_1/2, 1ℓ1/2, 1 group sparse regularization for compressive sensing. Issue 5 (July 2016) - Liu, Shengcai; Zhang, Jiangshe; Liu, Junmin; Yin, Qingyan
Abstract 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 _1, 2 ℓ 1, 2 and _1, ∞ 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 _1, 2 ℓ 1, 2 regularization is superior to the _1 ℓ 1 regularization only for strongly group sparse signals. This means the sparsity of intra-group is useless for _1, 2 ℓ 1, 2 regularization. Our experiments show that recovering signals with intra-group sparse needs more measurements than those without, by the _1, ∞ ℓ 1, ∞ regularization. In this paper, we propose a novel group sparsity-inducing regularization defined as a mixture of the _1/2 ℓ 1 / 2 norm and the _1 ℓ 1 norm, referred to as _1/2, 1 ℓ 1 / 2, 1 regularization, which can overcome these shortcomings of _1, 2 ℓ 1, 2 and _1, ∞ ℓ 1, ∞ regularization. We define a new null space property for _1/2, 1 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 _1/2, 1 ℓ 1 / 2, 1 regularization is superior to _1, 2 ℓ 1, 2 and _1, ∞ ℓ 1, ∞ regularization.