Advertisement

Manifold Embedding of Graphs Using the Heat Kernel

  • Xiao Bai
  • Richard C. Wilson
  • Edwin R. Hancock
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3604)

Abstract

In this paper, we investigate the use of heat kernels as a means of embedding the individual nodes of a graph on a manifold in a vector space. The heat kernel of the graph is found by exponentiating the Laplacian eigen-system over time. We show how the spectral representation of the heat kernel can be used to compute both Euclidean and geodesic distances between nodes. We use the resulting pattern of distances to embed the nodes of the graph on a manifold using multidimensional scaling. The distribution of embedded points can be used to characterise the graph, and can be used for the purposes of graph clustering. Here for the sake of simplicity, we use spatial moments. We experiment with the resulting algorithms on the COIL database, and they are demonstrated to offer a useful margin of advantage over existing alternatives.

Keywords

Heat Kernel Geodesic Distance Laplacian Matrix Rand Index Laplacian Eigenvalue 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some its algorithmic application. Combinatorica 15, 215–245 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alexandrov, A.D., Zalgaller, V.A.: Intrinsic geometry of surfaces. Transl. Math. Monographs 15 (1967)Google Scholar
  3. 3.
    Busemann, H.: The geometry of geodesics. Academic Press, London (1955)zbMATHGoogle Scholar
  4. 4.
    Weinberger, S.: Review of algebraic l-theory and topological manifolds by a.ranicki. BAMS 33, 93–99 (1996)CrossRefGoogle Scholar
  5. 5.
    Ranicki, A.: Algebraic l-theory and topological manifolds. Cambridge University Press, Cambridge (1992)zbMATHGoogle Scholar
  6. 6.
    Tenenbaum, J.B., Silva, V.D., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 586–591 (2000)CrossRefGoogle Scholar
  7. 7.
    Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)CrossRefGoogle Scholar
  8. 8.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. Neural Information Processing Systems 14, 634–640 (2002)Google Scholar
  9. 9.
    Chung, F.R.K.: Spectral graph theory. American Mathematical Society, Providence (1997)zbMATHGoogle Scholar
  10. 10.
    Atkins, J.E., Bowman, E.G., Hendrickson, B.: A spectral algorithm for seriation and the consecutive ones problem. SIAM J. Comput. 28, 297–310 (1998)zbMATHCrossRefGoogle Scholar
  11. 11.
    Shokoufandeh, A., Dickinson, S., Siddiqi, K., Zucker, S.: Indexing using a spectral encoding of topological structure. In: IEEE Conf. on Computer Vision and Pattern Recognition, pp. 491–497 (1999)Google Scholar
  12. 12.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE PAMI 22, 888–905 (2000)Google Scholar
  13. 13.
    Umeyama, S.: An eigen decomposition approach to weighted graph matching problems. IEEE PAMI 10, 695–703 (1988)zbMATHGoogle Scholar
  14. 14.
    Luo, B., Hancock, E.: Structural graph matching using the em algorithm and singular value decomposition. IEEE PAMI 23, 1120–1136 (2001)Google Scholar
  15. 15.
    Yau, S.T., Schoen, R.M.: Differential geometry. Science Publication (1988)Google Scholar
  16. 16.
    Gilkey, P.B.: Invariance theory, the heat equation, and the atiyah-singer index theorem. Publish or Perish Inc. (1984)Google Scholar
  17. 17.
    Grigor’yan, A.: Heat kernels on manifolds, graphs and fractals. European Congress of Mathematics: Barcelona I, 393–406 (2001)MathSciNetGoogle Scholar
  18. 18.
    Lafferty, J., Lebanon, G.: Diffusion kernels on statistical manifolds. Technical Report CMU-CS-04-101 (2004)Google Scholar
  19. 19.
    Barlow, M.T.: Diffusions on fractals. In: TPHOLs 1999. Lecture Notes Math, vol. 1690, pp. 1–121 (1998)Google Scholar
  20. 20.
    Smola, A.J., Bartlett, P.L., Schölkopf, B., Schuurmans, D.: Advances in large margin classifiers, vol. 354, pp. 5111–5136. MIT Press, Cambridge (2000)zbMATHGoogle Scholar
  21. 21.
    de Verdiere, C.: Spectra of graphs. Math of France 4 (1998)Google Scholar
  22. 22.
    Rosenberg, S.: The laplacian on a Riemannian manifold. Cambridge University Press, Cambridge (2002)Google Scholar
  23. 23.
    Flusser, J., Suk, T.: Pattern recognition by affine moment invariants. Pattern Recognition 26, 167–174 (1993)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Harris, C., Stephens, M.: A combined corner and edge detector. In: Fourth Alvey Vision Conference, pp. 147–151 (1994)Google Scholar
  25. 25.
    Rand, W.M.: Objective criteria for the evaluation of clustering. Journal of the American Statistical Association 66, 846–850 (1971)CrossRefGoogle Scholar
  26. 26.
    Lindman, H., Caelli, T.: Constant curvature Riemannian scaling. Journal of Mathematical Psychology 17, 89–109 (1978)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Xiao Bai
    • 1
  • Richard C. Wilson
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkYorkUK

Personalised recommendations