pith. sign in

arxiv: 2604.22405 · v1 · submitted 2026-04-24 · 💻 cs.LG · cs.NA· math.NA

Robust Fuzzy local k-plane clustering with mixture distance of hinge loss and L1 norm

Junjun Huang , Xiliang Lu , Xuelin Xie , Jerry Zhijian Yang This is my paper

Pith reviewed 2026-05-08 12:24 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA
keywords fuzzy k-plane clusteringrobust clusteringoutlier handlinghinge lossL1 normfinite-area constraintlinear manifold clusteringmixture regression
0
0 comments X

The pith

A new fuzzy k-plane clustering method bounds each plane to a finite area and replaces L2 projection distance with a hinge-loss plus L1-norm mixture to resist outliers.

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

The paper introduces RFLkPC as a fuzzy clustering procedure for data lying near k planes. Traditional approaches suffer from outlier sensitivity because they measure distances with squared Euclidean projection and treat planes as extending to infinity. The new model instead uses a mixture distance combining hinge loss and L1 norm while adding explicit bounds that limit each cluster to a finite region. This change is claimed to produce stable partitions on both clean and contaminated data. The authors derive the corresponding optimization scheme and report improved results against prior k-plane methods on simulated and real data sets.

Core claim

The RFLkPC model assumes that each plane cluster is bounded to a finite area, which can flexibly and robustly handle plane clustering tasks with outliers or not. It replaces the conventional L2 projection distance with a mixture of hinge loss and L1 norm, supplies the full optimization procedure, and demonstrates higher accuracy than related models on both simulated and real data.

What carries the argument

The RFLkPC objective function that combines a hinge-loss–L1-norm mixture distance with explicit finite-area bounds on each plane cluster.

If this is right

  • The finite-area constraint allows the same model to be applied whether outliers are present or absent.
  • The derived optimization algorithm solves the resulting non-convex problem for the given distance mixture.
  • Clustering quality on simulated data with controlled outliers exceeds that of L2-based fuzzy k-plane baselines.
  • Experiments on real data sets confirm the same accuracy advantage without additional preprocessing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The bounded-plane formulation could be tested on other linear-manifold tasks such as motion segmentation where infinite extent is unrealistic.
  • Because the source code is released, direct replication on new data sets is possible without re-deriving the solver.
  • If the local minima reached by the algorithm are consistently good, the method may reduce reliance on manual outlier cleaning in preprocessing pipelines.

Load-bearing premise

Minimizing the proposed loss under the finite-area constraints produces a useful local solution without requiring extensive hyperparameter tuning or separate outlier removal.

What would settle it

On synthetic data generated from known finite planes plus added outliers, compare recovered cluster assignments and plane parameters against ground truth and against standard fuzzy k-plane clustering; if accuracy does not improve, the robustness claim is falsified.

Figures

Figures reproduced from arXiv: 2604.22405 by Jerry Zhijian Yang, Junjun Huang, Xiliang Lu, Xuelin Xie.

Figure 1
Figure 1. Figure 1: A toy example with three linear-structure clusters view at source ↗
Figure 2
Figure 2. Figure 2: Line clusters may be too expanding! III. THE PROPOSED METHOD A. RFLkPC model This section introduces our RFLkPC model, a novel robust approach to improve the performance of fuzzy plane cluster￾ing. Inspired by the work of the LkF model by Wang et al. [31], we generalize the local idea to fuzzy plane clustering. In addition, in order to better alleviate the effect of outliers, we use a mixture of hinge loss… view at source ↗
Figure 4
Figure 4. Figure 4: The simulated datasets:three types of plane clustering with different noise or outliers. The dataset Si ( view at source ↗
Figure 5
Figure 5. Figure 5: Tone perception data. -5 0 5 10 15 20 25 30 RW 15 20 25 30 35 40 45 CL cluster1 cluster2 Dataset: Crab view at source ↗
Figure 7
Figure 7. Figure 7: Results on Tone and Crab data for all competing view at source ↗
Figure 8
Figure 8. Figure 8: Results on UCI datasets for all competing methods. view at source ↗
Figure 9
Figure 9. Figure 9: The point cloud data contains 3 planes. has successfully achieved our original intention of improving the clustering effectiveness of FkPC. C. Point cloud data In computer vision applications such as motion segmen￾tation and robot path planning, it is often necessary to pre￾identify structural elements in indoor scenes, such as walls, floors, and desktops. This task can be formulated as a plane detection o… view at source ↗
Figure 11
Figure 11. Figure 11: Sensitivity of RFLkPC model to different regulariza view at source ↗
read the original abstract

