Pith. sign in

REVIEW 4 major objections 5 minor 52 references

Bi-Sparse Unsupervised Feature Selection

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Adding an elementwise sparsity penalty to row-sparse PCA improves unsupervised feature selection.

desk verdict A coherent bi-sparse PCA extension with honest limitations, but the claimed empirical edge over existing PCA-based UFS methods is not backed by the paper's own statistics. read the letter →

arxiv 2412.16819 v2 pith:UO3OGIIA submitted 2024-12-22 math.OC cs.LG

Xianchao Xiu , Chenyi Huang , Pan Shang , Wanquan Liu This is my paper · ORCID
classification math.OCcs.LG MSC 90C2690C3062H25
keywords unsupervisedfeatureselectionbi-sparseoptimizationsparsePCAℓ2p-normℓq-normproximalalternatingminimizationStiefelmanifoldnonconvex
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes that unsupervised feature selection can be improved by penalizing a PCA transformation matrix both row-wise and element-wise at the same time. Its BSUFS model adds an $\ell_{2,p}$-norm term, which drives whole rows of the transformation matrix to zero and thereby selects features, and an $\ell_q$-norm term, which drives individual entries to zero and thereby suppresses noise. With $p$ and $q$ allowed to range over $[0,1)$, the model subsumes earlier single-penalty sparse PCA formulations and the convex double-sparse model as limiting cases. The authors support the proposal with a proximal alternating minimization scheme and experiments on synthetic and eight real-world datasets, where BSUFS reports the best average clustering accuracy and competitive normalized mutual information.

What carries the argument

The load-bearing mechanism is a proximal alternating minimization loop whose three subproblems split the two sparsity structures and the orthogonality constraint. The $W$-update is a trust-region problem on the Stiefel manifold using the projected Riemannian gradient and Hessian; the $U$-update applies the proximal map of $|x|^q$ elementwise; and the $V$-update applies the proximal map of $\|x\|_p$ row-wise, with closed forms for the tested values $p,q\in\{0,1/2,2/3\}$. Alternating these steps is what lets BSUFS enforce row sparsity and entrywise sparsity simultaneously while keeping $W^\top W=I_m$.

What would settle it

Run Algorithm 1 on a small fixed dataset from many random initializations and record the limiting objective values and selected feature sets; if different initializations converge to markedly different objective values or features, the claimed numerical convergence is not robust to initialization. A sharper test is to search for a single iteration in which the full objective increases, which would directly contradict the sufficient-decrease property the missing lemma was meant to provide.

Watch

Extended reading notes

Core claim

The paper's central claim is that a PCA transformation matrix with two simultaneous sparsity structures selects more discriminative features and tolerates noise better than any single-sparsity PCA variant. BSUFS solves $\min_{W^\top W=I_m} -\operatorname{Tr}(W^\top S W)+\lambda_1\|W\|_{2,p}^p+\lambda_2\|W\|_q^q$ with $p,q\in[0,1)$, where the $\ell_{2,p}$ term pushes whole rows of $W$ to zero for feature selection and the $\ell_q$ term pushes individual entries to zero for noise filtering. In its numerical study the method achieves the highest average clustering accuracy on eight real-world datasets, and ablation experiments indicate that removing either penalty lowers performance.

Load-bearing premise

The convergence argument assumes that each $W$-update decreases the objective by an amount comparable to a sufficiently accurate solution of the Stiefel subproblem, but the trust-region method is only proved to reach a zero Riemannian gradient point, so the paper cannot supply the needed descent lemma and instead presents numerical evidence of decreasing objective values.

