Skip to main content
SpringerLink
Log in

\(L_{p}\)-norm probabilistic K-means clustering via nonlinear programming

  • Original Article
  • Published:
International Journal of Machine Learning and Cybernetics Aims and scope Submit manuscript

Abstract

Generalized fuzzy c-means (GFCM) is an extension of fuzzy c-means using \(L_{p}\)-norm distances. However, existing methods cannot solve GFCM with m = 1. To solve this problem, we define a new kind of clustering models, called \(L_{p}\)-norm probabilistic K-means (\(L_{p}\)-PKM). Theoretically, \(L_{p}\)-PKM is equivalent to GFCM at m = 1, and can have nonlinear programming solutions based on an efficient active gradient projection (AGP) method, namely, inverse recursion maximum-step active gradient projection (IRMSAGP). On synthetic and UCI datasets, experimental results show that \(L_{p}\)-PKM performs better than GFCM (m > 1) in terms of initialization robustness, p-influence, and clustering performance, and the proposed IRMSAGP also achieves better performance than the traditional AGP in terms of convergence speed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Liu A, Su Y, Nie W, Kankanhalli M (2017) Hierarchical clustering multi-task learning for joint human action grouping and recognition. IEEE Trans Pattern Anal Mach Intell 39(1):102–114

    Article  Google Scholar 

  2. Cheng D, Zhu Q, Huang J, Wu Q, Yang L (2019) A hierarchical clustering algorithm based on noise removal. Int J Mach Learn Cybern 10(7):1591–1602

    Article  Google Scholar 

  3. Iam-On N (2020) Clustering data with the presence of attribute noise: a study of noise completely at random and ensemble of multiple k-means clusterings. Int J Mach Learn Cybern 11(3):491–509

    Article  Google Scholar 

  4. Ronan T, Qi Z, Naegle KM (2016) Avoiding common pitfalls when clustering biological data. Sci Signal 9(432):re6

    Article  Google Scholar 

  5. Qin J, Fu W, Gao H, Zheng WX (2017) Distributed \(k\)-means algorithm and fuzzy \(c\)-means algorithm for sensor networks based on multiagent consensus theory. IEEE Trans Cybern 47(3):772–783

    Article  Google Scholar 

  6. Otto C, Wang D, Jain AK (2018) Clustering millions of faces by identity. IEEE Trans Pattern Anal Mach Intell 40(2):289–303

    Article  Google Scholar 

  7. Fan J, Wang J (2018) A two-phase fuzzy clustering algorithm based on neurodynamic optimization with its application for PolSAR image segmentation. IEEE Trans Fuzzy Syst 26(1):72–83

    Article  MathSciNet  Google Scholar 

  8. Wang C, Pedrycz W, Yang J, Zhou M, Li Z (2020) Wavelet frame-based fuzzy c-means clustering for segmenting images on graphs. IEEE Trans Cybern 50(9):3938–3949

    Article  Google Scholar 

  9. Bezdek JC (1981) Pattern recognition with fuzzy objective function algorithms. Adv Appl Pattern Recogn 22(1171):203–239

    MATH  Google Scholar 

  10. Pal NR, Bezdek JC (1995) On cluster validity for the fuzzy C-means model. IEEE Trans Fuzzy Syst 3(3):370–379

    Article  Google Scholar 

  11. Krishnapuram R, Keller JM (1996) The possibilistic c-means algorithm: insights and recommendations. IEEE Trans Fuzzy Syst 4(3):385–393

    Article  Google Scholar 

  12. Xenaki SD, Koutroumbas KD, Rontogiannis AA (2016) Sparsity-aware possibilistic clustering algorithms. IEEE Trans Fuzzy Syst 24(6):1611–1626

    Article  Google Scholar 

  13. Gu J, Jiao L, Yang S, Liu F (2018) Fuzzy double c-means clustering based on sparse self-representation. IEEE Trans Fuzzy Syst 26(2):612–626

    Article  Google Scholar 

  14. Maji P, Garai P (2019) Rough hypercuboid based generalized and robust IT2 fuzzy c-means algorithm. IEEE Trans Cybern. https://doi.org/10.1109/TCYB.2019.2925130

    Article  Google Scholar 

  15. Koutroumbas KD, Xenaki SD, Rontogiannis AA (2018) On the convergence of the sparse possibilistic c-means algorithm. IEEE Trans Fuzzy Syst 26(1):324–337

    Article  Google Scholar 

  16. Hung C, Kulkarni S, Kuo B (2011) A new weighted fuzzy c-means clustering algorithm for remotely sensed image classification. IEEE J Select Top Signal Process 5(3):543–553