K-plane clustering (KPC), hyperplane clustering, and mixture regression all essentially fall within the same class of problems. This problem can be conceptualized as clustering in relatively high-dimensional K subspaces or K linear manifolds. Traditional KPC or fuzzy KPC models demonstrate a pronounced susceptibility to outliers, as they presuppose that the projection distance between data points and the plane normal vector adheres to the L2 distance. Meanwhile, the assumption of infinitely extending clusters adversely affects clustering performance. To solve these problems, this paper proposed a new robust fuzzy local k-plane clustering (RFLkPC) method that combines the mixture distance of hinge loss and L1 norm. The RFLkPC model assumes that each plane cluster is bounded to a finite area, which can flexibly and robustly handle plane clustering tasks with outliers or not. The corresponding model and optimization algorithms of RFLkPC were provided. Compared to other related models on this topic, a large number of experiments verify the efficiency of RFLkPC on simulated data and real data. The source code for the proposed RFLkPC method is publicly available at https://github.com/xuelin-xie/RFLkPC.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a robust fuzzy local k-plane clustering (RFLkPC) method that replaces the standard L2 projection distance with a mixture of hinge loss and L1 norm while imposing a finite-area bounding assumption on each plane cluster. The paper derives the corresponding objective, supplies an (alternating) optimization procedure, and asserts that experiments on simulated and real data demonstrate superior efficiency and outlier robustness relative to prior k-plane and fuzzy k-plane models. Source code is stated to be publicly available.

Significance. If the experimental support and optimization behavior can be rigorously established, the combination of a robust distance with an explicit bounded-cluster constraint would address two well-known weaknesses of classical k-plane clustering (outlier sensitivity and infinite extent). Public code release is a clear positive for reproducibility.

major comments (3)
  1. [Abstract] Abstract: the central claim that 'a large number of experiments verify the efficiency of RFLkPC' is unsupported by any quantitative results, baseline tables, outlier-injection protocol, or performance metrics. This absence directly weakens the robustness assertion that is the paper's main contribution.
  2. [Model formulation] Model section (description of finite-area assumption): the bounded-cluster mechanism is introduced only at the conceptual level; no explicit penalty term, indicator constraint, or reformulation of the objective is supplied that would enforce finite extent independently of initialization or data scaling. Without such a formulation, the robustness claim reduces to an unverified modeling assumption.
  3. [Optimization algorithm] Optimization section: no convergence analysis, basin-of-attraction study, or initialization-robustness experiment is referenced for the alternating procedure. Given the non-convexity introduced by the hinge loss and the bounded-area term, it is unclear whether the algorithm reliably reaches solutions in which the robust distance dominates rather than degenerate or outlier-sensitive local minima.
minor comments (2)
  1. [Abstract] The abstract is overly long and contains several redundant phrases; a tighter version would improve readability.
  2. [Model formulation] Notation for the mixture coefficient and the finite-area radius parameter should be introduced once and used consistently throughout the model equations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below, indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract] the central claim that 'a large number of experiments verify the efficiency of RFLkPC' is unsupported by any quantitative results, baseline tables, outlier-injection protocol, or performance metrics.

    Authors: We agree that the abstract would be strengthened by including concrete quantitative highlights. In the revised version we will add specific metrics (e.g., clustering accuracy and robustness scores under controlled outlier injection) together with brief baseline comparisons, while retaining the summary nature of the abstract. The full experimental protocol, tables, and outlier-injection details already appear in Section 5 of the manuscript. revision: yes

  2. Referee: [Model formulation] the bounded-cluster mechanism is introduced only at the conceptual level; no explicit penalty term, indicator constraint, or reformulation of the objective is supplied that would enforce finite extent independently of initialization or data scaling.

    Authors: The referee correctly notes that the finite-area assumption was stated conceptually without an explicit enforcement mechanism. We will revise the model section to introduce a concrete penalty term (a quadratic constraint on the maximum radial extent of each plane) that is added to the objective and is independent of initialization and data scaling. The updated objective function, its derivation, and a discussion of how the term guarantees bounded clusters will be provided. revision: yes

  3. Referee: [Optimization algorithm] no convergence analysis, basin-of-attraction study, or initialization-robustness experiment is referenced for the alternating procedure. Given the non-convexity introduced by the hinge loss and the bounded-area term, it is unclear whether the algorithm reliably reaches solutions in which the robust distance dominates.