Editorial extensions

If this is right

  • If BSUFS is correct, unsupervised feature selection on high-dimensional unlabeled data can be improved by adding an entrywise sparsity penalty to an already row-sparse PCA model, without changing the feature-scoring procedure.
  • Because BSUFS reduces to existing single-penalty methods at $\lambda_2=0$ and to the earlier double-sparse model as $p,q$ approach $1$, a correct BSUFS gives a continuum of models between those baselines, so tuning $p$ and $q$ interpolates between them.
  • The reported ablation results imply that the $\ell_{2,p}$ term is the dominant selector while the $\ell_q$ term is complementary, suggesting that datasets with fine-grained noise benefit most from nonzero $q$.
  • The sparser transformation matrices shown for BSUFS suggest the method can serve as a label-free preprocessing step for clustering pipelines on image and gene data.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension: on a dataset whose features are a known union of informative, redundant, and noisy coordinates, the optimal $\lambda_2$ at fixed $\lambda_1$ should trace an interior peak in clustering accuracy; the paper's parameter-sensitivity plots hint at this but do not isolate noise type.
  • Because the paper only proves a zero-gradient result for the $W$ update, the practical convergence of BSUFS is likely initialization-dependent; comparing random and warm starts would quantify how much the reported gains depend on initialization.
  • The restriction of $p,q$ to $\{0,1/2,2/3\}$ is computational, not conceptual; if efficient proximal maps for other values in $[0,1)$ become available, the claimed benefits of the full range could be tested more finely.
  • The same bi-sparse penalty could be attached to other projection-based objectives, such as tensor PCA or supervised discriminative projections, where the row term would select variables and the entrywise term would denoise.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes BSUFS, an unsupervised feature selection method built from PCA with two nonconvex sparsity regularizers: an ℓ_{2,p} row-sparsity term for feature selection and an ℓ_q elementwise sparsity term for noise suppression, with p,q ∈ [0,1). The model in problem (8) unifies SPCAFS and the Lagrangian form of FSPCA as special cases. The algorithm is a proximal alternating minimization scheme (Algorithm 1) in which the W-subproblem is solved by trust-region Riemannian optimization and the U and V subproblems by closed-form proximal operators. The paper reports synthetic and real-world clustering experiments, ablation studies, statistical tests, parameter analyses, and complexity estimates, and it makes code publicly available. The central claims are that BSUFS 'consistently selects discriminative features and performs robustly on different noises' and that the bi-sparse formulation improves over single-sparse PCA-based alternatives.

Significance. If the claimed empirical advantage were established, BSUFS would be a useful contribution: it gives a unified nonconvex bi-sparse PCA framework, includes existing row-sparse PCA methods as special cases, provides closed-form proximal updates for commonly used p and q, and comes with public code and synthetic-data demonstrations of robustness to noise. The ablation and feature-visualization material is also a genuine attempt to attribute the effect to the two regularizers. However, the significance is currently limited because the paper's own statistical analysis does not support the headline advantage over the closest PCA-based baselines, the evaluation protocol appears to select hyperparameters and feature numbers on the test data, and the PAM algorithm lacks a convergence theorem. These are load-bearing issues for the central empirical claim, not presentation defects.

major comments (4)
  1. [Section IV-A.2 and Tables II–III] The evaluation protocol appears to fit the model to the test data. Section IV-A.2 selects regularization parameters from a candidate set, and Tables II and III report 'the best result with the number of features shown in brackets,' meaning the number of selected features is chosen to maximize ACC on the same data used for final evaluation. No separate validation split or nested tuning procedure is described. Because the abstract and conclusion claim that BSUFS 'consistently selects discriminative features,' this selection-on-test-set protocol can inflate the reported advantage and should be replaced by a held-out validation protocol or otherwise justified.
  2. [Section IV-E, Figures 9 and 10] The paper's own post-hoc Nemenyi test shows no statistically significant difference between BSUFS and FSPCA, SPCAFS, SPCA-PSD, or FEN-PCAFS on either ACC or NMI; only the graph-based methods are significantly worse. The raw average gaps in Tables II and III (about 2.2 percentage points in ACC and 0.5 percentage points in NMI over the closest baselines) are within the noise of that comparison. Since the central claim of the paper is that BSUFS improves feature selection over existing PCA-based UFS methods, this statistical result directly undermines the headline empirical conclusion as currently worded.
  3. [Remark 3.4 and Section IV-G-4] Algorithm 1 is presented as an efficient proximal alternating minimization algorithm, but no convergence theorem is provided. Remark 3.4 states that the sufficient-decrease lemma required by the PAM framework cannot be proved because Algorithm 2 is only guaranteed to converge to a zero Riemannian gradient point, and Section IV-G-4 explicitly says 'the convergence theorem cannot be derived.' The abstract and introduction do not qualify the algorithmic claim accordingly. Either the missing sufficient-decrease argument should be supplied, or the theoretical status of Algorithm 1 should be stated accurately in the main text rather than deferred to future work.
  4. [Section IV-D, Tables IV and V] The text states that 'Case IV consistently achieves the top performance,' but the NMI results do not support this. In Table V, Case IV is worse than Case III on USPS (60.16 vs. 61.14) and umist (67.62 vs. 69.45), and worse than Case II on GLIOMA (45.14 vs. 52.11). Since the added value of the ℓ_q-norm is the core novelty of BSUFS, the ablation evidence needs a more careful quantitative treatment, including the variability across the 50 k-means runs and statistical significance tests, rather than only averages over eight datasets.
minor comments (5)
  1. [Section II-A and problem (8)] The norms ∥W∥_{2,p} and ∥W∥_q are defined only for p,q ∈ (0,1), but the model (8) allows p,q ∈ [0,1); the p=0 and q=0 cases should be defined precisely (for example via ∥·∥_{2,0} and ∥·∥_0) before they appear in the model.
  2. [Section IV-E] The text refers to 'PEN-PCAFS' in the discussion of the Nemenyi test, but the method is FEN-PCAFS; this typo should be corrected.
  3. [Figure 1 caption] The caption contains the typo 'tranformation matrix' and should read 'transformation matrix.'
  4. [Section V] The conclusion says 'respected to ACC and NMI'; this should be 'with respect to ACC and NMI.'
  5. [Section IV-C] The sentence 'Here is another compared method called ALLfea' is awkward; ALLfea is not a feature-selection method but the baseline of using all features, so the description should be reworded for clarity.

