## 2 Preliminaries
### Basic Ide of Weighted Group Sparse Classification (WGSC)  
As a relatively simple but typical scenario for the application of the proposed idea in this paper, we introduce the main idea of weighted group sparse classification (WGSC). Classification is one of fundamental tasks in the field of the signal and image processing and pattern recognition. For a classification problem with \(g\) classes of subjects, the training samples can formulate a dictionary matrix \(\mathbf{A}=[\mathbf{A}_{1},\mathbf{A}_{2},\cdots,\mathbf{A}_{g}]\in\mathbb{R}^{m\times n}\), where \(\mathbf{A}_{i}=[\mathbf{a}_{i1},\mathbf{a}_{i2},\cdots,\mathbf{a}_{in_{i}}]\in\mathbb{R}^{m \times n_{i}}\) is the subset of the training samples from subject \(i\), \(\mathbf{a}_{ij}\) is the \(j\)-th training sample from the \(i\)-th class, \(n_{i}\) is the number of training samples from class \(i\), and \(n=\sum_{i=1}^{g}n_{i}\) is the number of total training samples. The aim is to correctly determine which class the input test sample \(\mathbf{y}\in\mathbb{R}^{m}\) belongs to. Although deep learning is very popular and powerful for classification tasks, it requires a very large-scale training set and computation resources for numerous parameters training with complicated back-propagation. [PERSON] et al. proposed the sparse representation-based classification (SRC) [34] for face recognition. With the assumption that samples of a specific subject lie in a linear subspace, a valid test sample \(\mathbf{y}\) is expected to be approximated well by a linear combination of the training samples from the same class, which leads to a sparse representation coefficient over all training samples. Specifically, the test sample \(\mathbf{y}\) is approximated by the linear combination of the dictionary items, i.e., \(\mathbf{y}\approx\mathbf{A}\mathbf{x}\), where \(\mathbf{x}\) is the coefficient vector. A simple minimization model with sparse representation can be \(\text{minimize}_{\mathbf{x}\in\mathbb{R}^{n}}\)\(\frac{1}{2}\|\mathbf{y}-\mathbf{A}\mathbf{x}\|_{2}^{2}+\lambda\|\mathbf{x}\|_{0}\). In most SRC-based approaches, \(\ell_{0}\) regularizer is relaxed to \(\ell_{1}\), and the model becomes the well-known LASSO model [35] in statistics. The label information of the dictionary atoms is not used in the simple model of SRC, hence the regression is based solely on the structure of each sample. When the subspaces spanned by different classes are not independent, SRC may lead the test image to be represented by training samples from multiple different classes. Considering ideal situation where the test image should only be approximated well by the training samples from one class corresponding to the correct one, in [8; 9; 10], the authors divided training samples into groups by prior label information and used group-sparsity regularizers. Naturally, the coefficient vector \(\mathbf{x}\) has group structure \(\mathbf{x}=[\mathbf{x}_{1}^{\top},\mathbf{x}_{2}^{\top},\cdots,\mathbf{x}_{g}^{\top}]^{\top} \in\mathbb{R}^{n}\), where \(\mathbf{x}_{i}=[x_{i1},x_{i2},\cdots,x_{in_{i}}]^{\top}\in\mathbb{R}^{n_{i}}\)\((i=1,2,\cdots,g)\). This kind of group sparse classification (GSC) approach aims to represent the test image using the minimum number of groups, and thus an ideal model is (1) which is NP-hard. As stated in Section 1, a convex approximation of \(\ell_{2,0}\), i.e., \(\ell_{2,1}\)-norm, has been used widely as a best convex regularizer to incorporate the class labels. More generally, the non-separable weighted \(\ell_{2,1}\)-norm, i.e., \(\|\mathbf{W}\cdot\|_{2,1}\) has also been used as the regularizer in GSC [11; 36; 37]. For example, [PERSON] et al.