    Authors: We acknowledge the absence of formal convergence guarantees and empirical robustness checks for the alternating optimizer. In the revision we will add (i) a brief convergence analysis based on the block-coordinate descent structure and (ii) additional experiments that vary initialization and report the frequency with which the robust (hinge+L1) distance dominates versus degenerate local minima. These results will be summarized in a new subsection of the experimental evaluation. revision: yes

Circularity Check

0 steps flagged

No circularity: model and assumptions defined independently of results

full rationale

The RFLkPC formulation introduces a mixture distance (hinge loss + L1) and finite-area cluster bounding as explicit modeling choices in the objective. These are stated as direct definitions rather than derived from or fitted to the evaluation data. No equation reduces a reported performance metric to a quantity computed on the same data by construction, and no self-citation chain is invoked to justify the central robustness claim. Experiments are presented only as empirical verification after the model is fully specified.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The method builds on standard fuzzy clustering and subspace clustering assumptions while introducing one new modeling choice (finite cluster extent) and one new distance construction; no new physical entities are postulated.

free parameters (3)
  • number of clusters k
    User-specified or selected by validation; standard hyperparameter in all k-plane methods.
  • fuzziness exponent m
    Controls softness of cluster membership; conventional in fuzzy c-means style algorithms.
  • mixture coefficient between hinge loss and L1 norm
    Controls robustness trade-off; must be chosen or tuned for each dataset.
axioms (2)
  • domain assumption Data points lie approximately on a small number of linear manifolds that can be treated as locally bounded planes.
    Core modeling assumption inherited from k-plane clustering literature and invoked to justify the finite-area constraint.
  • domain assumption An iterative optimization procedure can locate useful local minima of the non-convex objective formed by the mixed distance.
    Required for the supplied algorithm to produce the claimed clusters.

pith-pipeline@v0.9.0 · 5522 in / 1532 out tokens · 34759 ms · 2026-05-08T12:24:08.800078+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

66 extracted references · 66 canonical work pages

  1. [1]

    Some methods for classification and analysis of multi- variate observation,

    J. MacQuuen, “Some methods for classification and analysis of multi- variate observation,” in Proceedings of the 5th Berkley Symposium on Mathematical Statistics and Probability , 1967, pp. 281–297

    work page 1967

  2. [2]

    Fuzzy models for pattern recognition,

    J. C. Bezdek and S. K. Pal, “Fuzzy models for pattern recognition,” 1994

    work page 1994

  3. [3]

    The estimation of structural shifts by switching regressions,

    S. Goldfeld and R. Quandt, “The estimation of structural shifts by switching regressions,” in Annals of Economic and Social Measurement, V olume 2, number 4 . NBER, 1973, pp. 475–485

    work page 1973

  4. [4]

    Regression in heterogeneous problems,

    H. Huang, “Regression in heterogeneous problems,” Statistica Sinica , pp. 71–88, 2017

    work page 2017

  5. [5]

    Robust mixture regression model fitting by laplace distribution,

    W. Song, W. Y ao, and Y . Xing, “Robust mixture regression model fitting by laplace distribution,” Computational Statistics and Data Analysis , vol. 71, pp. 128–137, 2014

    work page 2014

  6. [6]

    Bayesian inference for finite mixture regression model based on non-iterative algorithm,

    A. Shan and F. Y ang, “Bayesian inference for finite mixture regression model based on non-iterative algorithm,” Mathematics, vol. 9, no. 6, p. 590, 2021

    work page 2021

  7. [7]

    Algorithmus 39 classenweise lineare regression,

    H. Späth, “Algorithmus 39 classenweise lineare regression,” Computing, vol. 22, pp. 367–373, 1979

    work page 1979

  8. [8]

    A weighted least-squares approach to clusterwise regres- sion,

    R. Schlittgen, “A weighted least-squares approach to clusterwise regres- sion,” AStA Advances in Statistical Analysis , vol. 95, no. 2, pp. 205–217, 2011

    work page 2011

  9. [9]

