Block-sparse solutions using kernel block RIP and its application to Group Lasso

Abstract

We propose kernel block restricted isometry property (KB-RIP) as a generalization of the well-studied RIP and prove a variety of results. First, we present a "sum-of-norms"-minimization based formulation of the sparse recovery problem and prove that under suitable conditions on KB-RIP, it recovers the optimal sparse solution exactly. The Group Lasso formulation, widely used as a good heuristic, arises naturally from the Lagrangian relaxation of our formulation. We present an efficient combinatorial algorithm for provable sparse recovery under similar assumptions on KB-RIP. This result improves the previously known assumptions on RIP under which a combinatorial algorithm was known. Finally, we provide numerical evidence to illustrate that not only are our sum-of-norms-minimization formulation and combinatorial algorithm significantly faster than Lasso, they also outperforms Lasso in terms of recovery. Copyright 2011 by the authors.