Circularity Check

0 steps flagged · score 1.0 of 10

No derivation assumes its conclusion; BSUFS is a defined optimization model, the optimization steps are algebraic reformulations, and comparisons are against external baselines. Minor self-citations are not load-bearing, and the admitted convergence-proof gap is a correctness risk, not circularity.

full rationale

The formal derivation is self-contained: model (8) is a stated objective, the reformulations (9)-(11) and PAM updates (12)-(14) follow by algebra, and the U/V updates use proximal operators stated in Lemmas 3.1-3.2. No equation is defined in terms of the intended conclusion; the containment claims, such as 'When λ2 = 0 in (8), our proposed BSUFS unifies SPCAFS and the Lagrangian form of FSPCA' and 'When p and q tend to 1 in (8), our proposed BSUFS equals the model in [31],' are algebraic identities rather than circular predictions. The self-citations [43] and [47] for Lq proximal operators and for the parameter choices p,q in {0,1/2,2/3} are not load-bearing, because the needed lemmas are restated in full in this paper and are standard mathematical facts. The paper explicitly flags an omitted proof in Remark 3.4: 'the main challenge is to obtain a sufficient decent lemma, because of the fact that the update of W^{k+1} relies on Algorithm 2... Therefore, it is deferred as a future work to tackle this challenge.' Section IV-G-4 similarly states: 'Although the convergence theorem cannot be derived as in Remark 3.4, the algorithm has good convergence in the numerical perspective.' I weigh this as a proof gap and correctness risk, not circularity, because no convergence theorem is being passed off as a derived result. The empirical advantage is weakened by the paper's own statistical test: 'no significant differences are observed in the performance of BSUFS relative to FSPCA, SPCAFS, SPCA-PSD, and PEN-PCAFS,' and by the protocol of reporting the best result with the number of features chosen on the test data. Those are evidence-strength issues, not reductions of the central claim to its own inputs. No circular step is present.

Assumptions & free parameters 6 free parameters · 3 assumptions · 0 invented entities

The central claim depends on one algorithm-convergence assumption (explicitly unproven), standard proximal operator results, and per-dataset tuning of λ1, λ2, p, q, and feature count. No new physical or mathematical entities are introduced.

