Abstract

This paper describes a new approach for embedding graphs on pseudo-Riemannian manifolds based on the wave kernel. The wave kernel is the solution of the wave equation on the edges of a graph. Under the embedding, each edge becomes a geodesic on the manifold. The eigensystem of the wave-kernel is determined by the eigenvalues and the eigenfunctions of the normalized adjacency matrix and can be used to solve the edge-based wave equation. By factorising the Gram-matrix for the wave-kernel, we determine the embedding co-ordinates for nodes under the wave-kernel. We investigate the utility of this new embedding as a means of gauging the similarity of graphs. We experiment on sets of graphs representing the proximity of image features in different views of different objects. By applying multidimensional scaling to the similarity matrix we demonstrate that the proposed graph representation is capable of clustering different views of the same object together.

Keywords

Wave Equation Pseudo Riemannian manifolds Edge-based Laplacian Graph Embedding 

References

  1. 1.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Leen, T.K., Dietterich, T.G., Tresp, V. (eds.) Advances in neural information processing systems, vol. 14. MIT Press, Cambridge (2002)Google Scholar
  2. 2.
    Clarke, C.J.S.: On the global isometric embedding of pseudo-riemannian manifolds. In: Proceedings of Royal Society of London. A, vol. 314, pp. 417–428 (1970)Google Scholar
  3. 3.
    Cox, T., Cox, M.: Multidimensional Scaling. Chapman-Hall, Boca Raton (1994)MATHGoogle Scholar
  4. 4.
    Dubuisson, M., Jain, A.: A modified hausdorff distance for object matching, pp. 566–568 (1994)Google Scholar
  5. 5.
    ElGhawalby, H., Hancock, E.R.: Measuring graph similarity using spectral geometry. In: Campilho, A., Kamel, M.S. (eds.) ICIAR 2008. LNCS, vol. 5112, pp. 517–526. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Friedman, J., Tillich, J.P.: Wave equations for graphs and the edge based laplacian. Pacific Journal of Mathematics 216(2), 229–266 (2004)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Heidelberg (1983)MATHGoogle Scholar
  8. 8.
    Hurt, N.E.: Mathematical physics of quantum wires and devices. Kluwer Academic Publishers, Dordrecht (2000)MATHGoogle Scholar
  9. 9.
    Luo, B., Wilson, R.C., Hancock, E.R.: Spectral embedding of graphs. Pattern Recogintion [36], 2213–2230 (2003)MATHCrossRefGoogle Scholar
  10. 10.
    Nash, J.F.: C1-isometric imbeddings. Ann. Math. 60, 383–396 (1954)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Nash, J.F.: The imbedding problem for riemannian manifolds. Ann. Math. 63, 20–63 (1956)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Pekalska, E., Haasdonk, B.: Kernel discriminant analysis for positive definite and indefinite kernels. IEEE transactions on pattern analysis and machine intelligence 31(6), 1017–1032 (2009)CrossRefGoogle Scholar
  13. 13.
    Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)CrossRefGoogle Scholar
  14. 14.
    Whitney, H.: Differentiable manifolds. Ann. of Math. 37(2), 645–680 (1936)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Xiao, B., Hancock, E.R., Wilson, R.C.: Graph characteristics from the heat kernel trace. Pattern Recognition 42(11), 2589–2606 (2009)MATHCrossRefGoogle Scholar
  16. 16.
    Xiao, B., Hancock, E.R., Yu, H.: Manifold embeddingforshapeanalysis. Neurocomputing 73, 1606–1613 (2010)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Hewayda ElGhawalby
    • 1
    • 2
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkUK
  2. 2.Faculty of EngineeringSuez Canal universityEgypt

Personalised recommendations