Unsupervised Feature Selection

High dimensionality degrades both the efficiency and the effectiveness of learning algorithms, and in many applications no class labels are available to guide dimensionality reduction. This paper proposes LIDFS, an unsupervised feature selection method based on Local Intrinsic Dimensionality (LID). The method builds on the formal equivalence between LID and the indiscriminability of distance measures. The data matrix is transposed so that each feature is treated as a point, and the LID of each feature is estimated by maximum likelihood from the distances to its $k$ nearest neighboring features. Features with lower LID exhibit greater local discriminability and receive higher importance scores. A refinement step then retains one representative per feature neighborhood and demotes locally redundant features. The result is a complete ranking from which any number of features can be selected without rerunning the procedure. LIDFS does not use any information about labels or the number of classes, and runs in $\mathcal{O}(d^2 n)$ time for $n$ instances and $d$ features. LIDFS is compared with classic and state-of-the-art methods and a random baseline on 23 high-dimensional datasets, the classic benchmark collection. The evaluation uses classification accuracy and AUC with five downstream classifiers, as well as clustering accuracy and normalized mutual information, over both full and aggressive selection ranges. LIDFS ranks first in the supervised evaluations and first or tied second in the clustering-based evaluations. It belongs to the top statistical group together with methods that require the number of classes, and it runs faster than its closest competitor. Several recent methods, by contrast, fail to outperform the random baseline. A sensitivity analysis shows that small neighborhood sizes yield the best results.

Unsupervised feature selection (UFS) is of great significance in image processing and pattern recognition. Nevertheless, existing model-based methods often rely on manually tuned hyperparameters, while deep learning-based methods generally require massive training data. To address these issues, this paper proposes an efficient manifold deep unfolding feature selection network (MDUFS-Net). Technically, we first incorporate the $\ell_{2,0}$-norm into the classic principal component analysis (PCA) framework, then unfold the alternating direction method of multipliers (ADMM)-based iterative algorithm into an end-to-end trainable network, thereby enabling automatic parameter learning. Moreover, a manifold optimization strategy is developed to improve the numerical robustness. On this basis, we further extend MDUFS-Net into an enhanced variant called MDUFS-Net+ by introducing the $\ell_{0}$-norm for more discriminative feature representations. Extensive experiments validate that our proposed MDUFS-Net and MDUFS-Net+ outperform state-of-the-art methods in the UFS task, with promising performance in clustering. Our code is available at https://github.com/xianchaoxiu/MDUFS.

Unsupervised feature selection seeks a compact subset of informative features without access to class labels, making feature utility difficult to define. Existing UFS methods therefore rely on indirect structural criteria, such as similarity preservation, locality, sparsity, cluster geometry, or reconstruction quality. In this paper, we instead study UFS through representation consistency and propose Inverted Contrastive Learning for Unsupervised Feature Selection (ICLFS), a feature-wise contrastive framework that reformulates UFS as a representation learning problem over features rather than samples. ICLFS first inverts the data matrix so that each feature is represented by its sample-profile vector, then constructs multiple masked positive views together with a shuffled negative view, and learns projector-space representations that remain consistent across these structured perturbations under an InfoNCE-based objective. Motivated by recent findings that cosine-based and InfoNCE-based training affect embedding norms, we use projector-space embedding magnitude as the saliency signal for ranking features. The resulting norm-based ranking is subsequently refined through Laplacian-Gated Ranking Correction, which suppresses locally redundant candidates while preserving salient ones. Extensive experiments on 12 benchmark datasets show that ICLFS achieves the best clustering accuracy on 10 datasets against both classical and neural baselines under the standard clustering-based UFS evaluation protocol, while remaining competitive on the other two. These results show that feature-wise contrastive representation consistency provides a strong and effective alternative to neighborhood, cluster, and reconstruction-based UFS formulations.

Feature selection is one of the most important and fundamental tasks in data mining, tackled by a family of methods with an established set of evaluation techniques to measure the quality of a specific method. Most of the methods commonly used for the unsupervised evaluation of feature selection algorithms suffer from critical design flaws which question their unsupervised nature. In this paper, we provide a critical discussion on the established allegedly unsupervised evaluation techniques, and shed light on the reasons why they are not truly unsupervised but, at best, supervised evaluation under an unsupervised downstream task. We also propose a novel, truly unsupervised evaluation framework to measure the quality of the feature selection algorithms without any form of information about the labels. The proposed framework utilizes unsupervised Principal Component Analysis, and optimal transport to measure the quality of the feature selection methods in a truly unsupervised manner.

We present Variance-Preserving Orthogonal Selection (VPOS), an unsupervised feature-selection method that performs sequential orthogonal deflation in the variance-weighted principal component analysis (PCA) loading space $\mathbf{V}_d\mathbfΛ_d^{1/2}$. After each feature is selected, its loading direction is projected out of all remaining candidates, so subsequent selections cover complementary directions of the rank-$d$ covariance approximation while returning original variables. We establish rank-reduction guarantees and a determinant-growth interpretation, and distinguish VPOS from greedy selection on raw data, unweighted eigenvector pivoting, Principal Feature Analysis (PFA), and Principal Variable Selection (PVS). Experiments enforce $k\leq d$, tune method-specific parameters on validation observations, and evaluate on unseen outer folds. Across seven labelled benchmarks, VPOS improves held-out normalised reconstruction error over matched PCA without deflation on every dataset, with reductions of 1--78%. It obtains the lowest mean reconstruction error on Wine, Breast Cancer, and MNIST and is within 1.7% of the lowest error on CIFAR-10 and HighDim. On CIFAR-10, VPOS is approximately 24$\times$ faster than the closely related PVS baseline while incurring a 1.7% reconstruction gap. These results establish VPOS as an efficient covariance-coverage method, particularly when correlated high-dimensional data must be represented by a small set of identifiable original variables.