free parameters (6)
  • lambda1 = grid {1e-6, 1e-4, 1e-2, 1, 1e2, 1e4, 1e6}, best per dataset
    Regularization weight for the ℓ2,p row-sparsity penalty; selected per dataset on the evaluation data.
  • lambda2 = grid {1e-6, 1e-4, 1e-2, 1, 1e2, 1e4, 1e6}, best per dataset
    Regularization weight for the ℓq elementwise penalty; selected per dataset on the evaluation data.
  • p = chosen from {0, 1/2, 2/3} per dataset
    Exponent of the ℓ2,p norm; tuned per dataset, affecting the selected features.
  • q = chosen from {0, 1/2, 2/3} per dataset
    Exponent of the ℓq norm; tuned per dataset, affecting noise filtering.
  • number_of_selected_features = 10 to 100, best per dataset
    The paper reports the best result over a range of selected feature counts, which optimizes the reported metric.
  • beta1, beta2, tau1, tau2, tau3 = not reported in the paper
    Penalty and proximal parameters in Algorithm 1; their values are not stated, hampering exact reproduction.
assumptions (3)
  • standard math Trust-region algorithm on the Stiefel manifold converges to a stationary point of the W subproblem.
    Used in Algorithm 2 and the W update; cited to Absil et al. [42]. Not sufficient for PAM global convergence, as Remark 3.4 notes.
  • standard math Proximal operators for |x|^q and for ||x||_p are correct and available for the chosen exponent values.
    Used in U and V updates; cited to [43], [27], [44], [45], and [46].
  • domain assumption Data is centralized or handled through S = X H X^T.
    PCA formulation in Section II-B assumes centered data; standard preprocessing for PCA-based methods.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bi-Sparse Unsupervised Feature Selection." pith.science (2026). https://pith.science/paper/UO3OGIIA

@misc{pith2026241216819,
  author       = {Pith},
  title        = {Pith review of: Bi-Sparse Unsupervised Feature Selection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UO3OGIIA}},
  note         = {Machine review of arXiv:2412.16819}
}
abstract

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.

Figures

Figures reproduced from arXiv: 2412.16819 by the authors.

Figure 1
Figure 1. Visualization of feature selection results by [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Visual comparisons on the Diamond9 dataset, where (a) is the dataset distribution and (b)-(j) are the feature selection results. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Visual comparisons on the Dartboard1 dataset, where (a) is the dataset distribution and (b)-(j) are the feature selection results. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Visual comparisons on the Dartboard1 dataset corrupted by 0.01 Gaussian noise, where (a) is the dataset distribution and (b)-(j) are the feature [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Visual comparisons on the Dartboard1 dataset corrupted by 0.03 salt-and-pepper noise, where (a) is the dataset distribution and (b)-(j) are the feature [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Visual comparisons of the ACC metric under different real-world datasets with different number of selected features. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: Visual comparisons of the NMI metric under different real-world datasets with different number of selected features. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Sparse visualization of the transformation matrix, where (a)-(d) are the results on the USPS dataset and (e)-(h) are the results on the umist dataset. [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]
Figure 9
Figure 9. Figure 9: Post-hoc Nemenyi test in terms of ACC. ACC and NMI results of Case IV are improved, which shows that the introduction of ℓq-norm to Case III is meaningful for feature selection. 2) Feature Visualization: Table VI presents visual compar￾isons of feature selection result…
Figure 11
Figure 11. Figure 11: Effects of p and q on eight real-world datasets in terms of ACC (%). NMI (a) COIL20 NMI (b) Isolet NMI (c) USPS NMI (d) umist NMI (e) GLIOMA NMI (f) pie NMI (g) LUNG NMI (h) MSTAR [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: Effects of p and q on eight real-world datasets in terms of NMI (%). in terms of ACC and NMI under these different values of p and q, respectively. In these two figures, the x-axis represents various values of p and the color variations in the bars indicate different …
Figure 13
Figure 13. Figure 13: Heatmap visualizations of correlations for 10 selected features, where (a)-(d) are the results of SPCAFS and (e)-(h) are the results of BSUFS. [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]
Figure 14
Figure 14. Figure 14: Model stability comparisons of all compared methods on four real-world datasets, where (a)-(d) are the ACC results and (e)-(h) are the NMI results. [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]
Figure 15
Figure 15. Figure 15: Effects of λ1 and λ2 on four real-world datasets, where (a)-(d) are the ACC results and (e)-(h) are the NMI results. 100 200 300 400 500 Iteration number 2 4 6 8 10 Objective function 105 (a) COIL20 100 200 300 400 500 Iteration number 2 4 6 8 10 Objective function 10…
Figure 16
Figure 16. Figure 16: Convergence curve of BSUFS on four real-world datasets. [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 42 canonical work pages

