Spectral clustering via half-quadratic optimization

  • Xiaofeng Zhu
  • Jiangzhang Gan
  • Guangquan Lu
  • Jiaye Li
  • Shichao ZhangEmail author
Part of the following topical collections:
  1. Computational Social Science as the Ultimate Web Intelligence


Spectral clustering has been demonstrated to often outperform K-means clustering in real applications because it improves the similarity measurement of K-means clustering. However, previous spectral clustering method still suffers from the following issues: 1) easily being affected by outliers; 2) constructing the affinity matrix from original data which often contains redundant features and outliers; and 3) unable to automatically specify the cluster number. This paper focuses on address these issues by proposing a new clustering algorithm along with the technique of half-quadratic optimization. Specifically, the proposed method learns the affinity matrix from low-dimensional space of original data, which is obtained by using a robust estimator to remove the influence of outliers as well as a sparsity regularization to remove redundant features. Moreover, the proposed method employs the 2,1-norm regularization to automatically learn the cluster number according to the data distribution. Experimental results on both synthetic and real data sets demonstrated that the proposed method outperforms the state-of-the-art methods in terms of clustering performance.


Spectral clustering Subspace learning Feature selection M-estimation Half-quadratic optimization 



This work is partially supported by the China Key Research Program (Grant No: 2016YFB1000905), the Natural Science Foundation of China (Grants No: 61836016, 61876046, 61573270, and 61672177), the Project of Guangxi Science and Technology (GuiKeAD17195062), the Guangxi Collaborative Innovation Center of Multi-Source Information Integration and Intelligent Processing, the Guangxi High Institutions Program of Introducing 100 High-Level Overseas Talents, the Strategic Research Excellence Fund at Massey University, the Marsden Fund of New Zealand (Grant No: MAU1721), and the Research Fund of Guangxi Key Lab of Multisource Information Mining and Security (18-A-01-01).


