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Kernel Spectral Clustering and Applications

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Unsupervised Learning Algorithms

Abstract

In this chapter we review the main literature related to kernel spectral clustering (KSC), an approach to clustering cast within a kernel-based optimization setting. KSC represents a least-squares support vector machine-based formulation of spectral clustering described by a weighted kernel PCA objective. Just as in the classifier case, the binary clustering model is expressed by a hyperplane in a high dimensional space induced by a kernel. In addition, the multi-way clustering can be obtained by combining a set of binary decision functions via an Error Correcting Output Codes (ECOC) encoding scheme. Because of its model-based nature, the KSC method encompasses three main steps: training, validation, testing. In the validation stage model selection is performed to obtain tuning parameters, like the number of clusters present in the data. This is a major advantage compared to classical spectral clustering where the determination of the clustering parameters is unclear and relies on heuristics. Once a KSC model is trained on a small subset of the entire data, it is able to generalize well to unseen test points. Beyond the basic formulation, sparse KSC algorithms based on the Incomplete Cholesky Decomposition (ICD) and L 0, \(L_{1},L_{0} + L_{1}\), Group Lasso regularization are reviewed. In that respect, we show how it is possible to handle large-scale data. Also, two possible ways to perform hierarchical clustering and a soft clustering method are presented. Finally, real-world applications such as image segmentation, power load time-series clustering, document clustering, and big data learning are considered.

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Notes

  1. 1.

    In this case the given data points represent the node of the graph and their similarity the corresponding edges.

  2. 2.

    This is a considerable novelty, since SVMs are typically known as classifiers or function approximation models rather than clustering techniques.

  3. 3.

    By choosing V = I, problem (3) is identical to kernel PCA [48, 58, 62].

  4. 4.

    http://www.esat.kuleuven.be/stadius/ADB/alzate/softwareKSClab.php.

  5. 5.

    http://www.esat.kuleuven.be/stadius/ADB/langone/softwareSKSClab.php.

  6. 6.

    A C implementation of the algorithm can be downloaded at: http://www.esat.kuleuven.be/stadius/ADB/novak/softwareKSCICD.php.

  7. 7.

    The images have been extracted from the Berkeley image database [45].

  8. 8.

    Here we use the cosine kernel described in Table 1.

  9. 9.

    In our experiments we used the mean silhouette value (MSV) as an internal cluster validation criterion to select the value of ρ which gives more coherent clusters.

  10. 10.

    A Matlab implementation of the algorithm can be downloaded at: http://www.esat.kuleuven.be/stadius/ADB/mall/softwareKSCnet.php.

  11. 11.

    In [40] this model selection step has been eliminated by proposing a self-tuned method where the structure of the projections in the eigenspace is exploited to automatically identify an optimal cluster structure.

References

  1. Alzate, C., Sinn, M.: Improved electricity load forecasting via kernel spectral clustering of smart meters. In: ICDM, pp. 943–948 (2013)

    Google Scholar 

  2. Alzate, C., Suykens, J.A.K.: Multiway spectral clustering with out-of-sample extensions through weighted kernel PCA. IEEE Trans. Pattern Anal. Mach. Intell. 32(2), 335–347 (2010)

    Article  Google Scholar 

  3. Alzate, C., Suykens, J.A.K.: Sparse kernel spectral clustering models for large-scale data analysis. Neurocomputing 74(9), 1382–1390 (2011)

    Article  Google Scholar 

  4. Alzate, C., Suykens, J.A.K.: Hierarchical kernel spectral clustering. Neural Networks 35, 21–30 (2012)

    Article  MATH  Google Scholar 

  5. Alzate, C., Espinoza, M., De Moor, B., Suykens, J.A.K.: Identifying customer profiles in power load time series using spectral clustering. In: Proceedings of the 19th International Conference on Neural Networks (ICANN 2009), pp. 315–324 (2009)

    Google Scholar 

  6. Baeza-Yates, R., Ribeiro-Neto, B.: Modern Information Retrieval. Addison-Wesley, Boston (1999)

    Google Scholar 

  7. Ben-Israel, A., Iyigun, C.: Probabilistic d-clustering. J. Classif. 25(1), 5–26 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bishop, C.M.: Pattern Recognition and Machine Learning (Information Science and Statistics). Springer, New York (2006)

    MATH  Google Scholar 

  9. Blondel, V.D., Guillaume, J.L., Lambiotte, R., Lefebvre, E.: Fast unfolding of communities in large networks. J. Stat. Mech. Theory Exp. 2008(10), P10,008 (2008)

