On Variational Block Sparse Recovery With Unknown Partition and $\ell _{0}$-Norm Constraint

This letter considers the estimation of block sparse signal with unknown partition. We take a variational approach to separate the estimates of support partition and signal amplitudes. Instead of using any surrogate, we propose to apply the exact
$\ell _{0}$
-norm constraint to promote the desired block sparsity. As for the companion algorithmic design, we exploit the ADMM framework to iteratively update our partition and signal estimates. Our main contribution lies in the novel convergence guarantee of the proposed non-convex and non-smooth ADMM algorithm. The insight brought up by our theoretical analysis sheds light on other programs involving
$\ell _{0}$
-norm constraints. The superior performance of the proposed method is empirically demonstrated by extensive experiments with the state-of-art competing methods.