A nonconvex ADMM algorithm for group sparsity with sparse groups

We present an efficient algorithm for computing sparse representations whose nonzero coefficients can be divided into groups, few of which are nonzero. In addition to this group sparsity, we further impose that the nonzero groups themselves be sparse. We use a nonconvex optimization approach for this purpose, and use an efficient ADMM algorithm to solve the nonconvex problem. The efficiency comes from using a novel shrinkage operator, one that minimizes nonconvex penalty functions for enforcing sparsity and group sparsity simultaneously. Our numerical experiments show that combining sparsity and group sparsity improves signal reconstruction accuracy compared with either property alone. We also find that using nonconvex optimization significantly improves results in comparison with convex optimization.