## 2 Related Work and Methods
### Dual-Graph Regularization  
The presence of sparsity in high-dimensional data implies the abundance of structural information within. It is crucial to explore the concealed information. Typically, the Laplacian matrix is commonly utilized to maintain the local structural information inherent in the data. The main idea is that if two data points, \(x_{i}\) and \(x_{j}\), in the original space, are near to one another, then the two data points in the projected subspace should also demonstrate proximity. The following expression can be used to convey this idea:  
\[\min_{x}\frac{1}{2}\frac{n}{i}\Big{|}\Big{|}W^{T}\textbf{x}_{i}-W^{T}\textbf{ x}_{j}\Big{|}\Big{|}_{2}^{2}S_{ij}=Tr(W^{T}XL^{x}X^{T}W) \tag{4}\]  
In this equation, \(L^{X}\) is the Laplacian matrix of the data matrix \(X\), while the similarity between sample \(x_{i}\) and \(x_{j}\) is denoted as \(S_{ij}\). \(L\) is obtained using the similarity matrix \(S\) and the degree matrix \(D\):  
\[L=D-S \tag{5}\]  
where \(D\) is a diagonal matrix, defined as follows:  
\[D=diag[\sum_{j=1}^{d}\left(S_{1j}+S_{j1}\right)/2,\sum_{j=1}^{d}\left(S_{2j}+S _{j2}\right)/2,\ldots,\sum_{j=1}^{d}\left(S_{nj}+S_{jn}\right)/2] \tag{6}\]  
In this paper, the similarity is formulated by utilizing the Gaussian kernel function. \[S_{ij}=\left\{ \begin{array}{l}\exp(-\big{|}\big{|}x_{i}-x_{j}\big{|}\big{|} _{2}^{2}),if\ x_{j}\in\aleph(x_{j})\ or\ x_{j}\in\aleph(x_{i})\\ 0,else\end{array} \right. \tag{7}\]  
For \(X\in R^{d\times n}\), \(\{x_{1},x_{2},\ldots,x_{n}\}\) is a collection of n data points, with each being a column of \(X\), and \(\{x_{1},x_{2},\ldots,x_{d}\}\) is the row of \(X\), representing a set of d features. \(\aleph(x_{i})\) represents the collection of the K nearest neighbors of x. Nevertheless, the aforementioned methods primarily focus on preserving the manifold structure of the data space alone. Inspired by the duality between features (e.g., words) and data instances (e.g., documents)--where data instances can be characterized by their distributions over features, and features can be grouped based on their distributions across instances--researchers have proposed methods that capture more comprehensive structural information. For example, [PERSON] and [PERSON] [16] introduced a Dual Regularized Co-Clustering (DRCC) technique based on semi-non-negative matrix tri-factorization, employing two graph structures to retain the geometric relationships of both data and feature manifolds. Their experimental results demonstrated improved clustering performance. [PERSON] et al. [17] incorporated dual-graph regularization into non-negative matrix factorization (NMF), proposing the dual-graph regularization NMF (DNMF) algorithm to better exploit intrinsic structural information. [PERSON] et al. [27] proposed a dual-graph regularized low-rank representation (DGLRR) model by integrating dual-graph constraints into a low-dimensional representation framework. [PERSON] et al. [28] further developed a dual-graph regularization feature selection approach, combining non-negative spectral learning with sparse regression (NSSRD). A growing body of work highlights that much real-world high-dimensional data--including its features--can be viewed as residing on nonlinear low-dimensional manifolds embedded in a high-dimensional space. Dual-graph regularization, which simultaneously preserves both the data and feature manifold structures, has shown great potential to enhance learning performance. However, its application remains largely confined to matrix factorization techniques and is seldom extended to other frameworks. Equations (3) and (4) are the adjacency matrices of corresponding data and features. Then, \(\aleph(X_{j})\) represents the collection of K neighbors. \[S_{ij}^{n}=\left\{ \begin{array}{l}\exp(-\big{|}|x_{i.i}-x_{j.i}\big{|}\big{|}_{2}^{2},if\ x_{i.i}\in\aleph(x_{j.i})\ or\ x_{j.i}\in \aleph(x_{i.i})\\ 0,else\end{array} \right. \tag{8}\] \[S_{ij}^{d}=\left\{