Block sparsity and sampling over a union of subspaces

Sparse signal representations have gained wide popularity in recent years. In many applications the data can be expressed using only a few nonzero elements in an appropriate expansion. In this paper, we study a block-sparse model, in which the nonzero coefficients are arranged in blocks. To exploit this structure, we redefine the standard (NP-hard) sparse recovery problem, based on which we propose a convex relaxation in the form of a mixed ℓ2/ℓ1 program. Isometry-based analysis is used to prove equivalence of the solution to that of the optimal program, under certain mild conditions. We further establish the robustness of our algorithm to mismodeling and bounded noise. We then present theoretical arguments and numerical experiments demonstrating the improved recovery performance of our method in comparison with sparse reconstruction that does not incorporate a block structure. The results are then applied to two related problems. The first is that of simultaneous sparse approximation. Our results can be used to prove isometry-based equivalence properties for this setting. In addition, we propose an alternative approach to acquire the measurements, that leads to performance improvement over standard methods. Finally, we show how our results can be used to sample signals in a finite structured union of subspaces, leading to robust and efficient recovery algorithms.