In this paper, we consider an unconstrained ${\ell }_{2,q}$ 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.

在本文中，我们认为不受约束 ${\ell }_{2，q}$最小化组稀疏信号恢复。对于此非凸和非Lipschitz问题，我们主要关注其局部极小值。首先，给出了局部最小化器的非零组的统一下界。其次，在组受限等距特性（GRIP）假设下，我们为目标函数的子级集中的点提供了全局恢复范围，并为局部极小值提供了局部恢复范围。第三，示出了足以使固定点成为局部极小值的条件。第四，受表示解决方案稀疏性的下界理论的启发，我们提出了一种新的有效的迭代加权最小二乘（IRLS）阈值化算法，其组支持集不扩张。与具有平滑算法的经典IRLS相比，