# RMRRASL: Robust Matrix Factorization with Robust Adaptive Structure Learning for Feature Selection
###### Abstract  
In this paper, we present a novel unsupervised feature selection method termed robust matrix factorization with robust adaptive structure learning (RMRRASL), which can select discriminative features from a large amount of multimedia data to improve the performance of classification and clustering tasks. RMRRASL integrates three models (robust matrix factorization, adaptive structure learning, and structure regularization) into a unified framework. More specifically, a robust matrix factorization-based feature selection (RMRFS) model is proposed by introducing an indicator matrix to measure the importance of features, and the \(L_{21}\)-norm is adopted as a metric to enhance the robustness of feature selection. Furthermore, a robust adaptive structure learning (RASL) model based on the self-representation capability of the samples is designed to discover the geometric structure relationships of original data. Lastly, a structure regularization (SR) term is designed on the learned graph structure, which constrains the selected features to preserve the structure information in the selected feature space. To solve the objective function of our proposed RMRRASL, an iterative optimization algorithm is proposed. By comparing our method with some state-of-the-art unsupervised feature selection approaches on several publicly available databases, the advantage of the proposed RMRRASL is demonstrated. feature selection; matrix factorization; adaptive structure learning; structure regularization 20232003203164.14.  
unsupervised feature selection is a significant challenge research because few labeled data are available in real life [13]. Unsupervised feature selection can be divided into three categories: (1) filtering methods, (2) wrapping methods, and (3) embedding-based methods. The filtering method involves performing feature selection on the original dataset in advance and then training a learner. In other words, the procedure of feature selection is independent of the learner training [14]. The wrapping method continuously selects feature subsets from the initial dataset and trains the learner until the optimal feature subset is selected [15; 16]. Embedding automatically performs feature selection during the learning process. Compared with the filtering and wrapping methods, embedded methods can significantly reduce the computational cost and achieve good results [17; 18]. [PERSON] et al. [19] proposed an unsupervised feature selection method named Laplacian score (LS), which selects optimal feature subsets to preserve the local manifold structure of the original data. However, the LS method ignores the correlation between features. In order to overcome this disadvantage, [PERSON] et al. [20] proposed an unsupervised feature selection method with robust neighborhood embedding (RNEFS). RNEFS adopts the local linear embedding (LLE) algorithm to compute the feature weight matrix. Then, the \(L_{1}\)-norm-based reconstructing error was utilized to suppress the influence of noise and outliers. Subsequently, [PERSON] et al. [21] proposed an efficient soft label feature selection (SLFS) method, which first performed soft label learning and then selected features on the basis of the learned soft labels. However, weight matrices of the above methods were defined in advance by considering the similarity of features, which may have affected their performances due to the quality of weight matrices. Therefore, [PERSON] et al. [22] proposed an adaptive graph convex non-negative matrix analysis method for unsupervised feature selection (CNAFS) which embedded self-expression and pseudo-label information into a joint model. CNAFS can select the most representative features by computing top-ranked features. [PERSON] et al. [23] proposed a non-negative matrix factorization adaptive constraint-based feature selection (NNSAFS) method, which introduced a feature map into matrix factorization to combine manifold learning and feature selection.