    Hyperplane clustering via dual principal component pursuit,

    M. C. Tsakiris and R. Vidal, “Hyperplane clustering via dual principal component pursuit,” in International conference on machine learning . PMLR, 2017, pp. 3472–3481

    work page 2017

  10. [10]

    Dual principal component pursuit for learning a union of hyperplanes: Theory and algorithms,

    T. Ding, Z. Zhu, M. Tsakiris, R. Vidal, and D. Robinson, “Dual principal component pursuit for learning a union of hyperplanes: Theory and algorithms,” in International Conference on Artificial Intelligence and Statistics. PMLR, 2021, pp. 2944–2952

    work page 2021

  11. [11]

    A benchmark for the comparison of 3-d motion segmentation algorithms,

    R. Tron and R. Vidal, “A benchmark for the comparison of 3-d motion segmentation algorithms,” in 2007 IEEE conference on computer vision and pattern recognition . IEEE, 2007, pp. 1–8

    work page 2007

  12. [12]

    Spectral curvature clustering (scc),

    G. Chen and G. Lerman, “Spectral curvature clustering (scc),” Interna- tional Journal of Computer Vision , vol. 81, pp. 317–330, 2009

    work page 2009

  13. [13]

    Identification of switched linear systems via sparse optimiza- tion,

    L. Bako, “Identification of switched linear systems via sparse optimiza- tion,” Automatica, vol. 47, no. 4, pp. 668–677, 2011

    work page 2011

  14. [14]

    Hybrid system identification by incremental fuzzy c-regression clustering,

    S. Blaži ˇc and I. Škrjanc, “Hybrid system identification by incremental fuzzy c-regression clustering,” in 2020 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE) . IEEE, 2020, pp. 1–7

    work page 2020

  15. [15]

    Alpha-cut implemented fuzzy clustering algorithms and switching regressions,

    M.-S. Y ang, K.-L. Wu, J.-N. Hsieh, and J. Y u, “Alpha-cut implemented fuzzy clustering algorithms and switching regressions,” IEEE Transac- tions on Systems, Man, and Cybernetics, Part B (Cybernetics) , vol. 38, no. 3, pp. 588–603, 2008

    work page 2008

  16. [16]

    On robust fuzzy c-regression models,

    J. M. Leski and M. Kotas, “On robust fuzzy c-regression models,” Fuzzy Sets and Systems , vol. 279, pp. 112–129, 2015

    work page 2015

  17. [17]

    Fuzzy k-means clustering with discriminative embedding,

    F. Nie, X. Zhao, R. Wang, X. Li, and Z. Li, “Fuzzy k-means clustering with discriminative embedding,” IEEE Transactions on Knowledge and Data Engineering , vol. 34, no. 3, pp. 1221–1230, 2020

    work page 2020

  18. [18]

    Incremental fuzzy c-regression clustering from streaming data for local-model-network identification,

    S. Blaži ˇc and I. Škrjanc, “Incremental fuzzy c-regression clustering from streaming data for local-model-network identification,” IEEE transac- tions on fuzzy systems , vol. 28, no. 4, pp. 758–767, 2019

    work page 2019

  19. [19]

    Robust fuzzy k-means clustering with shrunk patterns learning,

    X. Zhao, F. Nie, R. Wang, and X. Li, “Robust fuzzy k-means clustering with shrunk patterns learning,” IEEE Transactions on Knowledge and Data Engineering , vol. 35, no. 3, pp. 3001–3013, 2021

    work page 2021

  20. [20]

    Multi- view fuzzy classification with subspace clustering and information granules,

    X. Hu, X. Liu, W. Pedrycz, Q. Liao, Y . Shen, Y . Li, and S. Wang, “Multi- view fuzzy classification with subspace clustering and information granules,” IEEE Transactions on Knowledge and Data Engineering , vol. 35, no. 11, pp. 11 642–11 655, 2022

    work page 2022

  21. [21]

    Fuzzy double-ordered c-regression models based on fuzzy s-estimators,

    J. M. Leski, “Fuzzy double-ordered c-regression models based on fuzzy s-estimators,” Fuzzy Sets and Systems , p. 108531, 2023

    work page 2023

  22. [22]

    K-plane clustering,

    P . S. Bradley and O. L. Mangasarian, “K-plane clustering,” Journal of Global optimization , vol. 16, pp. 23–32, 2000

    work page 2000

  23. [23]

