ℓ _1/2,1 group sparse regularization for compressive sensing

Recently, the design of group sparse regularization has drawn much attention in group sparse signal recovery problem. Two of the most popular group sparsity-inducing regularization models are ℓ _1,2 and ℓ _1,∞ regularization. Nevertheless, they do not promote the intra-group sparsity. For example, Huang and Zhang (Ann Stat 38:1978–2004, 2010 ) claimed that the ℓ _1,2 regularization is superior to the ℓ _1 regularization only for strongly group sparse signals. This means the sparsity of intra-group is useless for ℓ _1,2 regularization. Our experiments show that recovering signals with intra-group sparse needs more measurements than those without, by the ℓ _1,∞ regularization. In this paper, we propose a novel group sparsity-inducing regularization defined as a mixture of the ℓ _1/2 norm and the ℓ _1 norm, referred to as ℓ _1/2,1 regularization, which can overcome these shortcomings of ℓ _1,2 and ℓ _1,∞ regularization. We define a new null space property for ℓ _1/2,1 regularization and apply it to establish a recoverability theory for both intra-group and inter-group sparse signals. In addition, we introduce an iteratively reweighted algorithm to solve this model and analyze its convergence. Comprehensive experiments on simulated data show that the proposed ℓ _1/2,1 regularization is superior to ℓ _1,2 and ℓ _1,∞ regularization.