Abstract
In this paper, we explore how the trace of the heat kernel can be used to characterise graphs for the purposes of measuring similarity and clustering. The heat-kernel is the solution of the heat-equation and may be computed by exponentiating the Laplacian eigensystem with time. To characterise the shape of the heat-kernel trace we use the zeta-function, which is found by exponentiating and summing the reciprocals of the Laplacian eigenvalues. From the Mellin transform, it follows that the zeta-function is the moment generating function of the heat-kernel trace. We explore the use of the heat-kernel moments as a means of characterising graph structure for the purposes of clustering. Experiments with the COIL and Oxford-Caltech databases reveal the effectiveness of the representation.
Chapter PDF
Similar content being viewed by others
Keywords
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
Chung, F.R.K.: Spectral Graph Theory American Mathematical Society (1997)
Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE PAMI 22, 888–905 (2000)
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)
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)
Coifman, R.R., Lafon, S.: Diffusion maps. Applied and Computational Harmonic Analysis (2004)
Lafferty, J., Lebanon, G.: Diffusion kernels on statistical manifolds. Technical Reports CMU-CS-04-101 (2004)
Yau, S.T., Schoen, R.M.: Differential geometry. Science Publication (1988)
Gilkey, P.B.: Invariance theory, the heat equation, and the atiyah-singer index theorem. Perish Inc. (1984)
Roweis, S., Saul, L.: Nonlinear dimensionality reduction by locally linear embedding. Science 290(5500), 2323–2326 (2000)
Luo, B., Wilson, R.C., Hancock, E.R.: Spectral embedding of graphs. Pattern Recognition 36, 2213–2230 (2003)
Rosenberg, S.: The laplacian on a Riemannian manifold. Cambridge University Press, Cambridge (2002)
Mohar, B.: Laplace eigenvalues of graphs - a Survey. Discrete Math. 109, 171–183 (1992)
Lovaz, L.: Random Walks on Graphs: A Survey. Combinatorics, Paul Erds is eighty 2, 353–397 (1996)
Hein, M., Audibert, J.Y., von Luxburg, U.: From graphs to manifolds – Weak and strong pointwise consistency of graph laplacian. In: Auer, P., Meir, R. (eds.) COLT 2005. LNCS, vol. 3559, pp. 470–485. Springer, Heidelberg (2005)
Belkin, M., Niyogi, P.: Laplacian Eigenmaps for Dimensionality Reduction and Data Representation. Neural Computation 15, 1373–1396 (2003)
Tenenbaum, J.B., Silva, V.D., Langford, J.C.: A global geometric framework for non-linear dimensionality reduction. Science 290, 586–591 (2000)
He, X., Niyogi, P.: Locality preserving projections. In: NIPS 2003 (2003)
Harris, C.G., Stephens, M.J.: A Combined Corner and Edge Detector. In: Fourth Alvey Vision Conference, pp. 147–151 (1994)
Sarkar, S., Boyer, K.L.: Quantitative measures of change based on feature organization: Eigenvalues and eigenvectors. In: Computer Vision and Image Understanding, pp. 110–136 (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Xiao, B., Hancock, E.R. (2006). Trace Formula Analysis of Graphs. In: Yeung, DY., Kwok, J.T., Fred, A., Roli, F., de Ridder, D. (eds) Structural, Syntactic, and Statistical Pattern Recognition. SSPR /SPR 2006. Lecture Notes in Computer Science, vol 4109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11815921_33
Download citation
DOI: https://doi.org/10.1007/11815921_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37236-3
Online ISBN: 978-3-540-37241-7
eBook Packages: Computer ScienceComputer Science (R0)