Skip to main content

SSC-\(l_0\): Sparse Subspace Clustering with the \(l_0\) Inequality Constraint

  • Conference paper
  • First Online:
Pattern Recognition (ACPR 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14408))

Included in the following conference series:

  • 407 Accesses

Abstract

Self-expression learning methods often obtain a coefficient matrix to measure the similarity between pairs of samples. However, directly using all points to represent a fixed sample in a class under the self-expression framework may not be ideal, as points from other classes participate in the representing process. To alleviate this issue, this study attempts to achieve representation learning between points only coming from the same class. In practice, it is easier for data points from the same class to represent each other than that from different classes. So, when reconstructing a point, if the number of non-zero elements in the coefficient vector is limited, a model is more likely to select data points from the class where the reconstructed point lies to complete the reconstruction work. Based on this idea, we propose Sparse Subspace Clustering with the \(l_0\) inequality constraint (SSC-\(l_0\)). In SSC-\(l_0\), the \(l_0\) inequality constraint determines the maximum number of non-zero elements in the coefficient vector, which helps SSC-\(l_0\) to conduct representation learning among the points in the same class. After introducing the simplex constraint to ensure the translation invariance of the model, an optimization method concerning \(l_0\) inequality constraint is formed to solve the proposed SSC-\(l_0\), and its convergence is theoretically analyzed. Extensive experiments on well-known datasets demonstrate the superiority of SSC-\(l_0\) compared to several state-of-the-art methods.

This work was supported by the National Natural Science Foundation of China (No. 62076164), Shenzhen Science and Technology Program (No. JCYJ20210324094601005), and Guangdong Basic and Applied Basic Research Foundation (No. 2021A1515011861).

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    https://cs.nyu.edu/~roweis/data/.

  2. 2.

    https://archive.ics.uci.edu/ml/index.php.

  3. 3.

    http://www.escience.cn/people/fpnie/index.html.

  4. 4.

    http://www.cad.zju.edu.cn/home/dengcai/Data/MLData.html.

References

  1. Zhou, J., Pedrycz, W., Wan, J., et al.: Low-rank linear embedding for robust clustering. IEEE Trans. Knowl. Data Eng. 35(5), 5060–5075 (2022)

    Google Scholar 

  2. Li, X., Chen, M., Wang, Q.: Discrimination-aware projected matrix factorization. IEEE Trans. Knowl. Data Eng. 32, 809–814 (2019)

    Article  Google Scholar 

  3. Lu, J., Wang, H., Zhou, J., et al.: Low-rank adaptive graph embedding for unsupervised feature extraction. Pattern Recogn. 113, 107758 (2021)

    Article  Google Scholar 

  4. Wang, Y., Gao, C., Zhou, J.: Geometrical structure preservation joint with self-expression maintenance for adaptive graph learning. Neurocomputing 501, 436–450 (2022)

    Article  Google Scholar 

  5. Lu, J., Lai, Z., Wang, H., et al.: Generalized embedding regression: a framework for supervised feature extraction. IEEE Trans. Neural Netw. Learn. Syst. 33, 185–199 (2022)

    Article  MathSciNet  Google Scholar 

  6. Ng, A., Jordan, M., Weiss, Y.: On spectral clustering: analysis and an algorithm. In: Advances in Neural Information Processing Systems, Vancouver, pp. 849–856. MIT (2002)

    Google Scholar 

  7. Stella, X., Shi, J.: Multiclass spectral clustering. In: Proceedings of the 9th IEEE International Conference on Computer Vision, Nice, pp. 313–319. IEEE Computer Society (2003)

    Google Scholar 

  8. MacQueen, J.: Some methods for classification and analysis of multivariate observations. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, pp. 281–297 (1967)

    Google Scholar 

  9. Nie, F., Wang, X., Huang, H.: Clustering and projected clustering with adaptive neighbors. In: Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, New York, pp. 977–986. ACM (2014)