  1. [31]

    Double sparse- representation feature selection algorithm for classification,

    Y . Zhu, X. Zhang, G. Wen, W. He, and D. Cheng, “Double sparse- representation feature selection algorithm for classification,” Multimedia Tools and Applications, vol. 76, pp. 17 525–17 539, 2017

    work page 2017

  2. [1]

    Machine learning: Trends, perspec- tives, and prospects,

    M. I. Jordan and T. M. Mitchell, “Machine learning: Trends, perspec- tives, and prospects,” Science, vol. 349, no. 6245, pp. 255–260, 2015

    work page 2015

  3. [2]

    A comprehensive review of dimensionality reduction techniques for feature selection and feature extraction,

    R. Zebari, A. Abdulazeez, D. Zeebaree, D. Zebari, and J. Saeed, “A comprehensive review of dimensionality reduction techniques for feature selection and feature extraction,” Journal of Applied Science and Technology Trends, vol. 1, no. 1, pp. 56–70, 2020

    work page 2020

  4. [3]

    Feature selection in image anal- ysis: a survey,

    V . Bolon-Canedo and B. Remeseiro, “Feature selection in image anal- ysis: a survey,” Artificial Intelligence Review , vol. 53, no. 4, pp. 2905– 2931, 2020

    work page 2020

  5. [4]

    Unsupervised feature selection via nonnegative spectral analysis and redundancy control,

    Z. Li and J. Tang, “Unsupervised feature selection via nonnegative spectral analysis and redundancy control,” IEEE Transactions on Image Processing, vol. 24, no. 12, pp. 5343–5355, 2015

    2015

  6. [5]

    Bi-level spectral feature selection,

    Z. Hu, J. Wang, K. Zhang, W. Pedrycz, and N. R. Pal, “Bi-level spectral feature selection,” IEEE Transactions on Neural Networks and Learning Systems, vol. 36, no. 4, pp. 6597–6611, 2025

    work page 2025

  7. [6]

    Supervised, unsupervised, and semi-supervised feature selection: A review on gene selection,

    J. C. Ang, A. Mirzal, H. Haron, and H. N. A. Hamed, “Supervised, unsupervised, and semi-supervised feature selection: A review on gene selection,” IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 13, no. 5, pp. 971–989, 2016

    work page 2016

  8. [7]

    Unsupervised feature selection algorithm for multiclass cancer classification of gene expression RNA-Seq data,

    P. Garc ´ıa-D´ıaz, I. S ´anchez-Berriel, J. A. Mart ´ınez-Rojas, and A. M. Diez-Pascual, “Unsupervised feature selection algorithm for multiclass cancer classification of gene expression RNA-Seq data,” Genomics, vol. 112, no. 2, pp. 1916–1925, 2020

    work page 1916

Show all 52 references
  1. [8]

    A review of machine learning methods of feature selection and classification for autism spectrum disorder,

    M. M. Rahman, O. L. Usman, R. C. Muniyandi, S. Sahran, S. Mo- hamed, and R. A. Razak, “A review of machine learning methods of feature selection and classification for autism spectrum disorder,” Brain Sciences, vol. 10, no. 12, p. 949, 2020

    2020

  2. [9]

    Feature selection in machine learning: A new perspective,

    J. Cai, J. Luo, S. Wang, and S. Yang, “Feature selection in machine learning: A new perspective,” Neurocomputing, vol. 300, pp. 70–79, 2018. 15

    2018

  3. [10]

    Feature selection for unsupervised learning,

    J. G. Dy and C. E. Brodley, “Feature selection for unsupervised learning,” Journal of Machine Learning Research , vol. 5, no. Aug, pp. 845–889, 2004

    2004

  4. [11]

    Unsupervised feature learning with emergent data-driven prototypicality,

    Y . Guo, Y . Zhang, Y . Chen, and S. X. Yu, “Unsupervised feature learning with emergent data-driven prototypicality,” in Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2024, pp. 23 199–23 208

    2024

  5. [12]

    A review of unsupervised feature selection methods,

    S. Solorio-Fern ´andez, J. A. Carrasco-Ochoa, and J. F. Mart´ınez-Trinidad, “A review of unsupervised feature selection methods,” Artificial Intelli- gence Review, vol. 53, no. 2, pp. 907–948, 2020

