A Convex Penalty for Block-Sparse Signals with Unknown Structures

We propose a novel convex penalty for block-sparse signals whose block partitions are unknown a priori. We first introduce a nonconvex penalty function, where the block partition is adjusted for the signal of interest by minimizing the mixed ℓ
2
/ℓ
1
norm over all possible block partitions. Then, by exploiting a variational representation of the ℓ
2
norm, we derive the proposed penalty function as a suitable convex relaxation of the nonconvex penalty. For the resulting regularization model, we provide a proximal splitting-based algorithm which is guaranteed to converge to an optimal solution. Numerical experiments show the effectiveness of the proposed penalty.