## 1 Introduction  
In recent decades, sparse reconstruction has become an active topic in many areas, such as in fields of signal processing, statistics, and machine learning [1]. By reconstructing a sparse solution from a linear measurement, we can obtain a certain expression of high-dimensional data as a vector with only a small number of nonzero entries. In practical applications, the data of interest can often be assumed to have a certain special structure. For example, in microarray analysis of gene expression [2, 3], hyperspectral image unmixing [4, 5, 6], force identification in industrial applications [7], classification problems [8, 9, 10, 11, 12, 13], etc., the solution of interest often possesses group-sparsity structure, namely the solution has a natural grouping of its coefficients and nonzero entries only occur in few groups. This paper focuses on the estimation of group sparse solution, which is related to the Group LASSO (least absolute shrinkage and selection operator) [14]. Suppose \(\mathbf{x}=[\mathbf{x}_{1}^{\top},\mathbf{x}_{2}^{\top},\cdots,\mathbf{x}_{g}^{\top}]^{\top} \in\mathbb{R}^{n}\) is a group sparse signal, where \(\mathbf{x}_{i}\in\mathbb{R}^{n_{i}}\), \(\sum_{i=1}^{g}n_{i}=n\) and \(g\) is the number of groups. Just as with the use of \(\ell_{0}\) pseudo-norm for evaluation of the sparsity, the group sparsity of \(\mathbf{x}\) can be evaluated with the \(\ell_{2,0}\) pseudo-norm, i.e., \(\|\mathbf{x}\|_{2,0}=\big{\|}\big{(}\|\mathbf{x}_{1}\|_{2},\|\mathbf{x}_{2}\|_{2},\cdots,\| \mathbf{x}_{g}\|_{2}\big{)}\big{\|}_{\mathcal{Y}}\), where \(\|\cdot\|_{2}\) is the Euclidean norm, and \(\|\cdot\|_{0}\) is the \(\ell_{0}\) pseudo-norm which counts the number of nonzero entries in the vector in \(\mathbb{R}^{g}\). The group sparse regularized least squares problem can be modeled as  
\[\underset{\mathbf{x}\in\mathbb{R}^{n}}{\text{minimize}}\ \frac{1}{2}\|\mathbf{y}-\mathbf{A}\mathbf{x} \|_{2}^{2}+\lambda\|\mathbf{x}\|_{2,0}, \tag{1}\]where \(\mathbf{y}\in\mathbb{R}^{m}\) and \(\mathbf{A}\in\mathbb{R}^{m\times n}\) are known, and \(\lambda>0\) is the regularization parameter. However, the employment of the pseudo-norm \(\ell_{2,0}\) makes (1) NP-hard [15]. Most studies in the application replace the nonconvex regularizer \(\ell_{2,0}\) with its tightest convex envelope \(\ell_{2,1}\)[16] (or its weighted variants), and the following regularized least squares problem has been proposed known as the Group LASSO [14],  
\[\underset{\mathbf{x}\in\mathbb{R}^{n}}{\text{minimize}}\ \frac{1}{2}\|\mathbf{y}-\mathbf{A} \mathbf{x}\|_{2}^{2}+\lambda\sum_{i=1}^{g}w_{i}\|\mathbf{x}_{i}\|_{2}, \tag{2}\]  
where \(w_{i}>0\) (\(i=1,\cdots,g\)) in the regularization term  
\[\|\mathbf{x}\|_{w,2,1}:=\sum_{i=1}^{g}w_{i}\|\mathbf{x}_{i}\|_{2} \tag{3}\]  
(this is \(\ell_{w,2,1}\)-norm of \(\mathbf{x}\), i.e., a separable weighted version [17] of \(\ell_{2,1}\)-norm \(\|\mathbf{x}\|_{2,1}=\sum_{i=1}^{g}\|\mathbf{x}_{i}\|_{2}\)) are used to adjust for group sizes with \(w_{i}=\sqrt{n_{i}}\) in [14; 18]. We give a simple but clear explanation in Appendix A, to show the bias of \(\ell_{2,1}\)-norm caused by group size in the application of group sparse classification (GSC). Although the convex optimization problem (2) has been used as a standard model for group sparse estimation applications, the convex regularizer \(\ell_{w,2,1}\) does not necessarily promote group sparsity sufficiently, mainly due to the fact that \(\ell_{w,2,1}\)-norm is just an approximation of \(\ell_{2,0}\) pseudo-norm within the severe restriction of the convexity. To promote the group sparsity more effectively than convex regularizers, nonconvex regularizers such as group SCAD (smoothly clipped absolute deviation) [3], group MCP (minimax concave penalty) [18; 19], \(\ell_{p,q}\) regularization (\(\|\mathbf{x}\|_{p,q}:=\left(\sum_{i=1}^{g}\|\mathbf{x}_{i}\|_{p}^{q}\right)^{1/q}\), \(0<q<1\leq p\)) [20], iterative weighted group minimization [21], and \(\ell_{2,0}\)[22] have been used for group sparse estimation problems.