    Proximal plane clustering via eigenvalues,

    Y .-H. Shao, L. Bai, Z. Wang, X.-Y . Hua, and N.-Y . Deng, “Proximal plane clustering via eigenvalues,” Procedia Computer Science , vol. 17, pp. 41–47, 2013

    work page 2013

  24. [24]

    Local k-proximal plane clustering,

    Z.-M. Y ang, Y .-R. Guo, C.-N. Li, and Y .-H. Shao, “Local k-proximal plane clustering,” Neural Computing and Applications , vol. 26, pp. 199– 211, 2015

    work page 2015

  25. [25]

    Twin support vector machine for clustering,

    Z. Wang, Y .-H. Shao, L. Bai, and N.-Y . Deng, “Twin support vector machine for clustering,” IEEE transactions on neural networks and learning systems , vol. 26, no. 10, pp. 2583–2588, 2015

    work page 2015

  26. [26]

    L1- norm distance minimization-based fast robust twin support vector k- plane clustering,

    Q. Y e, H. Zhao, Z. Li, X. Y ang, S. Gao, T. Yin, and N. Y e, “L1- norm distance minimization-based fast robust twin support vector k- plane clustering,” IEEE transactions on neural networks and learning systems, vol. 29, no. 9, pp. 4494–4503, 2018

    work page 2018

  27. [27]

    Clustering by twin support vector machine and least square twin support vector classifier with uniform output coding,

    L. Bai, Y .-H. Shao, Z. Wang, and C.-N. Li, “Clustering by twin support vector machine and least square twin support vector classifier with uniform output coding,” Knowledge-Based Systems , vol. 163, pp. 227– 240, 2019

    work page 2019

  28. [28]

    Least squares pro- jection twin support vector clustering (lsptsvc),

    B. Richhariya, M. Tanveer, A. D. N. Initiative et al., “Least squares pro- jection twin support vector clustering (lsptsvc),” Information Sciences , vol. 533, pp. 1–23, 2020

    work page 2020

  29. [29]

    Sparse pinball twin bounded support vector clustering,

    M. Tanveer, M. Tabish, and J. Jangir, “Sparse pinball twin bounded support vector clustering,” IEEE Transactions on Computational Social Systems, vol. 9, no. 6, pp. 1820–1829, Dec 2022

    work page 2022

  30. [30]

    Nearest q-flat to m points,

    P . Tseng, “Nearest q-flat to m points,” Journal of Optimization Theory and Applications , vol. 105, pp. 249–252, 2000

    work page 2000

  31. [31]

    Localized k-flats,

    Y . Wang, Y . Jiang, Y . Wu, and Z.-H. Zhou, “Localized k-flats,” in Proceedings of the AAAI Conference on Artificial Intelligence , vol. 25, 2011, pp. 525–530

    work page 2011

  32. [32]

    Improved fuzzy partitions for k-plane clustering algorithm and its robustness research,

    L. Zhu, S.-t. Wang, Y .-h. Pan, and B. Han, “Improved fuzzy partitions for k-plane clustering algorithm and its robustness research,” Journal of Electronics and Information Technology , vol. 30, no. 8, pp. 1923–1927, 2008. This Manuscript is accepted by IEEE TKDE, 2025. 14

    work page 1923

  33. [33]

    Fuzzy style k-plane clustering,

    S. Gu, Y . Nojima, H. Ishibuchi, and S. Wang, “Fuzzy style k-plane clustering,” IEEE Transactions on Fuzzy Systems , vol. 29, no. 6, pp. 1518–1532, 2020

    work page 2020

  34. [34]

    Fuzzy k-plane clustering method with local spatial information for segmentation of human brain mri image,

    P . Kumar, D. Kumar, and R. K. Agrawal, “Fuzzy k-plane clustering method with local spatial information for segmentation of human brain mri image,” Neural Computing and Applications , pp. 1–20, 2022

    work page 2022

  35. [35]

    Support vector machine classifier with pinball loss,

    X. Huang, L. Shi, and J. A. Suykens, “Support vector machine classifier with pinball loss,” IEEE transactions on pattern analysis and machine intelligence, vol. 36, no. 5, pp. 984–997, 2014

    work page 2014

  36. [36]

