Efficient Direct Structured Subspace Clustering

  • Wen-ming Cao
  • Rui Li
  • Sheng Qian
  • Si Wu
  • Hau-San WongEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11304)


Subspace clustering splits data instances that are drawn from special low-dimensional subspaces via utilizing similarities between them. Traditional methods contain two steps: (1) learning the affinity matrix and (2) clustering on the affinity matrix. Although these two steps can alternatively contribute to each other, there exist heavy dependencies between the performance and the initial quality of affinity matrix. In this paper, we propose an efficient direct structured subspace clustering approach to reduce the quality effects of the affinity matrix on performances. We first analyze the connection between the affinity and partition matrices, and then fuse the computation of affinity and partition matrices. This fusion allows better preserving the subspace structures which help strengthen connections between data points in the same subspaces. In addition, we introduce an algorithm to optimize our proposed method. We conduct comparative experiments on multiple data sets with state-of-the-art methods. Our method achieves better or comparable performances.


Subspace clustering Unsupervised learning 



The work described in this paper was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 11300715), and a grant from City University of Hong Kong [Project No. 7004884].


  1. 1.
    Zhao, C., Zhang, J., Ma, S., Fan, X., Zhang, Y., Gao, W.: Reducing image compression artifacts by structural sparse representation and quantization constraint prior. IEEE Trans. Circuits Syst. Video Technol. 27(10), 2057–2071 (2017)CrossRefGoogle Scholar
  2. 2.
    Lai, T., Wang, H., Yan, Y., Chin, T., Zhao, W.: Motion segmentation via a sparsity constraint. IEEE Trans. Intell. Transp. Syst. 18(4), 973–983 (2017)CrossRefGoogle Scholar
  3. 3.
    Li, C., You, C., Vidal, R.: Structured sparse subspace clustering: a joint affinity learning and subspace clustering framework. IEEE Trans. Image Process. 26(6), 2988–3001 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Elhamifar, E., Vidal, R.: Sparse subspace clustering: algorithm, theory and applications. IEEE Trans. Patt. Anal. Mach. Intell. 35(11), 2765–2781 (2013)CrossRefGoogle Scholar
  5. 5.
    Ma, Y., Shang, C., Yang, F., Huang, D.: Latent subspace clustering based on deep neural networks. In: 6th International Symposium on Advanced Control of Industrial Processes, pp. 502–507. IEEE Press, Hiroshima (2014)Google Scholar
  6. 6.
    Tian, F., Gao, B., Cui, Q., Chen, E., Liu, T.: Learning deep representations for graph clustering. In: 28th AAAI Conference on Artificial Intelligence, pp. 1293–1299. AAAI Press, Québec (2014)Google Scholar
  7. 7.
    Xie, J., Girshick, R., Farhadi, A.: Unsupervised deep embedding for clustering analysis. In: 33rd International Conference on Machine Learning, New York, pp. 478–487 (2016)Google Scholar
  8. 8.
    Peng, X., Xiao, S., Feng, J., Yau, W., Yi, Z.: Deep subspace clustering with sparsity prior. In: 25th International Joint Conference on Artificial Intelligence, New York, pp. 1925–1931 (2016)Google Scholar
  9. 9.
    Guo, X., Gao, L., Liu, X., Yin, J.: Improved deep embedded clustering with local structure preservation. In: 26th International Joint Conference on Artificial Intelligence, Melbourne, pp. 1753–1759 (2017)Google Scholar
  10. 10.
    Elhamifar, E., Vidal, R.: Sparse subspace clustering. In: 22nd IEEE Conference on Computer Vision and Pattern Recognition, pp. 2790–2797. IEEE Press, Florida (2009)Google Scholar
  11. 11.
    Liu, G., Lin, Z., Yu, Y.: Robust subspace segmentation by low-rank representation. In: 27th International Conference on Machine Learning, Haifa, pp. 663–670 (2010)Google Scholar
