A framework for low-complexity signal recovery and its application to structured sparsity

We study the problem of estimating an unknown vector from noisy underdetermined observations, with recovery guarantees. In such context, a regularity model on the unknown is needed to obtain recovery guarantees. We show that we can guarantee the recovery of generic models (cones) with the minimization of an arbitrary regularizer subject to a data-fit constraint (generalized robust basis pursuit) under a restricted isometry property (RIP) hypothesis on the observations. In the classical cases of sparse vectors and low rank matrix recovery, our framework yields sharp recovery guarantees. For the more refined model of structured sparsity in levels, our framework extends and improves existing RIP recovery guarantees.