###### Abstract  
Feature selection has been widely used in machine learning and data mining since it can alleviate the burden of the so-called curse of dimensionality of high-dimensional data. However, in previous works, researchers have designed feature selection methods with the assumption that all the information from a data set can be observed. In this paper, we propose unsupervised and supervised feature selection methods for use with incomplete data, further introducing an L2,1 norm and a reconstruction error minimization method. Specifically, the proposed feature selection objective functions take advantage of an indicator matrix reflecting unobserved information in incomplete data sets, and we present pairwise constraints, minimizing the L2,1-norm-robust loss functionand performing error reconstruction simultaneously. Furthermore, we derive two alternative iterative algorithms to effectively optimize the proposed objective functions and the convergence of the proposed algorithms is proven theoretically. Extensive experimental studies were performed on both real and synthetic incomplete data sets to demonstrate the performance of the proposed methods.  
unsupervised feature selection; supervised feature selection; incomplete data; L2,1 norm; reconstruction error +
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process. This can cause some useful information from missing instances to be discarded, which may degrade the data analysis performance. Furthermore, the scale of missing instances in the incomplete data is usually so large that the instance-deletion method becomes invalid. Therefore, imputation methods have been proposed to handle these issues, in which the values of missing instances are estimated using the traditional machine learning techniques [11; 12], such as the K-nearest neighbor technique (KNN). However, in feature selection with imputation methods, when using all instances, noise or non-useful information may be introduced;thus, a margin-based feature selection method [13] was designed without imputing missing values, in which the uncertainty of each instance was considered.  
In this work, the reconstruction of incomplete data is designed as the feature selection criterion, in which the selected feature approximates the original missing instance by means of a weigh matrix.Specifically, we propose unsupervised and supervised feature selection methods for incomplete data by further introducing the L\({}_{2,1}\) norm and through reconstruction error minimization. Specifically, the proposed feature selection objective functions take advantage of an indicator matrix for the unobserved information relating to an incomplete data set. We design pairwise constraints, minimizing the L\({}_{2,1}\)-norm-robust loss function and performing error reconstruction simultaneously.We further derive an alternative iterative algorithm to effectively optimize the proposed objective functions and the convergence of the proposed algorithms is proven theoretically. Consequently, extensive experimental studies were performed on both real and synthetic incomplete data sets to demonstrate the performance of the proposed methods.  
This paper is organized as follows. In Section 2, we review the related works. Then, in Section 3, we present the details of our approach, design the pairwise constraints objection function, optimize the proposed objective functions, and prove the convergence. Next, in Section 4, we display the results of extensive experimental studies and compare the performance of our proposed approach with that of other approaches. Finally, Section 5 concludes the paper.