Effective feature selection is essential for high-dimensional data analysis and machine learning. Unsupervised feature selection (UFS) aims to simultaneously cluster data and identify the most discriminative features. Most existing UFS methods linearly project features into a pseudo-label space for clustering, but they suffer from two critical limitations: (1) an oversimplified linear mapping that fails to capture complex feature relationships, and (2) an assumption of uniform cluster distributions, ignoring outliers prevalent in real-world data. To address these issues, we propose the Robust Autoencoder-based Unsupervised Feature Selection (RAEUFS) model, which leverages a deep autoencoder to learn nonlinear feature representations while inherently improving robustness to outliers. We further develop an efficient optimization algorithm for RAEUFS. Extensive experiments demonstrate that our method outperforms state-of-the-art UFS approaches in both clean and outlier-contaminated data settings.

Multi-view unsupervised feature selection (MUFS) has recently emerged as an effective dimensionality reduction method for unlabeled multi-view data. However, most existing methods mainly use first-order similarity graphs to preserve local structure, often overlooking the global structure that can be captured by second-order similarity. In addition, a few MUFS methods leverage predefined second-order similarity graphs, making them vulnerable to noise and outliers and resulting in suboptimal feature selection performance. In this paper, we propose a novel MUFS method, termed Structure-aware Hybrid-order sImilarity learNing for multi-viEw unsupervised Feature Selection (SHINE-FS), to address the aforementioned problem. SHINE-FS first learns consensus anchors and the corresponding anchor graph to capture the cross-view relationships between the anchors and the samples. Based on the acquired cross-view consensus information, it generates low-dimensional representations of the samples, which facilitate the reconstruction of multi-view data by identifying discriminative features. Subsequently, it employs the anchor-sample relationships to learn a second-order similarity graph. Furthermore, by jointly learning first-order and second-order similarity graphs, SHINE-FS constructs a hybrid-order similarity graph that captures both local and global structures, thereby revealing the intrinsic data structure to enhance feature selection. Comprehensive experimental results on real multi-view datasets show that SHINE-FS outperforms the state-of-the-art methods.

Multi-view unsupervised feature selection (MUFS) has recently received increasing attention for its promising ability in dimensionality reduction on multi-view unlabeled data. Existing MUFS methods typically select discriminative features by capturing correlations between features and clustering labels. However, an important yet underexplored question remains: \textit{Are such correlations sufficiently reliable to guide feature selection?} In this paper, we analyze MUFS from a causal perspective by introducing a novel structural causal model, which reveals that existing methods may select irrelevant features because they overlook spurious correlations caused by confounders. Building on this causal perspective, we propose a novel MUFS method called CAusal multi-view Unsupervised feature Selection leArning (CAUSA). Specifically, we first employ a generalized unsupervised spectral regression model that identifies informative features by capturing dependencies between features and consensus clustering labels. We then introduce a causal regularization module that can adaptively separate confounders from multi-view data and simultaneously learn view-shared sample weights to balance confounder distributions, thereby mitigating spurious correlations. Thereafter, integrating both into a unified learning framework enables CAUSA to select causally informative features. Comprehensive experiments demonstrate that CAUSA outperforms several state-of-the-art methods. To our knowledge, this is the first in-depth study of causal multi-view feature selection in the unsupervised setting.

Incomplete multi-view unsupervised feature selection (IMUFS), which aims to identify representative features from unlabeled multi-view data containing missing values, has received growing attention in recent years. Despite their promising performance, existing methods face three key challenges: 1) by focusing solely on the view-missing problem, they are not well-suited to the more prevalent mixed-missing scenario in practice, where some samples lack entire views or only partial features within views; 2) insufficient utilization of consistency and diversity across views limits the effectiveness of feature selection; and 3) the lack of theoretical analysis makes it unclear how feature selection and data imputation interact during the joint learning process. Being aware of these, we propose CLIM-FS, a novel IMUFS method designed to address the mixed-missing problem. Specifically, we integrate the imputation of both missing views and variables into a feature selection model based on nonnegative orthogonal matrix factorization, enabling the joint learning of feature selection and adaptive data imputation. Furthermore, we fully leverage consensus cluster structure and cross-view local geometrical structure to enhance the synergistic learning process. We also provide a theoretical analysis to clarify the underlying collaborative mechanism of CLIM-FS. Experimental results on eight real-world multi-view datasets demonstrate that CLIM-FS outperforms state-of-the-art methods.

