Hierarchical Sparsity Within And Across Overlapping Groups

Recently, different penalties have been proposed for signals whose non-zero coefficients reside in a small number of groups, where within each group, only few of the coefficients are active. In this paper, we extend such a penalty, and introduce an additional layer of grouping on the coefficients. Specifically, we first partition the signal into groups, and then apply the penalty on the $\ell _{2}$ norms of the groups. We discuss how this extended penalty can be used in energy minimization formulations, and demonstrate the effects of the proposed extension on a dereverberation experiment.