    Locally finite distance clustering with discriminative information,

    Y .-F. Qi, Y .-H. Shao, C.-N. Li, and Y .-R. Guo, “Locally finite distance clustering with discriminative information,” Information Sciences , vol. 623, pp. 607–632, 2023

    work page 2023

  37. [37]

    Regularization and variable selection via the elastic net,

    H. Zou and T. Hastie, “Regularization and variable selection via the elastic net,” Journal of the Royal Statistical Society Series B: Statistical Methodology, vol. 67, no. 2, pp. 301–320, 2005

    work page 2005

  38. [38]

    Estimating mixtures of normal distributions and switching regressions: Comment,

    D. W. Hosmer, “Estimating mixtures of normal distributions and switching regressions: Comment,” Journal of the American statistical association, vol. 73, no. 364, pp. 741–744, 1978

    work page 1978

  39. [39]

    Unsupervised learning of finite mixture models,

    M. Figueiredo and A. Jain, “Unsupervised learning of finite mixture models,” IEEE Transactions on Pattern Analysis and Machine Intelli- gence, vol. 24, no. 3, pp. 381–396, 2002

    work page 2002

  40. [40]

    Finite mixture models,

    G. J. McLachlan, S. X. Lee, and S. I. Rathnayake, “Finite mixture models,” Annual review of statistics and its application , vol. 6, pp. 355– 378, 2019

    work page 2019

  41. [41]

    Maximum likelihood from incomplete data via the em algorithm,

    A. P . Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” Journal of the royal statistical society: series B (methodological) , vol. 39, no. 1, pp. 1–22, 1977

    work page 1977

  42. [42]

    Robust mixture regression using the t- distribution,

    W. Y ao, Y . Wei, and C. Y u, “Robust mixture regression using the t- distribution,” Computational Statistics and Data Analysis , vol. 71, pp. 116–127, 2014

    work page 2014

  43. [43]

    Skew-normal mixture of experts,

    F. Chamroukhi, “Skew-normal mixture of experts,” in 2016 International Joint Conference on Neural Networks (IJCNN) . IEEE, 2016, pp. 3000– 3007

    work page 2016

  44. [44]

    Robust fitting of mixture regression models,

    X. Bai, W. Y ao, and J. E. Boyer, “Robust fitting of mixture regression models,” Computational Statistics and Data Analysis , vol. 56, no. 7, pp. 2347–2359, 2012

    work page 2012

  45. [45]

    Robust mixture regression modeling based on the normal mean-variance mixture dis- tributions,

    M. Naderi, E. Mirfarah, W.-L. Wang, and T.-I. Lin, “Robust mixture regression modeling based on the normal mean-variance mixture dis- tributions,” Computational Statistics and Data Analysis , vol. 180, p. 107661, 2023

    work page 2023

  46. [46]

    Robust feature selection via l2, 1-norm in finite mixture of regression,

    L. Xiangrui and Z. Dongxiao, “Robust feature selection via l2, 1-norm in finite mixture of regression,” Pattern Recognition Letters , vol. 108, pp. 15–22, 2018

    work page 2018

  47. [47]

    A selective overview and comparison of robust mixture regression estimators,

    C. Y u, W. Y ao, and G. Y ang, “A selective overview and comparison of robust mixture regression estimators,” International Statistical Review , vol. 88, no. 1, pp. 176–202, 2020

    work page 2020

  48. [48]

    Switching regression models and fuzzy clustering,

    R. J. Hathaway and J. C. Bezdek, “Switching regression models and fuzzy clustering,” IEEE Transactions on fuzzy systems , vol. 1, no. 3, pp. 195–204, 1993

    work page 1993

  49. [49]

    A possibilistic approach to clus- tering,

    R. Krishnapuram and J. M. Keller, “A possibilistic approach to clus- tering,” IEEE transactions on fuzzy systems , vol. 1, no. 2, pp. 98–110, 1993

    work page 1993

  50. [50]

    Improved possibilistic c-means clustering algorithms,

    Z. Jiangshe and L. Yiu-Wing, “Improved possibilistic c-means clustering algorithms,” IEEE transactions on fuzzy systems , vol. 12, pp. 209–217, 2004

    work page 2004

  51. [51]

    Possibilistic c-regression models clustering algorithm,

    C.-C. Kung, H.-C. Ku, and J.-Y . Su, “Possibilistic c-regression models clustering algorithm,” in 2013 International Conference on System Science and Engineering (ICSSE) . IEEE, 2013, pp. 297–302

    work page 2013

  52. [52]

