Abstract
This paper describes how heat-kernel asymptotics can be used to compute approximate Euclidean distances between nodes in a graph. The distances are used to embed the graph-nodes in a low-dimensional space by performing Multidimensional Scaling(MDS). We perform an analysis of the distances, and demonstrate that they are related to the sectional curvature of the connecting geodesic on the manifold. Experiments with moment invariants computed from the embedded points show that they can be used for graph clustering.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Witten, E., Green, M.B., Schwarz, J.H.: Superstring theory. Cambridge University Press, Cambridge (1988)
Chung, F.R.K.: Spectral graph theory. American Mathematical Society, Providence (1997)
Gilkey, P.B.: Invariance theory, the heat equation, and the atiyah-singer index theorem. Perish Inc (1984)
Harris, C.G., Stephens, M.J.: A combined corner and edge detector. In: Fourth Alvey Vision Conference, pp. 147–151 (1994)
Lindman, H., Caelli, T.: Constant curvature Riemannian scaling. Journal of Mathematical Psychology 17, 89–109 (1978)
Flusser, J., Suk, T.: Pattern recognition by affine moment invariants. Pattern recognition 26, 167–174 (1993)
Luo, B., Hancock, E.R.: Structural graph matching using the em algorithm and singular value decomposition. IEEE PAMI 23, 1120–1136 (2001)
Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE PAMI 22, 888–905 (2000)
Rosenberg, S.: The laplacian on a Riemannian manifold. Cambridge University Press, Cambridge (2002)
Yau, S.T., Schoen, R.M.: Differential geometry. Science Publication, Beijing (1988)
Umeyama, S.: An eigen decomposition approach to weighted graph matching problems. IEEE PAMI 10, 695–703 (1988)
Rand, W.M.: Objective criteria for the evaluation of clustering. Journal of the American Statistical Association 66, 846–850 (1971)
Bai, X., Hancock, E.R.: Heat kernels, manifolds and graph embedding. Structural,Syntactic, and Statistical Pattern Recognition, 198–206 (2004)
Bai, X., Yu, H., Hancock, E.R.: Graph matching using manifold embedding. In: Campilho, A.C., Kamel, M.S. (eds.) ICIAR 2004. LNCS, vol. 3211, pp. 198–206. Springer, Heidelberg (2004)
Bai, X., Yu, H., Hancock, E.R.: Graph matching using spectral embedding and semidefinite programming. In: British Machine Vision Conference, pp. 297–307 (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
Bai, X., Hancock, E.R. (2005). Recent Results on Heat Kernel Embedding of Graphs. In: Brun, L., Vento, M. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2005. Lecture Notes in Computer Science, vol 3434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31988-7_36
Download citation
DOI: https://doi.org/10.1007/978-3-540-31988-7_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25270-2
Online ISBN: 978-3-540-31988-7
eBook Packages: Computer ScienceComputer Science (R0)