Abstract

In this paper, we investigate the use of heat kernels as a means of embedding graphs in a pattern space. We commence by performing the spectral decomposition on the graph Laplacian. The heat kernel of the graph is found by exponentiating the resulting eigensystem over time. By equating the spectral heat kernel and its Gaussian form we are able to approximate the geodesic distance between nodes on a manifold. We use the resulting pattern of distances to embed the trees in a Euclidean space using multidimensional scaling. The arrangement of points in this space can be used to construct pattern vectors suitable for clustering the graphs. Here we compute a weighted proximity matrix, and from the proximity matrix a Laplacian matrix is computed. We use the eigenvalues of the Laplacian matrix to characterise the distribution of points representing the embedded nodes. Experiments on sets of shock graphs reveal the utility of the method on real-world data.

Keywords

Heat Kernel Geodesic Distance Laplacian Matrix Gaussian Form 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.

References

  1. 1.
    Alexandrov, A.D., Zalgaller, V.A.: Intrinsic geometry of surfaces. Transl. Math. Monographs 15 (1967)Google Scholar
  2. 2.
    Ranicki, A.: Algebraic l-theory and topological manifolds. Cambridge University Press, Cambridge (1992)MATHGoogle Scholar
  3. 3.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. Neural Information Processing Systems 14, 634–640 (2002)Google Scholar
  4. 4.
    Hjaltason, G.R., Samet, H.: Properties of embedding methods for similarity searching in metric spaces. PAMI 25, 530–549 (2003)Google Scholar
  5. 5.
    Grigor’yan, A.: Heat kernels on manifolds, graphs and fractals (2003) (preprint)Google Scholar
  6. 6.
    Busemann, H.: The geometry of geodesics. Academic Press, London (1955)MATHGoogle Scholar
  7. 7.
    Tenenbaum, J.B., Silva, V.D., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 586–591 (2000)CrossRefGoogle Scholar
  8. 8.
    Lafferty, J., Lebanon, G.: Diffusion kernels on statistical manifolds (2004) (CMU preprint)Google Scholar
  9. 9.
    Coulhon, T., Barlow, M., Grigor’yan, A.: Manifolds and graphs with slow heat kernel decay (2000) (imperial college preprint)Google Scholar
  10. 10.
    Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some its algorithmic application. Combinatorica 15, 215–245 (1995)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Smola, A.J., Kondor, R.: Kernels and regularisation of graphs (2004)Google Scholar
  12. 12.
    Weinberger, S.: Review of algebraic l-theory and topological manifolds by a.ranicki. BAMS 33, 93–99 (1996)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Xiao Bai
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkUK

Personalised recommendations