    Article  Google Scholar 

  17. Yang M, Nataliani Y (2018) A feature-reduction fuzzy clustering algorithm based on feature-weighted entropy. IEEE Trans Fuzzy Syst 26(2):817–835

    Article  Google Scholar 

  18. Chang X, Wang Q, Liu Y, Wang Y (2017) Sparse regularization in fuzzy \(c\)-means for high-dimensional data clustering. IEEE Trans Cybern 47(9):2616–2627

    Article  Google Scholar 

  19. Hathaway RJ, Bezdek JC, Hu Y (2000) Generalized fuzzy c-means clustering strategies using \(L_p\) norm distances. IEEE Trans Fuzzy Syst 8(5):576–582

    Article  Google Scholar 

  20. Wang T, Li H, Qian Y, Huang B, Zhou X (2020) A regret-based three-way decision model under interval type-2 fuzzy environment. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2020.3033448

    Article  Google Scholar 

  21. Zhang C, Li H, Chen C, Qian Y, Zhou X (2020) Enhanced group sparse regularized nonconvex regression for face recognition. IEEE Trans Pattern Anal Mach Intell. https://doi.org/10.1109/TPAMI.2020.3033994

    Article  Google Scholar 

  22. Nocedal J, Wright S (2006) Numerical optimization. Springer, Berlin, pp 455–456

    MATH  Google Scholar 

  23. Rosen JB (1961) The gradient projection method for nonlinear programming. Part I. Linear constraints. J Soc Ind Appl Math 9(4):514–532

    Article  Google Scholar 

  24. Goldfarb D, Lapidus L (1968) Conjugate gradient method for nonlinear programming problems with linear constraints. Ind Eng Chem Fundam 7(1):142–151

    Article  Google Scholar 

  25. Claerbout JF (1985) Fundamentals of geophysical data processing. Blackwell Scientific Publications, New York, pp 90–91

    Google Scholar 

  26. Zhang XD (2004) Matrix analysis and applications. Tsinghua University Press, Beijing, pp 70–71

    Google Scholar 

  27. Petersen KB, Pedersen MS (2012) The matrix cookbook. Technical University of Denmark, Lyngby, pp 18–19

    Google Scholar 

  28. Xia S, Peng D, Meng D et al (2020) A fast adaptive k-means with no bounds. IEEE Trans Pattern Anal Mach Intell. https://doi.org/10.1109/TPAMI.2020.3008694

    Article  Google Scholar 

  29. Honda K, Notsu A, Ichihashi H (2010) Fuzzy PCA-guided robust \(k\)-means clustering. IEEE Trans Fuzzy Syst 18(1):67–79

    Article  Google Scholar 

  30. Xia S, Zhang Z, Wang G et al (2020) GBNRS: a novel rough set algorithm for fast adaptive attribute reduction in classification. IEEE Trans Knowl Data Eng. https://doi.org/10.1109/TKDE.2020.2997039

    Article  Google Scholar 

  31. Xia S, Wang G, Chen Z et al (2019) Complete random forest based class noise filtering learning for improving the generalizability of classifiers. IEEE Trans Knowl Data Eng 31(11):2063–2078

    Article  Google Scholar 

  32. Xia S, Liu Y, Ding X et al (2019) Granular ball computing classifiers for efficient. Scalable Robust Learn Inf Sci 483:136–152

    Google Scholar 

  33. Davies DL, Bouldin DW (1979) A cluster separation measure. IEEE Trans Pattern Anal Mach Intell PAMI–1(2):224–227

    Article  Google Scholar 

  34. Vinh NX, Epps J, Bailey J (2010) Information theoretic measures for clusterings comparison: variants, properties, normalization and correction for chance. J Mach Learn Res 11(1):2837–2854

    MathSciNet  MATH  Google Scholar 

  35. Rosenberg A, Hirschberg J (2007) V-measure: a conditional entropy-based external cluster evaluation measure. In: Proceedings of the joint 2007 conference on empirical min natural language processing and computational natural language learning, Prague, Czech Republic, pp 410–420

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grant numbers: 61876010, 61806013, and 61906005.

Author information

Authors and Affiliations

  1. Beijing University of Technology, Beijing, China

    Bowen Liu, Yujian Li, Ting Zhang & Zhaoying Liu

  2. Guilin University of Electronic Technology, Guilin, China

    Yujian Li

Authors
  1. Bowen Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yujian Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ting Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhaoying Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yujian Li.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, B., Li, Y., Zhang, T. et al. \(L_{p}\)-norm probabilistic K-means clustering via nonlinear programming. Int. J. Mach. Learn. & Cyber. 12, 1597–1607 (2021). https://doi.org/10.1007/s13042-020-01257-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13042-020-01257-6

Keywords

Search

Navigation