To deal with high-dimensional unlabeled datasets in many areas, principal component analysis (PCA) has become a rising technique for unsupervised feature selection (UFS). However, most existing PCA-based methods only consider the structure of datasets by embedding a single sparse regularization or constraint on the transformation matrix. In this paper, we introduce a novel bi-sparse method called BSUFS to improve the performance of UFS. The core idea of BSUFS is to incorporate $\ell _{2,p}$ -norm and $\ell _{q}$ -norm into the classical PCA, which enables our method to select relevant features and filter out irrelevant noises, thereby obtaining discriminative features. Here, the parameters $p$ and $q$ are within the range of [ $0, 1$ ). Therefore, BSUFS not only constructs a unified framework for bi-sparse optimization, but also includes some existing works as special cases. To solve the resulting non-convex model, we propose an efficient proximal alternating minimization (PAM) algorithm using Stiefel manifold optimization and sparse optimization techniques. In addition, the computational complexity analysis is presented. Extensive numerical experiments on synthetic and real-world datasets demonstrate the effectiveness of our proposed BSUFS. The results reveal the advantages of bi-sparse optimization in feature selection and show its potential for other fields in image processing. Our code is available at https://github.com/xianchaoxiu/BSUFS.

$\ell _{1}$-norm quantile regression is a common choice if there exists outlier or heavy-tailed error in high-dimensional data sets. However, it is computationally expensive to solve this problem when the feature size of data is ultra high. As far as we know, existing screening rules can not speed up the computation of the$\ell _{1}$-norm quantile regression, which dues to the non-differentiability of the quantile function/pinball loss. In this paper, we introduce the dual circumscribed sphere technique and propose a novel$\ell _{1}$-norm quantile regression screening rule. Our rule is expressed as the closed-form function of given data and eliminates inactive features with a low computational cost. Numerical experiments on some simulation and real data sets show that this screening rule can be used to eliminate almost all inactive features. Moreover, this rule can help to reduce up to 23 times of computational time, compared with the computation without our screening rule.

To deal with outliers or heavy-tailed random errors in common high-dimensional data sets, robust regressions are preferable selections and Rank Lasso is a notable model among them. However, the large-scaled feature size in data set increases the computational cost of solving Rank Lasso. In this paper, we build up a safe feature screening rule for Rank Lasso, which can effectively and safely identify inactive features in data sets and reduce the computation time of this model. The advantage of our screening rule is that it can be expressed as the closed-form function of given data and is easily implemented. The proposed screening rule is evaluated on some simulation and real data sets, which show that it can safely discard inactive features with a small computational cost and reduce the time for solving Rank Lasso.