Abstract
In this paper, we investigate the use of invariants derived from the heat kernel as a means of clustering graphs. We turn to the heat-content, i.e. the sum of the elements of the heat kernel. The heat content can be expanded as a polynomial in time, and the co-efficients of the polynomial are known to be permutation invariants. We demonstrate how the polynomial co-efficients can be computed from the Laplacian eigensystem. Graph-clustering is performed by applying principal components analysis to vectors constructed from the polynomial co-efficients. We experiment with the resulting algorithm on the COIL database, where it is demonstrated to outperform the use of Laplacian eigenvalues.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alexandrov, A.D., Zalgaller, V.A.: Intrinsic geometry of surfaces. Transl. Math. Monographs 15 (1967)
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)
de Verdiere, C.: Spectra of graphs. Math of France, 4 (1998)
Chung, F.R.K.: Spectral graph theory. American Mathematical Society (1997)
Hjaltason, G.R., Samet, H.: Properties of embedding methods for similarity searching in metric spaces. PAMI 25, 530–549 (2003)
Harris, C.G., Stephens, M.J.: A combined corner and edge detector. In: Fourth Alvey Vision Conference, pp. 147–151 (1994)
Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some its algorithmic application. Combinatorica 15, 215–245 (1995)
Mcdonald, P., Meyers, R.: Diffusions on graphs, poisson problems and spectral geometry. Transactions on Amercian Mathematical Society 354, 5111–5136 (2002)
Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE PAMI 22, 888–905 (2000)
Yau, S.T., Schoen, R.M.: Differential geometry. Science Publication (1988)
Umeyama, S.: An eigen decomposition approach to weighted graph matching problems. IEEE PAMI 10, 695–703 (1988)
Weinberger, S.: Review of algebraic l-theory and topological manifolds by a.ranicki. BAMS 33, 93–99 (1996)
Bai, X., Hancock, E.R.: Heat kernels, manifolds and graph embedding. In: Fred, A., Caelli, T.M., Duin, R.P.W., Campilho, A.C., de Ridder, D. (eds.) SSPR&SPR 2004. LNCS, vol. 3138, pp. 198–206. Springer, Heidelberg (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Xiao, B., Hancock, E.R. (2005). Graph Clustering Using Heat Content Invariants. In: Marques, J.S., Pérez de la Blanca, N., Pina, P. (eds) Pattern Recognition and Image Analysis. IbPRIA 2005. Lecture Notes in Computer Science, vol 3523. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11492542_16
Download citation
DOI: https://doi.org/10.1007/11492542_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26154-4
Online ISBN: 978-3-540-32238-2
eBook Packages: Computer ScienceComputer Science (R0)