Block sparse vector recovery via weighted generalized range space property

In block sparse vector recovery problems we are interested in finding the vector with the least number of active blocks that best describes the observation. The convex relaxation of that problem, typically used to reduce complexity, is strictly equivalent with the original problem only when certain conditions are met, such as Restricted Isometry Property, Null Space Characterization, and Block Mutual Coherence. In practice, those conditions may not be satisfied, which implies that solving the relaxed problem may not retrieve the block sparsest solution. In this paper, we propose a weighted approach, which, in the noise free case and under certain conditions guarantees that the relaxed problem solution has the same support as the sparsest block vector. The weights can be obtained based on a low resolution estimate of the group sparse signal.