    Google Scholar 

  10. Candes, E.J., Wakin, M.B., Boyd, S.: Enhancing sparsity by reweighted l1 minimization. J. Fourier Anal. Appl. (Special Issue on Sparsity) 14(5), 877–905 (2008)

    Google Scholar 

  11. Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1997)

    MATH  Google Scholar 

  12. De Brabanter, K., De Brabanter, J., Suykens, J.A.K., De Moor, B.: Optimized fixed-size kernel models for large data sets. Comput. Stat. Data Anal. 54(6), 1484–1504 (2010)

    Google Scholar 

  13. Deerwester, S.C., Dumais, S.T., Landauer, T.K., Furnas, G.W., Harshman, R.A.: Indexing by latent semantic analysis. J. Am. Soc. Inf. Sci. 41(6), 391–407 (1990)

    Article  Google Scholar 

  14. Delvenne, J.C., Yaliraki, S.N., Barahona, M.: Stability of graph communities across time scales. Proc. Natl. Acad. Sci. 107(29), 12755–12760 (2010)

    Article  Google Scholar 

  15. Dhanjal, C., Gaudel, R., Clemenccon, S.: Efficient eigen-updating for spectral graph clustering (2013) [arXiv/1301.1318]

    Google Scholar 

  16. Dhillon, I., Guan, Y., Kulis, B.: Kernel k-means, spectral clustering and normalized cuts. In: 10th ACM Knowledge Discovery and Data Mining Conf., pp. 551–556 (2004)

    Google Scholar 

  17. Dhillon, I., Guan, Y., Kulis, B.: Weighted graph cuts without eigenvectors a multilevel approach. IEEE Trans. Pattern Anal. Mach. Intell. 29(11), 1944–1957 (2007)

    Article  Google Scholar 

  18. Espinoza, M., Joye, C., Belmans, R., De Moor, B.: Short-term load forecasting, profile identification and customer segmentation: a methodology based on periodic time series. IEEE Trans. Power Syst. 20(3), 1622–1630 (2005)

    Article  Google Scholar 

  19. Fowlkes, C., Belongie, S., Chung, F., Malik, J.: Spectral grouping using the Nyström method. IEEE Trans. Pattern Anal. Mach. Intell. 26(2), 214–225 (2004)

    Article  Google Scholar 

  20. Frederix, K., Van Barel, M.: Sparse spectral clustering method based on the incomplete cholesky decomposition. J. Comput. Appl. Math. 237(1), 145–161 (2013)

    Google Scholar 

  21. Friedman, J., Hastie, T., Tibshirani, R.: A note on the group lasso and a sparse group lasso (2010) [arXiv:1001.0736]

    Google Scholar 

  22. Huang, K., Zheng, D., Sun, J., Hotta, Y., Fujimoto, K., Naoi, S.: Sparse learning for support vector classification. Pattern Recogn. Lett. 31(13), 1944–1951 (2010)

    Article  Google Scholar 

  23. Hubert, L., Arabie, P.: Comparing partitions. J. Classif. 1(2), 193–218 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lancichinetti, A., Radicchi, F., Ramasco, J.J., Fortunato, S.: Finding statistically significant communities in networks. PLoS ONE 6(4), e18961 (2011)

    Article  Google Scholar 

  25. Langone, R., Suykens, J.A.K.: Community detection using kernel spectral clustering with memory. J. Phys. Conf. Ser. 410(1), 012100 (2013)

    Article  Google Scholar 

  26. Langone, R., Alzate, C., Suykens, J.A.K.: Modularity-based model selection for kernel spectral clustering. In: Proc. of the International Joint Conference on Neural Networks (IJCNN 2011), pp. 1849–1856 (2011)

    Google Scholar 

  27. Langone, R., Alzate, C., Suykens, J.A.K.: Kernel spectral clustering for community detection in complex networks. In: Proc. of the International Joint Conference on Neural Networks (IJCNN 2012), pp. 2596–2603 (2012)

    Google Scholar 

  28. Langone, R., Alzate, C., Suykens, J.A.K.: Kernel spectral clustering with memory effect. Phys. A Stat. Mech. Appl. 392(10), 2588–2606 (2013)

    Article  MathSciNet  Google Scholar 

  29. Langone, R., Alzate, C., De Ketelaere, B., Suykens, J.A.K.: Kernel spectral clustering for predicting maintenance of industrial machines. In: IEEE Symposium Series on Computational Intelligence and data mining SSCI (CIDM) 2013, pp. 39–45 (2013)