Many novel unsupervised feature selection methods are proposed each year, yet their empirical evaluation is limited to supervised and unsupervised evaluation metrics computed on selected datasets, along with comparisons to existing methods. However, in the absence of an established evaluation baseline, it is difficult to determine the value added to the existing literature by each of these methods, and how effective their underlying approaches are. In this evaluation study, we are using random feature selection as a baseline for evaluating unsupervised feature selection methods. While a random selection of features might seem to be a rather weak baseline, we show in this evaluation study that many of the state-of-the-art methods in unsupervised feature selection are outperformed by random feature selection in both performance and efficiency. Accordingly, we emphasize the strict requirement of considering random feature selection as a baseline in the development process of novel unsupervised feature selection methods, and we encourage the community to develop better evaluation measures for feature selection. Both would help to ensure consistent improvement and realistic assessment.

Unsupervised feature selection is commonly formulated as a multiobjective optimisation problem that jointly optimises subset quality and subset size. Yet the behaviour of this formulation depends critically on the choice of evaluation objective, the direction of subset-size regularisation, and the initialisation strategy. We study these factors in a controlled setting using a synthetic dataset with known informative, redundant, and irrelevant feature types. Six formulations are compared by combining three evaluation objectives: accuracy, silhouette score, and PCA reconstruction loss with subset-size minimisation or maximisation. The results show that formulation strongly affects both search dynamics and the quality of the resulting Pareto front. Silhouette-based formulations exhibit a strong bias toward trivial low-cardinality solutions and remain weak proxies for predictive performance. In contrast, the proposed PCA loss objective produces compact subsets with test accuracy comparable to subsets obtained by directly optimising supervised accuracy. Overall, the study shows that objective design is central to effective multiobjective unsupervised feature selection.

We propose Noise-Based Spectral Embedding (NBSE), a physics-informed framework for selecting informative features from high-dimensional data without greedy search. NBSE constructs a sparse similarity graph on the samples and identifies the Nishimori temperature $β_N$ the critical inverse temperature at which the Bethe Hessian becomes singular. The corresponding smallest eigenvector captures the dominant mode of an intrinsically degree-corrected diffusion process, naturally reweighting nodes to prevent hub dominance. By transposing the data matrix and applying NBSE in feature space, we obtain a one-dimensional spectral embedding that reveals groups of redundant or semantically related dimensions; balanced binning then selects one representative per group. We prove that coloured Gaussian perturbations shift $β_N$ by at most $O(\barσ^2)$, guaranteeing robustness to measurement noise. Experiments on ImageNet embeddings from MobileNetV2 and EfficientNet-B4 show that NBSE preserves classification accuracy even under aggressive compression: on EfficientNet-B4 the accuracy drop is below $1\%$ when retaining only $30\%$ of features, outperforming ANOVA $F$-test and random selection by up to $6.8\%$.

In this paper, we proposed Bayesian Tucker decomposition (BTuD) in which residual is supposed to obey Gaussian distribution analogous to linear regression. Although we have proposed an algorithm to perform the proposed BTuD, the conventional higher-order orthogonal iteration can generate Tucker decomposition consistent with the present implementation. Using the proposed BTuD, we can perform unsupervised feature selection successfully applied to various synthetic datasets, global coupled maps with randomized coupling strength, and gene expression profiles. Thus we can conclude that our newly proposed unsupervised feature selection method is promising. In addition to this, BTuD based unsupervised FE is expected to coincide with TD based unsupervised FE that were previously proposed and successfully applied to a wide range of problems.

Unsupervised learning of high-dimensional data is challenging due to irrelevant or noisy features obscuring underlying structures. It's common that only a few features, called the influential features, meaningfully define the clusters. Recovering these influential features is helpful in data interpretation and clustering. We propose i-IF-Learn, an iterative unsupervised framework that jointly performs feature selection and clustering. Our core innovation is an adaptive feature selection statistic that effectively combines pseudo-label supervision with unsupervised signals, dynamically adjusting based on intermediate label reliability to mitigate error propagation common in iterative frameworks. Leveraging low-dimensional embeddings (PCA or Laplacian eigenmaps) followed by $k$-means, i-IF-Learn simultaneously outputs influential feature subset and clustering labels. Numerical experiments on gene microarray and single-cell RNA-seq datasets show that i-IF-Learn significantly surpasses classical and deep clustering baselines. Furthermore, using our selected influential features as preprocessing substantially enhances downstream deep models such as DeepCluster, UMAP, and VAE, highlighting the importance and effectiveness of targeted feature selection. Code is available at: [https://github.com/mc25800852/i_if_learn].

Unsupervised feature selection is critical for improving clustering performance in high-dimensional data, where irrelevant features can obscure meaningful structure. In this work, we propose 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 identifying stable and informative features, and SFS-MWK++, a scalable variant based on subsampling. We support our approach with a theoretical analysis, demonstrating that, under explicit assumptions on noise features and cluster structure, relevant features are assigned consistently higher weights than noise features across a range of Minkowski exponents. Our software can be found at https://github.com/xzhang4-ops1/FSMWK.