$$\ell _{1/2,1}$$ 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 $$\ell _{1,2}$$ and $$\ell _{1,\infty }$$ regularization. Nevertheless, they do not promote the intra-group sparsity. For example, Huang and Zhang (Ann Stat 38:1978–2004, 2010) claimed that the $$\ell _{1,2}$$ regularization is superior to the $$\ell _1$$ regularization only for strongly group sparse signals. This means the sparsity of intra-group is useless for $$\ell _{1,2}$$ regularization. Our experiments show that recovering signals with intra-group sparse needs more measurements than those without, by the $$\ell _{1,\infty }$$ regularization. In this paper, we propose a novel group sparsity-inducing regularization defined as a mixture of the $$\ell _{1/2}$$ norm and the $$\ell _{1}$$ norm, referred to as $$\ell _{1/2,1}$$ regularization, which can overcome these shortcomings of $$\ell _{1,2}$$ and $$\ell _{1,\infty }$$ regularization. We define a new null space property for $$\ell _{1/2,1}$$ 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 $$\ell _{1/2,1}$$ regularization is superior to $$\ell _{1,2}$$ and $$\ell _{1,\infty }$$ regularization.