The ℓ2, regularized group sparse optimization: Lower bound theory, recovery bound and algorithms

Abstract

In this paper, we consider an unconstrained ℓ 2 , q minimization for group sparse signal recovery. For this nonconvex and non-Lipschitz problem, we mainly focus on its local minimizers. Firstly, a uniform lower bound for nonzero groups of the local minimizers is presented. Secondly, under group restricted isometry property (GRIP) assumption, we provide a global recovery bound for points in a sublevel set of the objective function, as well as a local recovery bound for local minimizers. Thirdly, a sufficient condition for a stationary point to be a local minimizer is shown. Fourthly, inspired by the lower bound theory which indicates the sparsity of solutions, we propose a new efficient iteratively reweighted least square (IRLS) with thresholding algorithm, with nonexpansiveness of the group support set. Compared with the classical IRLS with smoothing algorithm, our algorithm performs better in both theoretical global convergence guarantee and numerical computation.