    2020

  6. [13]

    Unsupervised discriminative feature selection via contrastive graph learning,

    Q. Zhou, Q. Wang, Q. Gao, M. Yang, and X. Gao, “Unsupervised discriminative feature selection via contrastive graph learning,” IEEE Transactions on Image Processing , vol. 33, pp. 972–986, 2024

    2024

  7. [14]

    Unsupervised adaptive feature selection with binary hashing,

    D. Shi, L. Zhu, J. Li, Z. Zhang, and X. Chang, “Unsupervised adaptive feature selection with binary hashing,” IEEE Transactions on Image Processing, vol. 32, pp. 838–853, 2023

    2023

  8. [15]

    Feature selection: A data perspective,

    J. Li, K. Cheng, S. Wang, F. Morstatter, R. P. Trevino, J. Tang, and H. Liu, “Feature selection: A data perspective,” ACM Computing Surveys, vol. 50, no. 6, pp. 1–45, 2017

    2017

  9. [16]

    Laplacian score for feature selection,

    X. He, D. Cai, and P. Niyogi, “Laplacian score for feature selection,” in Advances in Neural Information Processing Systems , vol. 18, 2005, pp. 507–514

    2005

  10. [17]

    Unsupervised feature selection for multi- cluster data,

    D. Cai, C. Zhang, and X. He, “Unsupervised feature selection for multi- cluster data,” in Proceedings of the 16th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining , 2010, pp. 333– 342

    2010

  11. [18]

    ℓ2,1-norm regularized discriminative feature selection for unsupervised learning,

    Y . Yang, H. T. Shen, Z. Ma, Z. Huang, and X. Zhou, “ ℓ2,1-norm regularized discriminative feature selection for unsupervised learning,” in IJCAI International Joint Conference on Artificial Intelligence , 2011, pp. 1589–1594

    2011

  12. [19]

    Unsupervised feature selection with structured graph optimization,

    F. Nie, W. Zhu, and X. Li, “Unsupervised feature selection with structured graph optimization,” in Proceedings of the AAAI Conference on Artificial Intelligence , vol. 30, no. 1, 2016

    2016

  13. [20]

    Robust neighborhood em- bedding for unsupervised feature selection,

    Y . Liu, D. Ye, W. Li, H. Wang, and Y . Gao, “Robust neighborhood em- bedding for unsupervised feature selection,” Knowledge-Based Systems, vol. 193, p. 105462, 2020

    2020

  14. [21]

    Precise feature selection via non- convex regularized graph embedding and self-representation for unsuper- vised learning,

    H. Bai, M. Huang, and P. Zhong, “Precise feature selection via non- convex regularized graph embedding and self-representation for unsuper- vised learning,” Knowledge-Based Systems, vol. 296, p. 111900, 2024

    2024

  15. [22]

    Principal component analysis,

    M. Greenacre, P. J. Groenen, T. Hastie, A. I. d’Enza, A. Markos, and E. Tuzhilina, “Principal component analysis,” Nature Reviews Methods Primers, vol. 2, no. 1, p. 100, 2022

    2022

  16. [23]

    A selective overview of sparse principal component analysis,

    H. Zou and L. Xue, “A selective overview of sparse principal component analysis,” Proceedings of the IEEE , vol. 106, no. 8, pp. 1311–1320, 2018

    2018

  17. [24]

    Sparse PCA via ℓ2,p-norm regularization for unsupervised feature selection,

    Z. Li, F. Nie, J. Bian, D. Wu, and X. Li, “Sparse PCA via ℓ2,p-norm regularization for unsupervised feature selection,” IEEE Transactions on Pattern Analysis and Machine Intelligence , vol. 45, no. 4, pp. 5322– 5328, 2023

    2023

  18. [25]

    A comprehensive survey on regularization strategies in machine learning,

    Y . Tian and Y . Zhang, “A comprehensive survey on regularization strategies in machine learning,” Information Fusion , vol. 80, pp. 146– 166, 2022

    2022

  19. [26]

