## 2 Related Work and Methods
### Norm  
For the matrix \(A\in R^{d\times n}\), the \(l_{q,p}\)-norm is characterized as  
\[\|A\|_{p,q}=\left(\sum_{i=1}^{n}\big{(}\sum_{i=1}^{d}\mid a_{ij \mid}\,^{p}\big{)}^{q/p}\right)\frac{1}{\mathfrak{q}} \tag{11}\]  
where \(a_{ij}\) represents the \(j\)-th column vector in row \(i\) of matrix \(A\). Two commonly used norms are \(\ell_{2,1}\)-norm and Frobenius norm (\(\ell_{2,2}\)).  
In order to gain a deeper understanding of the role and principles of the \(\ell_{2,1}\)-norm, [PERSON] and [PERSON] [29] introduced a novel vector outlier regularization (VOR) function and then found that the outlier in \(\|X-UV\|_{2,1}\) has much less impact on the final result than the square error. In other words, the \(\ell_{2,1}\)-norm is insensitive to noise and anomalies. For multi-class classification, [PERSON] et al. [2] used the \(\ell_{2,1}\)-norm to increase sparsity. [PERSON] et al. [30] proposed co-regularization unsupervised feature selection (CUFS). CUFS applies the \(\ell_{2,1}\)-norm to ensure sparsity in the projection matrix and cluster base matrix, which hold data reconstruction and cluster structure. [PERSON] et al. [31] introduced the robust structured non-negative matrix factorization (RSNMF), which incorporates both the global and local structures of the data. This method tackles the challenges posed by noise and outliers by employing the \(\ell_{2,1}\)-NMF optimization loss function. Furthermore, it enhances sparsity in the basis matrix through \(\ell_{2,1}\)-regularization. The incorporation of \(\ell_{2,1}\)-regularization improves the robustness of the algorithm suggested in this study.  
A great multitude of computational studies have shown that sparser solutions can be obtained by using the \(\ell_{p}\)-norm (\(0<p<1\)) in contrast to using the \(\ell_{1}\)-norm. Therefore,Wang et al. [32] further proved that the minimum solution based on generalized \(\ell_{2,p}\)-norm (\(p\in(0,1)\)) has greater sparsity compared to that grounded in \(\ell_{2,1}\)-norm and proposed a method to optimize the \(\ell_{2,p}\)-norm (\(p\in(0,1)\)) that is non-convex and non-Lipschitz continuous, which proved its optimality and convergence. Since then, we have discovered the good performance of \(\ell_{2,p}\)-norm (\(p\in(0,1)\)). [PERSON] et al. [33] found that in the minimization process, \(\ell_{2,1/2}\) and \(\ell_{2,1-2}\) can obtain sparser solutions compared with \(\ell_{2,1}\) and \(\ell_{2,2}\). As shown in Figure 2, in the unsupervised feature selection proposed by [PERSON] and [PERSON] [34] based on non-negative spectrum analysis and redundant control, \(\ell_{2,p}\)-norm constraints are imposed on the prediction matrix so that the proposed model is suitable for feature selection, and noisy or irrelevant features can be discarded to a certain extent. The semi-supervised NMF learning framework proposed by [PERSON] and [PERSON] [34] learns robust discriminant representation using block diagonal structure and applying \(\ell_{2,p}\)-norm (especially when \(0<p\leq 1\)) to the loss function. It is found that implementing the \(\ell_{2,p}\)-norm on the cost function can effectively address issues related to noise and outliers, especially when \(0<p<1\) and q = 2. [PERSON] et al. [35] adopted the \(\ell_{2,1/2}\)-norm regularization to induce sparsity of the matrix, thereby enhancing the stability of the algorithm and heightening the accuracy of clustering. For the same purpose, [PERSON] et al. [26] used \(\ell_{2,1/2}\)-norm for the cost function of non-negative matrix decomposition. [PERSON] et al. [36] used \(\ell_{2,1/2}\)-norm to enhance matrix sparsity in the peer-to-peer projection matrix to make better feature selection.  
In summary, \(\ell_{2,p}\)-norm (\(p\in(0,1)\)) can not only make the solution more sparse but also eliminate the impact of noise and outliers, so the robustness of the algorithm is enhanced. Different \(p\) values can be selected according to task requirements and characteristics, and \(\ell_{2,1/2}\)-norm is commonly used at present.