# RMRRASL: Robust Matrix Factorization with Robust Adaptive Structure Learning for Feature Selection
###### Abstract
By combining feature transformation with adaptive constraints, NNSAFS can significantly improve the accuracy of feature selection. However, the above methods take Euclidean distance as the metric. Euclidean distance is sensitive to noise and redundant data, which may cause the performance to be unstable and not robust. To address this issue, [PERSON] et al. [24] proposed a joint adaptive and discriminative unsupervised feature selection method, which adaptively learns similar graphs while employing irrelevance constraints to enhance the discriminability of the selected features. [PERSON] et al. [25] proposed a regularized self-representation (RSR) model, which efficiently performs feature selection by using the self-representation ability of features with the \(L_{21}\)-norm error to guarantee robustness. [PERSON] et al. [26] proposed a robust spectral learning framework, which utilized graph embedding and spectral regression to deal with noise during unsupervised feature selection. [PERSON] et al. [27] proposed a robust unsupervised feature selection based on the matrix factorization (RUFSM) method, which adopted the \(L_{21}\)-norm to improve robustness and preserve the local manifold structure, so that an optimal feature subset could be found to retain the manifold structure information. [PERSON] et al. [28] proposed an unsupervised feature selection method named graph-regularized local linear embedding (GRLLE), which combined linear embedding and graph regularization into a unified framework while maintaining local spatial structure. However, the correlation between the selected features was still ignored in the above methods. This paper proposes an unsupervised feature selection method called robust matrix factorization with robust adaptive structure learning (RMFRASL) to solve the above problems. First, this paper presents a robust matrix factorization-based feature selection model (RMFFS), which introduces a feature indicator matrix into matrix factorization with an \(L_{21}\)-norm-based error metric. Next, a robust adaptive structure learning (RASL) model based on self-representation capability is proposed to better describe the local geometry of the samples, which can reduce redundant and noisy features to learn a more discriminative graph structure. Then, to preserve the geometric structure information during the feature selection, a structure regularization (SR) term is constructed. Finally, the above three models are combined into a unified framework and shown in Figure 1. In summary, the main contributions of this paper are as follows:  
(1) In order to overcome the shortcoming of existing methods in that they neglect the correlation of features, we introduce an indicator matrix into matrix factorization and design a robust matrix factorization-based feature selection (RMFFS). (2) To consider the structure information during the process of feature selection, a structure regularization term is integrated into the RMFFS model. (3) To solve the disadvantages of the existing methods needing to predefine a graph structure, a robust adaptive structure learning model based on the self-representation capability of the sample is proposed, which adopts \(L_{21}\)-norm as an error metric rather than \(L_{2}\)-norm to reduce the influence of noise or outliers. (4) To further improve the performance of feature selection methods, an unfiled framework is proposed by combining the process of the feature selection and structure learning. (5) An iterative optimization algorithm is designed to solve the objective function, and the convergence of the optimization algorithm is verified from mathematical theory and numerical experiments. This paper is organized as follows: Section 2 briefly reviews the related work. The proposed RMFRASL model is introduced in Section 3. The convergence analysis is provided in Section 4. The experimental results and analysis are provided in Section 5. Section 6 gives the conclusions and future research works.