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Spectral clustering based on matrix perturbation theory

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Abstract

This paper exposes some intrinsic characteristics of the spectral clustering method by using the tools from the matrix perturbation theory. We construct a weight matrix of a graph and study its eigenvalues and eigenvectors. It shows that the number of clusters is equal to the number of eigenvalues that are larger than 1, and the number of points in each of the clusters can be approximated by the associated eigenvalue. It also shows that the eigenvector of the weight matrix can be used directly to perform clustering; that is, the directional angle between the two-row vectors of the matrix derived from the eigenvectors is a suitable distance measure for clustering. As a result, an unsupervised spectral clustering algorithm based on weight matrix (USCAWM) is developed. The experimental results on a number of artificial and real-world data sets show the correctness of the theoretical analysis.

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References

  1. Bach R, Jordan M I. Learning spectral clustering. University of California at Berkeley Technical report UCB/CSD-03-1249. 2003

  2. Xing E P, Jordan M I. On semidefinite relaxation for normalized k-cut and connections to spectral clustering. University of California at Berkeley Technical report UCB/CSD-3-1265. 2003

  3. Donath W E, Hoffman A J. Lower bounds for partitioning of graphs. IBM J Res Devel, 1973, 17(5): 420–425

    Article  MATH  MathSciNet  Google Scholar 

  4. Fiedler M. A property of eigenvectors of non-negative symmetric matrices and its application to graph theory. Czechoslovak Mathemat J, 1975, 25(100): 619–633

    MathSciNet  Google Scholar 

  5. Hagen L, Kahng A B. New spectral methods for ratio cut partitioning and clustering. IEEE Trans Comput-Aid Design, 1992, 11(9): 1074–1085

    Article  Google Scholar 

  6. Chan P K, Schlag M D F, Zien J Y. Spectral k-way ratio-cut partitioning and clustering. IEEE Trans Comput-Aid Design Integ Circ Syst, 1994, 13(9): 1088–1096

    Article  Google Scholar 

  7. Shi J, Malik J. Normalized cuts and image segmentation. IEEE Trans Patt Anal Mach Intel, 2000, 22(8): 888–905

    Article  Google Scholar 

  8. Fowlkes C, Belongie S, Chung F, et al. Spectral grouping using the Nyström method. IEEE Trans Patt Anal Mach Intel, 2004, 26(2): 214–225

    Article  Google Scholar 

  9. Ding C H Q, He X, Zha H, et al. A min-max cut algorithm for graph partitioning and data clustering. In: Cercone N, Lin T Y, Wu X, eds. ICDM 2001. Los Alamitos, California: IEEE Computer Society, 2001. 107–114

    Chapter  Google Scholar 

  10. Ding C H Q, He X, Zha H. A spectral method to separate disconnected and nearly-disconnected web graph components. In: Provost F, Srikant R, eds. Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. New York: Association for Computing Machinery, 2001. 275–280.

    Chapter  Google Scholar 

  11. Weiss Y. Segmentation using eigenvectors: a unifying view. In: Computer Vision, 1999, the proceedings of the Seventh IEEE International Conference on. Los Alamitos, California: IEEE Computer Society, 1999. 975–982

    Google Scholar 

  12. Dhillon I S, Guan Y, Kulis B. A unified view of kernel k-means, spectral clustering and graph cuts. University of Texas at Austin UTCS Technical Report TR-04-25. 2004

  13. Kannan R, Vempala S, Vetta A. On clusterings: good, bad and spectral. J ACM, 2004, 51(3): 597–515

    Article  MathSciNet  Google Scholar 

  14. 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. Cambridge, MA: MIT Press, 2002. 849–856

    Google Scholar 

  15. Brand M, Huang K. A unifying theorem for spectral embedding and clustering. Mitsubishi Electric Research Laboratory Technical Report TR2002-42. 2002.

  16. Sun J. Matrix Perturbation Analysis (in Chinese). 2nd ed. Beijing: Science Press, 2001. 252–272

    Google Scholar 

  17. Hettich S, Bay S D. The UCI KDD Archive [http://kdd.ics.uci.edu]. Irvine, CA: University of California, Department of Information and Computer Science, 1999

    Google Scholar 

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Correspondence to Tian Zheng.

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Supported by the National Natural Science Foundation of China (Grant No. 60375003) and the Aeronatical Science Foundation of China (Grant No. 03I53059)

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Tian, Z., Li, X. & Ju, Y. Spectral clustering based on matrix perturbation theory. SCI CHINA SER F 50, 63–81 (2007). https://doi.org/10.1007/s11432-007-0007-8

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  • DOI: https://doi.org/10.1007/s11432-007-0007-8

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