Unsupervised feature selection (FS) is essential for high-dimensional learning tasks where labels are not available. It helps reduce noise, improve generalization, and enhance in-terpretability. However, most existing unsupervised FS methods evaluate features in isolation, even though informative signals often emerge from groups of related features. For example, adjacent pixels, functionally connected brain regions, or correlated financial indicators tend to act together, making independent evaluation suboptimal. Although some methods attempt to capture group structure, they typically rely on predefined partitions or label supervision, limiting their applicability. We propose GroupFS, an end-to-end, fully differentiable framework that jointly discovers latent feature groups and selects the most informative groups among them, without relying on fixed a priori groups or label supervision. GroupFS enforces Laplacian smoothness on both feature and sample graphs and applies a group sparsity regu-larizer to learn a compact, structured representation. Across nine benchmarks spanning images, tabular data, and biological datasets, GroupFS consistently outperforms state-of-the-art unsupervised FS in clustering and selects groups of features that align with meaningful patterns.

Multi-view unsupervised feature selection (MUFS), which selects informative features from multi-view unlabeled data, has attracted increasing research interest in recent years. Although great efforts have been devoted to MUFS, several challenges remain: 1) existing methods for incomplete multi-view data are limited to handling missing views and are unable to address the more general scenario of missing variables, where some features have missing values in certain views; 2) most methods address incomplete data by first imputing missing values and then performing feature selection, treating these two processes independently and overlooking their interactions; 3) missing data can result in an inaccurate similarity graph, which reduces the performance of feature selection. To solve this dilemma, we propose a novel MUFS method for incomplete multi-view data with missing variables, termed Tensorized Reliable UnSupervised mulTi-view Feature Selection (TRUST-FS). TRUST-FS introduces a new adaptive-weighted CP decomposition that simultaneously performs feature selection, missing-variable imputation, and view weight learning within a unified tensor factorization framework. By utilizing Subjective Logic to acquire trustworthy cross-view similarity information, TRUST-FS facilitates learning a reliable similarity graph, which subsequently guides feature selection and imputation. Comprehensive experimental results demonstrate the effectiveness and superiority of our method over state-of-the-art methods.

The goal of unsupervised feature selection is to identify a small number of important features that can represent the data. We propose a new algorithm, a modification of the classical pivoted QR algorithm of Businger and Golub, that requires a small number of passes over the data. The improvements are based on two ideas: keeping track of multiple features in each pass, and skipping calculations that can be shown not to affect the final selection. Our algorithm selects the exact same features as the classical pivoted QR algorithm, and has the same favorable numerical stability. We describe experiments on real-world datasets which sometimes show improvements of {\em several orders of magnitude} over the classical algorithm. These results appear to be competitive with recently proposed randomized algorithms in terms of pass efficiency and run time. On the other hand, the randomized algorithms may produce better features, at the cost of small probability of failure.

It is important to identify the discriminative features for high dimensional clustering. However, due to the lack of cluster labels, the regularization methods developed for supervised feature selection can not be directly applied. To learn the pseudo labels and select the discriminative features simultaneously, we propose a new unsupervised feature selection method, named GlObal and Local information combined Feature Selection (GOLFS), for high dimensional clustering problems. The GOLFS algorithm combines both local geometric structure via manifold learning and global correlation structure of samples via regularized self-representation to select the discriminative features. The combination improves the accuracy of both feature selection and clustering by exploiting more comprehensive information. In addition, an iterative algorithm is proposed to solve the optimization problem and the convergency is proved. Simulations and two real data applications demonstrate the excellent finite-sample performance of GOLFS on both feature selection and clustering.

Unsupervised feature selection is critical for improving clustering performance in high-dimensional data, where irrelevant features can obscure meaningful structure. In this work, we introduce the Minkowski weighted $k$-means++, a novel initialisation strategy for the Minkowski Weighted $k$-means. Our initialisation selects centroids probabilistically using feature relevance estimates derived from the data itself. Building on this, we propose two new feature selection algorithms, FS-MWK++, which aggregates feature weights across a range of Minkowski exponents to identify stable and informative features, and SFS-MWK++, a scalable variant based on subsampling. We support our approach with a theoretical guarantee under mild assumptions and extensive experiments showing that our methods consistently outperform existing alternatives. Our software can be found at https://github.com/xzhang4-ops1/FSMWK.