    Learning feature-sparse principal subspace,

    F. Nie, L. Tian, R. Wang, and X. Li, “Learning feature-sparse principal subspace,” IEEE Transactions on Pattern Analysis and Machine Intelli- gence, vol. 45, no. 4, pp. 4858–4869, 2023

    2023

  20. [27]

    Struc- tured sparsity optimization with non-convex surrogates of ℓ2,0-norm: A unified algorithmic framework,

    X. Zhang, J. Zheng, D. Wang, G. Tang, Z. Zhou, and Z. Lin, “Struc- tured sparsity optimization with non-convex surrogates of ℓ2,0-norm: A unified algorithmic framework,” IEEE Transactions on Pattern Analysis and Machine Intelligence , vol. 45, no. 5, pp. 6386–6402, 2023

    2023

  21. [28]

    Fast sparse PCA via positive semidefinite projection for unsupervised feature selection,

    J. Zheng, X. Zhang, Y . Liu, W. Jiang, K. Huo, and L. Liu, “Fast sparse PCA via positive semidefinite projection for unsupervised feature selection,” arXiv preprint arXiv:2309.06202 , 2023

    2023 arXiv

  22. [29]

    Prin- cipal component analysis with fuzzy elastic net for feature selection,

    Y . Gao, Q. Wu, Z. Xu, C. Cao, J. Pan, G. Shao, F. Nie, and Q. Zhu, “Prin- cipal component analysis with fuzzy elastic net for feature selection,” IEEE Transactions on Fuzzy Systems , vol. 32, no. 12, pp. 6878–6890, 2024

    2024

  23. [30]

    Group sparse optimization via ℓp,q regularization,

    Y . Hu, C. Li, K. Meng, J. Qin, and X. Yang, “Group sparse optimization via ℓp,q regularization,” Journal of Machine Learning Research, vol. 18, no. 30, pp. 1–52, 2017

    2017

  24. [32]

    Identifying disease sensitive and quantitative trait-relevant biomarkers from multidimensional heterogeneous imaging genetics data via sparse multimodal multitask learning,

    H. Wang, F. Nie, H. Huang, S. L. Risacher, A. J. Saykin, L. Shen, and A. D. N. Initiative, “Identifying disease sensitive and quantitative trait-relevant biomarkers from multidimensional heterogeneous imaging genetics data via sparse multimodal multitask learning,” Bioinformat...

    2012

  25. [33]

    Multi-view clustering and feature learning via structured sparsity,

    H. Wang, F. Nie, and H. Huang, “Multi-view clustering and feature learning via structured sparsity,” inInternational Conference on Machine Learning. PMLR, 2013, pp. 352–360

    2013

  26. [34]

    Double sparsity: Learning sparse dictionaries for sparse signal approximation,

    R. Rubinstein, M. Zibulevsky, and M. Elad, “Double sparsity: Learning sparse dictionaries for sparse signal approximation,” IEEE Transactions on Signal Processing , vol. 58, no. 3, pp. 1553–1564, 2010

    2010

  27. [35]

    Double-sparsity recovery for adc- distorted compressive sensing,

    X. Bian, W. Xu, and S. Wang, “Double-sparsity recovery for adc- distorted compressive sensing,” in 2022 IEEE 33rd Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC). IEEE, 2022, pp. 1215–1220

    2022

  28. [36]

    DSTPCA: Double-sparse constrained tensor principal component analysis method for feature selection,

    Y . Hu, J.-X. Liu, Y .-L. Gao, and J. Shang, “DSTPCA: Double-sparse constrained tensor principal component analysis method for feature selection,” IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 18, no. 4, pp. 1481–1491, 2021

    2021

  29. [37]

    Micro-doppler effects removed sparse aperture isar imaging via low-rank and double sparsity constrained ADMM and linearized ADMM,

    S. Zhang, Y . Liu, and X. Li, “Micro-doppler effects removed sparse aperture isar imaging via low-rank and double sparsity constrained ADMM and linearized ADMM,” IEEE Transactions on Image Process- ing, vol. 30, pp. 4678–4690, 2021

    2021

  30. [38]

    Clustering-guided sparse structural learning for unsupervised feature selection,

    Z. Li, J. Liu, Y . Yang, X. Zhou, and H. Lu, “Clustering-guided sparse structural learning for unsupervised feature selection,” IEEE Transac- tions on Knowledge and Data Engineering , vol. 26, no. 9, pp. 2138– 2150, 2014