  1. 1.
    Alberto, P., Casbon, J.A., Saqi, M.A.S.: Spectral clustering of protein sequences. Nucleic Acids Res. 34(5), 1571–1580 (2006)CrossRefGoogle Scholar
  2. 2.
    Boyd, V., Faybusovich, L.: Convex optimization. IEEE Trans. Autom. Control 51(11), 1859–1859 (2006)CrossRefGoogle Scholar
  3. 3.
    Charbonnier, P., Blancfraud, L., Aubert, G., Barlaud, M.: Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Process. 6(2), 298–311 (1997)CrossRefGoogle Scholar
  4. 4.
    Edgar, R.C.: Search and clustering orders of magnitude faster than blast. Bioinformatics 26(19), 2460 (2010)CrossRefGoogle Scholar
  5. 5.
    Elhamifar, E., Vidal, R.: Sparse subspace clustering: Algorithm, theory, and applications. IEEE Trans Pattern Anal Mach Intell 35(11), 2765–2781 (2013)CrossRefGoogle Scholar
  6. 6.
    Goh, A., Vidal, R.: Locally linear manifold clustering. Journal of Machine Learning Research (2009)Google Scholar
  7. 7.
    Guangcan, L., Zhouchen, L., Shuicheng, Y., Ju, S., Yu, Y., Yi, M.: Robust recovery of subspace structures by low-rank representation. IEEE Trans. Pattern Anal. Mach. Intell. 35(1), 171–184 (2013)CrossRefGoogle Scholar
  8. 8.
    Hartigan, J.A., Wong, M.A.: A k-means clustering algorithm. In: ECCV, pp. 492–503 (2008)Google Scholar
  9. 9.
    He, R., Hu, B., Yuan, X., Wang, L.: M-Estimators and Half-Quadratic minimization (2014)Google Scholar
  10. 10.
    Hu, H., Lin, Z., Feng, J., Zhou, J.: Smooth representation clustering. In: CVPR, pp. 3834–3841 (2014)Google Scholar
  11. 11.
    Hu, M., Yang, Y., Shen, F., Xie, N., Hong, R., Shen, H.T.: Collective reconstructive embeddings for cross-modal hashing. IEEE Transactions on Image Processing. (2019)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Huber, P.J.: Robust statistics. Wiley, New York (2011)Google Scholar
  13. 13.
    Ji, Z., Tao, X., Wang, H.: Outlier detection from large distributed databases. World Wide Web 17(4), 539–568 (2014)CrossRefGoogle Scholar
  14. 14.
    Junye, G., Liu, M., Zhang, D.: Spectral clustering algorithm based on effective distance. Journal of Frontiers of Computer Science & Technology (2014)Google Scholar
  15. 15.
    Law, M.H.C., Figueiredo, J., Jain, A.K.: Simultaneous feature selection and clustering using mixture models. IEEE Trans Pattern Anal Mach Intell 26(9), 1154–1166 (2004)CrossRefGoogle Scholar
  16. 16.
    Lei, C., Zhu, X.: Unsupervised feature selection via local structure learning and sparse learning. Multimed Tools Appl 77(22), 29605–29622 (2018)CrossRefGoogle Scholar
  17. 17.
    Li, C.G., 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
  18. 18.
    Liu, X.Y., Jing-Wei, L.I., Hong, Y.U., You, Q.Z., Lin, H.F.: Adaptive spectral clustering based on shared nearest neighbors. J. Chin. Comput. Syst. 32(9), 1876–1880 (2011)Google Scholar
  19. 19.
    Liu, G., Lin, Z., Yan, S., Ju, S., Yu, Y., Ma, Y.: Robust recovery of subspace structures by low-rank representation. IEEE Trans Pattern Anal Mach Intell 35(1), 171 (2013)CrossRefGoogle Scholar
  20. 20.
    Ng, A.Y., Jordan, M.I., Weiss, Y.: On spectral clustering: analysis and an algorithm. In: NIPS, pp .849–856 (2001)Google Scholar
  21. 21.
    Nie, F., Wang, X., Jordan, M.I., Huang, H.: The constrained laplacian rank algorithm for graph-based clustering. In: AAAI, pp. 1969–1976 (2016)Google Scholar
  22. 22.
    Nikolova, M., Ng, MK.: Analysis of Half-Quadratic minimization methods for signal and image recovery. Society for Industrial and Applied Mathematics (2005)Google Scholar
  23. 23.
    Peng, Y., Zhu, Q., Huang, B.: Spectral clustering with density sensitive similarity function. Knowl.-Based Syst. 24(5), 621–628 (2011)CrossRefGoogle Scholar
  24. 24.
    Roth, V., Lange, T.: Feature selection in clustering problems. In: NIPS (2003)Google Scholar
  25. 25.
    Shah, S.A., Koltun, V.: Robust continuous clustering. Proc Natl Acad Sci U SA 114(37), 9814–9819 (2017)CrossRefGoogle Scholar
  26. 26.
    Szu-Hao, H., Yi-Hong, C., Shang-Hong, L., Novak, C.L.: Learning-based vertebra detection and iterative normalized-cut segmentation for spinal mri. IEEE Trans. Med. Imaging 28(10), 1595–1605 (2009)CrossRefGoogle Scholar
  27. 27.
    Tao, X., Li, Y., Zhong, N.: A personalized ontology model for Web information gathering. IEEE Trans Knowl Data Eng 23(4), 496–511 (2010)CrossRefGoogle Scholar
