Block-Sparse Recovery With Optimal Block Partition

This paper presents a convex recovery method for block-sparse signals whose block partitions are unknown a priori. We first introduce a nonconvex penalty function, where the block partition is adapted for the signal of interest by minimizing the mixed
$\ell _{2}/\ell _{1}$
norm over all possible block partitions. Then, by exploiting a variational representation of the
$\ell _{2}$
norm, we derive the proposed penalty function as a suitable convex relaxation of the nonconvex one. For a block-sparse recovery model designed with the proposed penalty, we develop an iterative algorithm which is guaranteed to converge to a globally optimal solution. Numerical experiments demonstrate the effectiveness of the proposed method.