    Google Scholar 

  10. Lu, J., Lin, J., Lai, Z., et al.: Target redirected regression with dynamic neighborhood structure. Inf. Sci. 544, 564–584 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhou, J., Gao, C., Wang, X., et al.: Typicality-aware adaptive similarity matrix for unsupervised learning. IEEE Trans. Neural Netw. Learn. Syst. (2023). https://doi.org/10.1109/TNNLS.2023.3243914

  12. Elhamifar, E., Vidal, R.: Sparse subspace clustering: algorithm, theory, and applications. IEEE Trans. Pattern Anal. Mach. Intell. 35, 2765–2781 (2013)

    Article  Google Scholar 

  13. Liu, G., Lin, Z., Yan, S., et al.: Robust recovery of subspace structures by low-rank representation. IEEE Trans. Pattern Anal. Mach. Intell. 35, 171–184 (2013)

    Article  Google Scholar 

  14. Lu, C., Feng, J., Lin, Z., et al.: Subspace clustering by block diagonal representation. IEEE Trans. Pattern Anal. Mach. Intell. 41, 487–501 (2019)

    Article  Google Scholar 

  15. Lu, C., Lin, Z., Yan, S.: Smoothed low rank and sparse matrix recovery by iteratively reweighted least squares minimization. IEEE Trans. Image Process. 24, 646–654 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lu, C.-Y., Min, H., Zhao, Z.-Q., Zhu, L., Huang, D.-S., Yan, S.: Robust and efficient subspace segmentation via least squares regression. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012. LNCS, vol. 7578, pp. 347–360. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-33786-4_26

    Chapter  Google Scholar 

  17. Huang, J., Nie, F., Huang, H.: A new simplex sparse learning model to measure data similarity for clustering. In: Proceedings of the 24th International Conference on Artificial Intelligence, Buenos Aires, pp. 3569–3575. AAAI (2015)

    Google Scholar 

  18. Maggu, J., Majumdar, A., Chouzenoux, E.: Transformed subspace clustering. IEEE Trans. Knowl. Data Eng. 33, 1796–1801 (2020)

    Article  Google Scholar 

  19. Zelnik-Manor, L., Perona, P.: Self-tuning spectral clustering, in: Advances in Neural Information Processing Systems, Montreal, pp. 1601–1608 (2004)

    Google Scholar 

  20. Condat, L.: Fast projection onto the simplex and the \(l\)-1 ball. Math. Program. 158, 575–585 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lyons, M., Budynek, J., Akamatsu, S.: Automatic classification of single facial images. IEEE Trans. Pattern Anal. Mach. Intell. 21, 1357–1362 (1999)

    Article  Google Scholar 

  22. Samaria, F., Harter, A.: Parameterisation of a stochastic model for human face identification. In: Proceedings of 1994 IEEE Workshop on Applications of Computer Vision, Sarasota, pp. 138–142. IEEE (1994)

    Google Scholar 

  23. Sim, T., Baker, S., Bsat, M.: The CMU Pose, illumination and expression database of human faces. Carnegie Mellon University Technical Report CMU-RI-TR-OI-02 (2001)

    Google Scholar 

  24. Phillips, P., Moon, H., Rizvi, S., et al.: The FERET evaluation methodology for face-recognition algorithms. IEEE Trans. Pattern Anal. Mach. Intell. 22, 1090–1104 (2000)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jie Zhou .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Wang, Y., Zhou, J., Lin, Q., Lu, J., Gao, C. (2023). SSC-\(l_0\): Sparse Subspace Clustering with the \(l_0\) Inequality Constraint. In: Lu, H., Blumenstein, M., Cho, SB., Liu, CL., Yagi, Y., Kamiya, T. (eds) Pattern Recognition. ACPR 2023. Lecture Notes in Computer Science, vol 14408. Springer, Cham. https://doi.org/10.1007/978-3-031-47665-5_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-47665-5_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-47664-8

  • Online ISBN: 978-3-031-47665-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics