## 1 Introduction
As previously discussed, high-dimensional data often exhibit significant noise, redundancy, and complex intrinsic correlations, making direct analysis and modeling in the original data space highly challenging. In response to these issues, increasing attention has been paid to feature selection algorithms that can effectively preserve the inherent structure of the data, as such preservation not only enhances feature selection performance but also improves model interpretability. To address structural preservation, [PERSON] et al. [13] presented MCFS, which combines sparse regression with spectral analysis to retain the local structural information of the original data. Building upon similar principles, [PERSON] et al. [14] introduced NDFS, which integrates spectral clustering with feature selection to capture the intrinsic geometric structure. [PERSON] et al. [13] also proposed GNMF, which constructs a nearest-neighbor graph to preserve the manifold structure. Further advancements include [PERSON] and [PERSON] [15], who incorporated spectral analysis of projected data into an autoencoder framework, enabling the compression of high-dimensional data into a low-dimensional space while maintaining local structure. Similarly, [PERSON] et al. [10] developed SLREO, which combines sparse latent representation with extended OLSDA. This method integrates OLSDA into a non-negative manifold structure to generate pseudo-labels, ensuring both structure preservation and non-negativity of the labels. Despite their effectiveness, these approaches primarily focus on preserving the manifold structure in the data space, often neglecting the structural information embedded in the feature space. To overcome this limitation, researchers have proposed methods that jointly preserve structures in both data and feature spaces. For example, [PERSON] and [PERSON] [16] introduced DRCC, while [PERSON] et al. [17] presented DNMF; both approaches simultaneously consider the geometric structures of the sample manifold and the feature manifold, leading to more comprehensive structure-aware feature selection. Furthermore, norm-based constraints are commonly employed on matrices to enhance robustness, suppress noise, and improve the sparsity of the selection matrix. For instance, [PERSON] et al. [18] presented NSSLFS, which incorporates \(\ell_{1}\)-norm and \(\ell_{2,1}\)-norm into the objective functions to achieve sparsity and robustness. [PERSON] et al. [17] applied an \(\ell_{2,1}\)-norm constraint to the residual matrix of latent representation learning to eliminate redundant connectivity information and enhance feature discriminability. [PERSON] et al. [18] introduced RDGDNMF, which integrates the \(\ell_{2,1/2}^{1/2}\)-norm into the cost function of NMF. This not only enforces sparsity in the basis matrix but also significantly improves the model's robustness against noise and outliers. Some feature selection methods are inherently limited by their reliance on linear transformations. In contrast, autoencoders have emerged as powerful tools in various applications due to their strong nonlinear mapping capabilities. As a result, many researchers have explored autoencoder-based approaches for unsupervised feature selection. For instance, [PERSON] et al. [19] utilized an autoencoder network to compress raw features into a low-dimensional space representation and selected features based on their contribution to this representation. [PERSON] et al. [20] enhanced feature selection performance by jointly optimizing the structure of the autoencoder and incorporating regularization and redundancy control penalties. [PERSON] et al. [5] proposed an unsupervised feature selection algorithm built upon a transform autoencoder (UFS-TAE), which addresses the generalized feature selection problem by learning a non-negative and orthogonal indicator matrix through the autoencoder framework.