    Google Scholar 

  30. Langone, R., Mall, R., Suykens, J.A.K.: Soft kernel spectral clustering. In: Proc. of the International Joint Conference on Neural Networks (IJCNN 2013), pp. 1–8 (2013)

    Google Scholar 

  31. Langone, R., Mall, R., Suykens, J.A.K.: Clustering data over time using kernel spectral clustering with memory. In: SSCI (CIDM) 2014, pp. 1–8 (2014)

    Google Scholar 

  32. Langone, R., Agudelo, O.M., De Moor, B., Suykens, J.A.K.: Incremental kernel spectral clustering for online learning of non-stationary data. Neurocomputing 139, 246–260 (2014)

    Google Scholar 

  33. Langone, R., Alzate, C., De Ketelaere, B., Vlasselaer, J., Meert, W., Suykens, J.A.K.: Ls-svm based spectral clustering and regression for predicting maintenance of industrial machines. Eng. Appl. Artif. Intell. 37, 268–278 (2015)

    Google Scholar 

  34. Leskovec, J., Kleinberg, J., Faloutsos, C.: Graph evolution: densification and shrinking diameters. ACM Trans. Knowl. Discov. Data 1(1), 2 (2007)

    Article  Google Scholar 

  35. Leskovec, J., Lang, K.J., Mahoney, M.: Empirical comparison of algorithms for network community detection. In: Proceedings of the 19th International Conference on World Wide Web, WWW ’10, pp. 631–640. ACM, New York (2010)

    Google Scholar 

  36. Liao, T.W.: Clustering of time series data - a survey. Pattern Recogn. 38(11), 1857–1874 (2005)

    Article  MATH  Google Scholar 

  37. Lin, F., Cohen, W.W.: Power iteration clustering. In: ICML, pp. 655–662 (2010)

    Google Scholar 

  38. Mall, R., Langone, R., Suykens, J.: FURS: fast and unique representative subset selection retaining large scale community structure. Soc. Netw. Anal. Min. 3(4), 1–21 (2013)

    Article  Google Scholar 

  39. Mall, R., Langone, R., Suykens, J.A.K.: Kernel spectral clustering for big data networks. Entropy (Special Issue on Big Data) 15(5), 1567–1586 (2013)

    Google Scholar 

  40. Mall, R., Langone, R., Suykens, J.A.K.: Self-tuned kernel spectral clustering for large scale networks. In: IEEE International Conference on Big Data (2013)

    Book  Google Scholar 

  41. Mall, R., Langone, R., Suykens, J.A.K.: Agglomerative hierarchical kernel spectral data clustering. In: Symposium Series on Computational Intelligence (SSCI-CIDM), pp. 1–8 (2014)

    Google Scholar 

  42. Mall, R., Langone, R., Suykens, J.A.K.: Multilevel hierarchical kernel spectral clustering for real-life large scale complex networks. PLoS ONE 9(6), e99966 (2014)

    Article  Google Scholar 

  43. Mall, R., Mehrkanoon, S., Langone, R., Suykens, J.A.K.: Optimal reduced sets for sparse kernel spectral clustering. In: Proc. of the International Joint Conference on Neural Networks (IJCNN 2014), pp. 2436–2443 (2014)

    Google Scholar 

  44. Mall, R., Jumutc, V., Langone, R., Suykens, J.A.K.: Representative subsets for big data learning using kNN graphs. In: IEEE International Conference on Big Data, pp. 37–42 (2014)

    Google Scholar 

  45. Martin, D., Fowlkes, C., Tal, D., Malik, J.: A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics. In: Proc. 8th Int’l Conf. Computer Vision, vol. 2, pp. 416–423 (2001)

    Google Scholar 

  46. Meila, M., Shi, J.: Learning segmentation by random walks. In: T.K. Leen, T.G. Dietterich, V. Tresp (eds.) Advances in Neural Information Processing Systems, vol. 13. MIT Press, Cambridge (2001)

    Google Scholar 

  47. Meila, M., Shi, J.: A random walks view of spectral segmentation. In: Artificial Intelligence and Statistics AISTATS (2001)

    Google Scholar 

  48. Mika, S., Schölkopf, B., Smola, A.J., Müller, K.R., Scholz, M., Rätsch, G.: Kernel PCA and de-noising in feature spaces. In: Kearns, M.S., Solla, S.A., Cohn, D.A. (eds.) Advances in Neural Information Processing Systems, vol. 11. MIT Press, Cambridge (1999)

    Google Scholar 

  49. Newman, M.E.J.: Modularity and community structure in networks. Proc. Natl. Acad. Sci. U. S. A. 103(23), 8577–8582 (2006)

