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Abstract

Commute time has proved to be a powerful attribute for clustering and characterising graph structure, and which is easily computed from the Laplacian spectrum. Moreover, commute time is robust to deletions of random edges and noisy edge weights. In this paper, we explore the idea of using convolution kernel to compare the distributions of commute time over pairs of graphs. We commence by computing the commute time distance in graphs. We then use a Gaussian convolution kernel to compare distributions. We use kernel kmeans for clustering and use kernel PCA for illustration using the COIL object recognition database.

Keywords

commute times laplacian graph kernel convolution kernel 

References

  1. 1.
    Kriegel, H.P., Borgwardt, K.M.: Shortest path kernels on graphs. In: Proceedings of the Fifth IEEE International Conference on Data Mining (ICDM 2005), pp. 74–81 (2005)Google Scholar
  2. 2.
    Ong, C.S., Schönauer, S., Vishwanathan, S.V.N., Smola, A.J., Kriegel, H.-P., Borgwardt, K.M.: Protein function prediction via graph kernels. In: In Proceedings of Intelligent Systems in Molecular Biology (ISMB), Detroit, USA (2005)Google Scholar
  3. 3.
    Bunke, K., Riesen, H.: A family of novel graph kernels for structural pattern recognition. LNCS, pp. 20–31. Springer, Heidelberg (2007)Google Scholar
  4. 4.
    Yau, S.T., Chung, F.: Discrete green’s function. Journal of Combinatorial Theory Series A 91(1-2), 191–214 (2004)MathSciNetGoogle Scholar
  5. 5.
    Duffy, N., Collins, M.: Convolution kernels for natural language. Advances in Neural Information Processing Systems 14 (2002)Google Scholar
  6. 6.
    Cox, M.A.A., Cox, T.F.: Multidimensional Scaling. Chapman and Hall, Boca Raton (2001)zbMATHGoogle Scholar
  7. 7.
    Jain, A., Dubuisson, M.: A modified haursdoff distance for object matching. In: Proc. 12th Int. Conf. Pattern Recognition, pp. 566–568 (1994)Google Scholar
  8. 8.
    Flach, P., Wrobel, S., Gartner, T.: On graph kernels: Hardness results and efficient alternatives. In: Scholkopf, B., Warmuth, M. (eds.) Sixteen Annual Conference on Computational Learning Theory and Seventh Kernel Workshop, COLT (2003)Google Scholar
  9. 9.
    Gartner, T.: Exponential and geometric kernels for graphs. In: NIP 2002 Workshop on Unreal Data, Volume Principles of Modelling Nonvectorial Data (2002)Google Scholar
  10. 10.
    Haussler, D.: Convolution kernels on discrete structure. Technical report, UCSC-CRL-99-10, UC Sansta Cruz (1999)Google Scholar
  11. 11.
    Gartner, T., Wrobel-S. Hovarth, T.: Cyclic pattern kernels for predictive graph mining. In: Proceedings of International Conference on Knowloedge Discovery and Data Mining (KDD), pp. 158–167 (2004)Google Scholar
  12. 12.
    Hancock, E.R., Huet, B.: Relational object recognition from large structural libraries. Pattern Recognition 35(9), 1895–1915 (2002)zbMATHCrossRefGoogle Scholar
  13. 13.
    Tsuda, K., Inokuchi-A. Kashima, H.: Marginalized kernels between labelled graphs. In: In Proceedings of the 20th International Conference on Machine Learning (ICML), Washington, DC, United States (2003)Google Scholar
  14. 14.
    Saunders, C., Shawe-Taylor, -J., Christianini, N., Watkins, C., Lodhi, H.: Text classification using string kernels. Journal of Machine Learning Research 2, 419–444 (2002)zbMATHCrossRefGoogle Scholar
  15. 15.
    Ueda, N., Akutsu, -T., -Perret, J.-L., Vert, J.-P., Mahe, P.: Extensions of marginalized graph kernels. In: ICML (2004)Google Scholar
  16. 16.
    Lafon, S., coifman, -R.R., -Kevrekidis, I.G., Nadler, B.: Diffusion maps, spectral clustering and eigenfunctions of fokker-planck operators. In: Advances in Neural Information Processing Systems, vol. 18 (2005)Google Scholar
  17. 17.
    Nayar, S., Murase, -H., Nene, S.: Columbia object image library: Coil (1996)Google Scholar
  18. 18.
    Bunke, H., Neuhaus, M.: Edit distance based kernel functions for structural pattern classification. Pattern Recognition 39(10), 1852–1863 (2006)zbMATHCrossRefGoogle Scholar
  19. 19.
    Hancock, E.R., Qiu, H.: Clustering & embedding using commute times. IEEE Transactions on Pattern Analysis & Machine Intelligence(PAMI) 29(11), 1873–1890 (2007)CrossRefGoogle Scholar
  20. 20.
    Smola, A., Muller, -K.R., Scholkopf, B.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation 10, 1299–1319 (1998)CrossRefGoogle Scholar
  21. 21.
    Smola, A.J., Muller, -K.-R., Scholkopf, B.: Kernel principal component analysis. In: Scholkopf, B., Burges, C.J.C., Smola, A.J. (eds.) Advances in Kernels Methods- Support Vector Learning, pp. 327–352 (1999)Google Scholar
  22. 22.
    Christianini, N., Shawe-Taylor, J.: Kernel Methods for Pattern Analysis. Cambridge University Press, New York (2004)Google Scholar
  23. 23.
    Borgwardt, K.M., Schraudolph, -N.N., Vishwanathan, S.: Fast computation of graph kernels. In: Adv. NIPS (2007)Google Scholar
  24. 24.
    Smola, -J., Vishwanathan, S.V.N.: Fast kernels for strings and tree matching. Advances in Neural Information Processing Systems 15 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Normawati A Rahman
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkYorkUK

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