    2014

  31. [39]

    Non-convex optimization for machine learning,

    P. Jain, P. Kar et al., “Non-convex optimization for machine learning,” Foundations and Trends® in Machine Learning , vol. 10, no. 3-4, pp. 142–363, 2017

    2017

  32. [40]

    Sparse SVM for sufficient data reduction,

    S. Zhou, “Sparse SVM for sufficient data reduction,” IEEE Transactions on Pattern Analysis and Machine Intelligence , vol. 44, no. 9, pp. 5560– 5571, 2022

    2022

  33. [41]

    A new insight on augmented lagrangian method with applications in machine learning,

    J. Bai, L. Jia, and Z. Peng, “A new insight on augmented lagrangian method with applications in machine learning,” Journal of Scientific Computing, vol. 99, no. 2, p. 53, 2024

    2024

  34. [42]

    Absil, R

    P.-A. Absil, R. Mahony, and R. Sepulchre, Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2008

    2008

  35. [43]

    Revisiting Lq(0 ≤ q <1) norm regularized optimization,

    S. Zhou, X. Xiu, Y . Wang, and D. Peng, “Revisiting Lq(0 ≤ q <1) norm regularized optimization,” arXiv:2306.14394, 2023

    2023 arXiv

  36. [44]

    Beck, First-order Methods in Optimization

    A. Beck, First-order Methods in Optimization . SIAM, 2017

    2017

  37. [45]

    L1/2 regularization: A thresholding representation theory and a fast solver,

    Z. Xu, X. Chang, F. Xu, and H. Zhang, “ L1/2 regularization: A thresholding representation theory and a fast solver,” IEEE Transactions on Neural Networks and Learning Systems , vol. 23, no. 7, pp. 1013– 1027, 2012

    2012

  38. [46]

    Fast image deconvolution using closed- form thresholding formulas of Lq (q = 1 2 , 2 3 ) regularization,

    W. Cao, J. Sun, and Z. Xu, “Fast image deconvolution using closed- form thresholding formulas of Lq (q = 1 2 , 2 3 ) regularization,” Journal of Visual Communication and Image Representation , vol. 24, no. 1, pp. 31–41, 2013

    2013

  39. [47]

    Efficient and robust sparse linear discriminant analysis for data classification,

    J. Liu, M. Feng, X. Xiu, W. Liu, and X. Zeng, “Efficient and robust sparse linear discriminant analysis for data classification,” IEEE Trans- actions on Emerging Topics in Computational Intelligence, vol. 9, no. 1, pp. 617–629, 2025

    2025

  40. [48]

    Proximal alternating linearized minimization for nonconvex and nonsmooth problems,

    J. Bolte, S. Sabach, and M. Teboulle, “Proximal alternating linearized minimization for nonconvex and nonsmooth problems,” Mathematical Programming, vol. 146, no. 1, pp. 459–494, 2014

    2014

  41. [49]

    Learning feature sparse principal subspace,

    L. Tian, F. Nie, R. Wang, and X. Li, “Learning feature sparse principal subspace,” Advances in Neural Information Processing Systems, vol. 33, pp. 14 997–15 008, 2020

    2020

  42. [50]

    Sparse tensor PCA via tensor decomposition for unsupervised feature selection,

    J. Zheng, X. Zhang, W. Jiang, X. Qiu, and M. Ren, “Sparse tensor PCA via tensor decomposition for unsupervised feature selection,” arXiv preprint arXiv:2407.16985, 2024

    2024 arXiv

  43. [51]

    Subspace Newton method for sparse group ℓ0 optimization problem,

    S. Liao, C. Han, T. Guo, and B. Li, “Subspace Newton method for sparse group ℓ0 optimization problem,” Journal of Global Optimization, vol. 90, no. 1, pp. 93–125, 2024

    2024

  44. [52]

    Physics-inspired com- pressive sensing: Beyond deep unrolling,

    J. Zhang, B. Chen, R. Xiong, and Y . Zhang, “Physics-inspired com- pressive sensing: Beyond deep unrolling,” IEEE Signal Processing Magazine, vol. 40, no. 1, pp. 58–72, 2023

    2023

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.