms in non-structured sparse recovery problems. We also provide a sufficient condition for a unique solution of the relaxed convex problem.