  12. 12.
    Liu, G., Yan, S.: Latent low-rank representation for subspace segmentation and feature extraction. In: 13th International Conference on Computer Vision, pp. 1615–1122. IEEE Press, Barcelona (2011)Google Scholar
  13. 13.
    Liu, G., Lin, Z., Yan, S., Sun, J., Yu, Y., Ma, Y.: Robust recovery of subspace structures by low-rank representation. IEEE Trans. Patt. Anal. Mach. Intell. 35(1), 171–184 (2013)CrossRefGoogle Scholar
  14. 14.
    Li, C., Vidal, R.: Structured sparse subspace clustering: a unified optimization framework. In: 28th IEEE Conference on Computer Vision and Pattern Recognition, pp. 277–286. IEEE Press, Boston (2015)Google Scholar
  15. 15.
    Vidal, R., Favaro, P.: Low rank subspace clustering (LRSC). Patt. Recogn. Lett. 43(1), 47–61 (2014)CrossRefGoogle Scholar
  16. 16.
    Feng, J., Lin, Z., Xu, H., Yan, S.: Robust subspace segmentation with block-diagonal prior. In: 27th IEEE Conference on Computer Vision and Pattern Recognition, pp. 3818–3825. IEEE Press, Ohio (2014)Google Scholar
  17. 17.
    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). Scholar
  18. 18.
    Lu, C., Lin, Z., Yan, S.: Correlation adaptive subspace segmentation by trace lasso. In: 14th International Conference on Computer Vision, pp. 1345–1352. IEEE Press, Sydney (2013)Google Scholar
  19. 19.
    Park, D., Caramanis, C, Sanghavi, S.: Greedy subspace clustering. In: 28th Conference on Neural Information Processing Systems, Montréal, pp. 2753–2761 (2014)Google Scholar
  20. 20.
    Yang, Y., Feng, J., Jojic, N., Yang, J., Huang, T.S.: \(\ell ^{0}\)-sparse subspace clustering. In: Leibe, B., Matas, J., Sebe, N., Welling, M. (eds.) ECCV 2016. LNCS, vol. 9906, pp. 731–747. Springer, Cham (2016). Scholar
  21. 21.
    Grave, E., Obozinski, G., Bach, F.: Trace lasso: a trace norm regularization for correlated designs. In: 25th Conference on Neural Information Processing Systems, Granada, pp. 2187–2195 (2011)Google Scholar
  22. 22.
    Patel, V., Vidal, R.: Kernel sparse subspace clustering. In: 22nd International Conference on Pattern Recognition, pp. 2849–2853. IEEE Press, Paris (2014)Google Scholar
  23. 23.
    De la Torre, F., Kanade, T.: Discriminative cluster analysis. In: 23rd International Conference on Machine Learning, Pittsburgh, pp. 241–248 (2006)Google Scholar
  24. 24.
    You, C., Robinson, D., Vidal, R.: Scalable sparse subspace clustering by orthogonal matching pursuit. In: 29th IEEE Conference on Computer Vision and Pattern Recognition, pp. 3918–3927. IEEE Press, Las Vegas (2016)Google Scholar
  25. 25.
    Ji, P., Salzmann, M., Li, H.: Efficient dense subspace clustering. In: IEEE Winter Conference on Application of Computer Vision, pp. 461–468. IEEE Press, Steamboat Springs (2014)Google Scholar
  26. 26.
    Ji, P., Zhang, T., Li, H., Salzmann, M., Reid, I.: Deep subspace clustering networks. In: 32nd Conference on Neural Information Processing Systems, Montréal, pp. 24–33 (2017)Google Scholar
  27. 27.
    Ren, Y., Domeniconi, C., Zhang, G., Yu, G.: A weighted adaptive mean shift clustering algorithm. In: SIAM International Conference on Data Mining, pp. 794–802. SIAM Press, Pennsylvania (2014)CrossRefGoogle Scholar
  28. 28.
    Friedman, J.H., Meulman, J.J.: Clustering objects on subsets of attributes. J. Royal Stat. Soc. Ser. B (Stat. Methodol.) 66, 815–849 (2004)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Wen-ming Cao
    • 1
  • Rui Li
    • 1
  • Sheng Qian
    • 1
  • Si Wu
    • 2
  • Hau-San Wong
    • 1
    Email author
  1. 1.Department of Computer ScienceCity University of Hong KongHong KongChina
  2. 2.School of Computer Science and EngineeringSouth China University of TechnologyGuangzhouChina

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