    Article  Google Scholar 

  50. Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E 69(2), 026113 (2004)

    Article  Google Scholar 

  51. Ng, A.Y., Jordan, M.I., Weiss, Y.: On spectral clustering: analysis and an algorithm. In: Dietterich, T.G., Becker, S., Ghahramani, Z. (eds.) Advances in Neural Information Processing Systems 14, pp. 849–856. MIT Press, Cambridge (2002)

    Google Scholar 

  52. Ning, H., Xu, W., Chi, Y., Gong, Y., Huang, T.S.: Incremental spectral clustering by efficiently updating the eigen-system. Pattern Recogn. 43(1), 113–127 (2010)

    Article  MATH  Google Scholar 

  53. Novak, M., Alzate, C., langone, R., Suykens, J.A.K.: Fast kernel spectral clustering based on incomplete cholesky factorization for large scale data analysis. Internal Report 14–119, ESAT-SISTA, KU Leuven (Leuven, Belgium) (2015)

    Google Scholar 

  54. Peluffo, D., Garcia, S., Langone, R., Suykens, J.A.K., Castellanos, G.: Kernel spectral clustering for dynamic data using multiple kernel learning. In: Proc. of the International Joint Conference on Neural Networks (IJCNN 2013), pp. 1085–1090 (2013)

    Google Scholar 

  55. Puzicha, J., Hofmann, T., Buhmann, J.: Non-parametric similarity measures for unsupervised texture segmentation and image retrieval. In: Computer Vision and Pattern Recognition, pp. 267–272 (1997)

    Google Scholar 

  56. Rosvall, M., Bergstrom, C.T.: Maps of random walks on complex networks reveal community structure. Proc. Natl. Acad. Sci. 105(4), 1118–1123 (2008)

    Article  Google Scholar 

  57. Rousseeuw, P.J.: Silhouettes: a graphical aid to the interpretation and validation of cluster analysis. J. Comput. Appl. Math. 20(1), 53–65 (1987)

    Article  MATH  Google Scholar 

  58. Schölkopf, B., Smola, A.J., Müller, K.R.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput. 10, 1299–1319 (1998)

    Article  Google Scholar 

  59. Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 22(8), 888–905 (2000)

    Article  Google Scholar 

  60. Strehl, A., Ghosh, J.: Cluster ensembles - a knowledge reuse framework for combining multiple partitions. J. Mach. Learn. Res. 3, 583–617 (2002)

    MathSciNet  MATH  Google Scholar 

  61. Suykens, J.A.K., Van Gestel, T., De Brabanter, J., De Moor, B., Vandewalle, J.: Least Squares Support Vector Machines. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

  62. Suykens, J.A.K., Van Gestel, T., Vandewalle, J., De Moor, B.: A support vector machine formulation to PCA analysis and its kernel version. IEEE Trans. Neural Netw. 14(2), 447–450 (2003)

    Article  Google Scholar 

  63. von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 17(4), 395–416 (2007)

    Article  MathSciNet  Google Scholar 

  64. Williams, C.K.I., Seeger, M.: Using the Nyström method to speed up kernel machines. In: Advances in Neural Information Processing Systems, vol. 13. MIT Press, Cambridge (2001)

    Google Scholar 

  65. Yuan, M., Lin, Y.: Model selection and estimation in regression with grouped variables. J. R. Stat. Soc. 68(1), 49–67 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  66. Zhu, J., Rosset, S., Hastie, T., Tibshirani, R.: 1-norm svms. In: Neural Information Processing Systems, vol. 16 (2003)

    Google Scholar 

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Acknowledgements

EU: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC AdG A-DATADRIVE-B (290923). This chapter reflects only the authors’ views, the Union is not liable for any use that may be made of the contained information. Research Council KUL: GOA/10/09 MaNet, CoE PFV/10/002 (OPTEC), BIL12/11T; PhD/Postdoc grants. Flemish Government: FWO: projects: G.0377.12 (Structured systems), G.088114N (Tensor-based data similarity); PhD/Postdoc grants. IWT: projects: SBO POM (100031); PhD/Postdoc grants. iMinds Medical Information Technologies SBO 2014. Belgian Federal Science Policy Office: IUAP P7/19 (DYSCO, Dynamical systems, control and optimization, 2012–2017.)

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Langone, R., Mall, R., Alzate, C., Suykens, J.A.K. (2016). Kernel Spectral Clustering and Applications. In: Celebi, M., Aydin, K. (eds) Unsupervised Learning Algorithms. Springer, Cham. https://doi.org/10.1007/978-3-319-24211-8_6

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