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 ℓ₁ 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 ℓ₁ 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.