Abstract
In this paper, we investigate the use of heat kernels as a means of embedding the individual nodes of a graph on a manifold in a vector space. The heat kernel of the graph is found by exponentiating the Laplacian eigen-system over time. We show how the spectral representation of the heat kernel can be used to compute both Euclidean and geodesic distances between nodes. We use the resulting pattern of distances to embed the nodes of the graph on a manifold using multidimensional scaling. The distribution of embedded points can be used to characterise the graph, and can be used for the purposes of graph clustering. Here for the sake of simplicity, we use spatial moments. We experiment with the resulting algorithms on the COIL database, and they are demonstrated to offer a useful margin of advantage over existing alternatives.
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References
Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some its algorithmic application. Combinatorica 15, 215–245 (1995)
Alexandrov, A.D., Zalgaller, V.A.: Intrinsic geometry of surfaces. Transl. Math. Monographs 15 (1967)
Busemann, H.: The geometry of geodesics. Academic Press, London (1955)
Weinberger, S.: Review of algebraic l-theory and topological manifolds by a.ranicki. BAMS 33, 93–99 (1996)
Ranicki, A.: Algebraic l-theory and topological manifolds. Cambridge University Press, Cambridge (1992)
Tenenbaum, J.B., Silva, V.D., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 586–591 (2000)
Roweis, S.T., Saul, L.K.: Nonlinear dimensionality reduction by locally linear embedding. Science 290, 2323–2326 (2000)
Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. Neural Information Processing Systems 14, 634–640 (2002)
Chung, F.R.K.: Spectral graph theory. American Mathematical Society, Providence (1997)
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)
Shi, J., Malik, J.: Normalized cuts and image segmentation. IEEE PAMI 22, 888–905 (2000)
Umeyama, S.: An eigen decomposition approach to weighted graph matching problems. IEEE PAMI 10, 695–703 (1988)
Luo, B., Hancock, E.: Structural graph matching using the em algorithm and singular value decomposition. IEEE PAMI 23, 1120–1136 (2001)
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. Publish or Perish Inc. (1984)
Grigor’yan, A.: Heat kernels on manifolds, graphs and fractals. European Congress of Mathematics: Barcelona I, 393–406 (2001)
Lafferty, J., Lebanon, G.: Diffusion kernels on statistical manifolds. Technical Report CMU-CS-04-101 (2004)
Barlow, M.T.: Diffusions on fractals. In: TPHOLs 1999. Lecture Notes Math, vol. 1690, pp. 1–121 (1998)
Smola, A.J., Bartlett, P.L., Schölkopf, B., Schuurmans, D.: Advances in large margin classifiers, vol. 354, pp. 5111–5136. MIT Press, Cambridge (2000)
de Verdiere, C.: Spectra of graphs. Math of France 4 (1998)
Rosenberg, S.: The laplacian on a Riemannian manifold. Cambridge University Press, Cambridge (2002)
Flusser, J., Suk, T.: Pattern recognition by affine moment invariants. Pattern Recognition 26, 167–174 (1993)
Harris, C., Stephens, M.: A combined corner and edge detector. In: Fourth Alvey Vision Conference, pp. 147–151 (1994)
Rand, W.M.: Objective criteria for the evaluation of clustering. Journal of the American Statistical Association 66, 846–850 (1971)
Lindman, H., Caelli, T.: Constant curvature Riemannian scaling. Journal of Mathematical Psychology 17, 89–109 (1978)
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Bai, X., Wilson, R.C., Hancock, E.R. (2005). Manifold Embedding of Graphs Using the Heat Kernel. In: Martin, R., Bez, H., Sabin, M. (eds) Mathematics of Surfaces XI. Lecture Notes in Computer Science, vol 3604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537908_3
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DOI: https://doi.org/10.1007/11537908_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28225-9
Online ISBN: 978-3-540-31835-4
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