    Stepwise possibilistic c- regressions,

    S.-T. Chang, K.-P . Lu, and M.-S. Y ang, “Stepwise possibilistic c- regressions,” Information Sciences , vol. 334, pp. 307–322, 2016

    work page 2016

  53. [53]

    Fuzzy weighted c-harmonic regressions clustering algorithm,

    Y . Zhao, P .-h. Wang, Y .-g. Li, and M.-y. Li, “Fuzzy weighted c-harmonic regressions clustering algorithm,” Soft Computing , vol. 22, pp. 4595– 4611, 2018

    work page 2018

  54. [54]

    Subspace clustering,

    R. Vidal, “Subspace clustering,” IEEE Signal Processing Magazine , vol. 28, no. 2, pp. 52–68, 2011

    work page 2011

  55. [55]

    Random sample consensus (ransac),

    H. Cantzler, “Random sample consensus (ransac),” Institute for Per- ception, Action and Behaviour , Division of Informatics, University of Edinburgh, vol. 3, 1981

    work page 1981

  56. [56]

    Algebraic clustering of affine subspaces,

    M. C. Tsakiris and R. Vidal, “Algebraic clustering of affine subspaces,” IEEE transactions on pattern analysis and machine intelligence , vol. 40, no. 2, pp. 482–489, 2017

    work page 2017

  57. [57]

    K-means projective clustering,

    P . K. Agarwal and N. H. Mustafa, “K-means projective clustering,” in Proceedings of the twenty-third ACM SIGMOD-SIGACT-SIGART symposium on Principles of database systems , 2004, pp. 155–165

    work page 2004

  58. [58]

    Dual principal component pursuit,

    M. C. Tsakiris and R. Vidal, “Dual principal component pursuit,” in Proceedings of the IEEE International Conference on Computer Vision Workshops, 2015, pp. 10–18

    work page 2015

  59. [59]

    Detection and characterization of cluster substructure ii. fuzzy c-varieties and convex combinations thereof,

    J. C. Bezdek, C. Coray, R. Gunderson, and J. Watson, “Detection and characterization of cluster substructure ii. fuzzy c-varieties and convex combinations thereof,” SIAM Journal on Applied Mathematics , vol. 40, no. 2, pp. 358–372, 1981

    work page 1981

  60. [60]

    Membership affinity lasso for fuzzy clustering,

    L. Guo, L. Chen, X. Lu, and C. L. P . Chen, “Membership affinity lasso for fuzzy clustering,” IEEE Transactions on Fuzzy Systems , vol. 28, no. 2, pp. 294–307, Feb 2020

    work page 2020

  61. [61]

    Comparing community structure identification,

    L. Danon, A. Diaz-Guilera, J. Duch, and A. Arenas, “Comparing community structure identification,” Journal of statistical mechanics: Theory and experiment , vol. 2005, no. 09, p. P09008, 2005

    work page 2005

  62. [62]

    Properties of the hubert-arable adjusted rand index

    D. Steinley, “Properties of the hubert-arable adjusted rand index.” Psychological methods , vol. 9, no. 3, p. 386, 2004

    work page 2004

  63. [63]

    A comparison of extrinsic clustering evaluation metrics based on formal constraints,

    E. Amigó, J. Gonzalo, J. Artiles, and F. V erdejo, “A comparison of extrinsic clustering evaluation metrics based on formal constraints,” Information retrieval, vol. 12, pp. 461–486, 2009

    work page 2009

  64. [64]

    Some effects of inharmonic partials on interval percep- tion,

    E. A. Cohen, “Some effects of inharmonic partials on interval percep- tion,” Music Perception, vol. 1, no. 3, pp. 323–349, 1984

    work page 1984

  65. [65]

    A multivariate study of variation in two species of rock crab of the genus leptograpsus,

    N. Campbell and R. Mahon, “A multivariate study of variation in two species of rock crab of the genus leptograpsus,” Australian Journal of Zoology, vol. 22, no. 3, pp. 417–425, 1974

    work page 1974

  66. [66]

    Principal component analysis,

    S. Wold, K. Esbensen, and P . Geladi, “Principal component analysis,” Chemometrics and Intelligent Laboratory Systems , vol. 2, no. 1, pp. 37–52, 1987

    work page 1987