This paper explores the use of commute-time preserving embedding as means of data-clustering. Commute time is a measure of the time taken for a random walk to set-out and return between a pair of nodes on a graph. It may be computed from the spectrum of the Laplacian matrix. Since the commute time is averaged over all potential paths between a pair of nodes, it is potentially robust to variations in graph structure due to edge insertions or deletions. Here we demonstrate how nodes of a graph can be embedded in a vector space in a manner that preserves commute time. We present a number of important properties of the embedding. We experiment with the method for separating object motions in image sequences.


Laplacian Matrix Graph Match Locality Preserve Projection Commute Time Eigenvector Matrix 
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.


  1. 1.
    Bai, X., Yu, H., Hancock, E.R.: Graph matching using spectral embedding and alignment. In: ICPR, pp. 398–401 (2004)Google Scholar
  2. 2.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. In: Advances in Neural Information Processing Systems, pp. 585–591 (2001)Google Scholar
  3. 3.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation 15(6), 1373–1396 (2003)MATHCrossRefGoogle Scholar
  4. 4.
    Caelli, T., Kosinov, S.: An eigenspace projection clustering method for inexact graph matching. IEEE Trans. Pattern Anal. Mach. Intell. 26(4), 515–519 (2004)CrossRefGoogle Scholar
  5. 5.
    Carcassoni, M., Hancock, E.R.: Spectral correspondence for point pattern matching. Pattern Recognition 36(1), 193–204 (2003)MATHCrossRefGoogle Scholar
  6. 6.
    Chung, F.R.K., Yau, S.-T.: Discrete green’s functions. J. Combin. Theory Ser., 191–214 (2000)Google Scholar
  7. 7.
    Costeira, J., Kanade, T.: A multi-body factorization method for motion analysis. In: ICCV, pp. 1071–1076 (1995)Google Scholar
  8. 8.
    Costeira, J., Kanade, T.: A multibody factorization method for independently moving objects. IJCV 29(3), 159–179 (1997)CrossRefGoogle Scholar
  9. 9.
    Harel, D., Koren, Y.: Graph drawing by high-dimensional embedding. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 207–219. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  10. 10.
    He, X., Niyogi, P.: Locality preserving projections. In: NIPS, pp. 585–591 (2003)Google Scholar
  11. 11.
    Luo, B., Wilson, R.C., Hancock, E.R.: Spectral embedding of graphs. In: 2002 Winter Workshop on Computer Vision (2002)Google Scholar
  12. 12.
    MacQueen, J.B.: Some methods for classification and analysis of multivariate observations. In: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, pp. 281–297 (1967)Google Scholar
  13. 13.
    Qiu, H., Hancock, E.R.: Image segmentation using commute times. In: BMVC, pp. 929–938 (2005)Google Scholar
  14. 14.
    Roweis, S., Saul, L.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)CrossRefGoogle Scholar
  15. 15.
    Coifman, R.R., Lafon, S., Lee, A.B., Maggioni, M., Nadler, B., Warner, F., Zucker, S.W.: Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps. National Academy of Sciences 102(21), 7426–7431 (2005)CrossRefGoogle Scholar
  16. 16.
    Sch, B., Smola, A., Muller, K.: Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation 10, 1299–1319 (1998)CrossRefGoogle Scholar
  17. 17.
    Shapiro, L., Brady, J.: Feature-based correspondence: an eigenvector approach. Image and Vision Computing 10(2), 283–288 (1992)CrossRefGoogle Scholar
  18. 18.
    Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE PAMI 22(8), 888–905 (2000)Google Scholar
  19. 19.
    Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319–2323 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Huaijun Qiu
    • 1
  • Edwin R. Hancock
    • 1
  1. 1.Department of Computer ScienceUniversity of YorkYorkUK

Personalised recommendations