  28. 28.
    Wang, R., Zong, M.: Joint self-representation and subspace learning for unsupervised feature selection. World Wide Web 21(6), 1745–1758 (2018)CrossRefGoogle Scholar
  29. 29.
    Wang, W.W., Xiao-Ping, L.I., Feng, X.C., Si, Qi W.: A survey on sparse subspace clustering. Acta Autom. Sin. 41(8), 1373–1384 (2015)Google Scholar
  30. 30.
    Wang, L., Zheng, K., Tao, X., Han, X.: Affinity propagation clustering algorithm based on large-scale data-set. International Journal of Computers & Applications (3), 1–6 (2018)Google Scholar
  31. 31.
    Wang, R., Ji, W., Liu, M., Wang, X., Weng, J., Deng, S., Gao, S., Yuan, C.-a.: Review on mining data from multiple data sources. Pattern Recogn. Lett. 109, 120–128 (2018)CrossRefGoogle Scholar
  32. 32.
    Wang, R., Ji, W., Song, B.: Durable relationship prediction and description using a large dynamic graph. World Wide Web 21(6), 1575–1600 (2018)CrossRefGoogle Scholar
  33. 33.
    Yan, J., Pollefeys, M.: A general framework for motion segmentation Independent, articulated, rigid, non-rigid, degenerate and non-degenerate. In: ECCV, pp. 94–106 (2006)Google Scholar
  34. 34.
    Yi, B., Yang, Y., Shen, F., Xie, N., Shen, H.T., Li, X.: Describing video with attention based bidirectional lstm. IEEE Transactions on Cybernetics (2018)Google Scholar
  35. 35.
    Zelnik-Manor, L., Perona, P.: Self-tuning spectral clustering. In: NIPS, pp. 1601–1608 (2005)Google Scholar
  36. 36.
    Zhang, S., Li, X., Ming, Z., Zhu, X., Cheng, D.: Learning k for knn classification. Acm Trans Intell Syst Technol 8(3), 43 (2017)Google Scholar
  37. 37.
    Zhang, S., Li, X., Zong, M., Zhu, X., Wang, R.: Efficient knn classification with different numbers of nearest neighbors. IEEE Trans Neural Netw Learn Syst 29 (5), 1774–1785 (2018)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Zheng, W., Zhu, X., Zhu, Y., Hu, R., Lei, Cong: Dynamic graph learning for spectral feature selection. Multimed Tools Appl 77(22), 29739–29755 (2018)CrossRefGoogle Scholar
  39. 39.
    Zhou, X., Shen, F., Liu, Li, Liu, W., Nie, L., Yang, Y., Shen, H.T.: Graph convolutional network hashing (2018)Google Scholar
  40. 40.
    Zhu, X., Li, X., Zhang, S.: Block-row sparse multiview multilabel learning for image classification. IEEE Trans Cybern 46(2), 450–461 (2016)CrossRefGoogle Scholar
  41. 41.
    Zhu, X., He, W., Li, Y., Yang, Y., Zhang, S., Hu, R., Xhu, Y.: One-step spectral clustering via dynamically learning affinity matrix and subspace, pp. 2963–2969 (2017)Google Scholar
  42. 42.
    Zhu, X., Li, X., Zhang, S., Ju, C., Wu, X.: Robust joint graph sparse coding for unsupervised spectral feature selection. IEEE Trans Neural Netw Learn Syst 28(6), 1263–1275 (2017)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Zhu, X., Li, X., Zhang, S., Xu, Z., Yu, L., Wang, C.: Graph pca hashing for similarity search. IEEE Trans Multimed 19(9), 2033–2044 (2017)CrossRefGoogle Scholar
  44. 44.
    Zhu, X., Shichao, Z., Li, Y., Jilian, Z., Lifeng, Y., Yue, F.: Low-rank sparse subspace for spectral clustering. IEEE Trans Knowl Data Eng 31(8), 1532–1543 (2019)CrossRefGoogle Scholar
  45. 45.
    Zhu, X., Zhang, S., Hu, R., He, W., Lei, C., Zhu, P.: One-step multi-view spectral clustering. IEEE Transactions on Knowledge and Data Engineering. (2018)CrossRefGoogle Scholar
  46. 46.
    Zhu, X., Zhang, S., Li, Y., Zhang, J., Yang, L., Fang, Y.: Low-rank sparse subspace for spectral clustering. IEEE Transactions on Knowledge and Data Engineering. CrossRefGoogle Scholar
  47. 47.
    Zhu, X., Zhu, Y., Zhang, S., Hu, R., He, W.: Adaptive hypergraph learning for unsupervised feature selection, 3581–3587, pp. (2018)Google Scholar
  48. 48.
    Zhu, Y., Zhu, X., Zheng, W.: Robust multi-view learning via half-quadratic minimization, pp. 3278–3284 (2018)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Xiaofeng Zhu
    • 1
    • 2
  • Jiangzhang Gan
    • 2
  • Guangquan Lu
    • 3
  • Jiaye Li
    • 3
  • Shichao Zhang
    • 4
    Email author
  1. 1.University of Electronic Science and Technology of ChinaChengduChina
  2. 2.SNCS of Massey University Auckland CampusAucklandNew Zealand
  3. 3.Guangxi Key Lab of MIMSGuangxi Normal UniversityGuilinChina
  4. 4.Central South UniversityChangshaChina

Personalised recommendations