Abstract
In this paper, we explore whether the geometric properties of the point distribution obtained by embedding the nodes of a graph on a manifold can be used for the purposes of graph clustering. The embedding is performed using the heat-kernel of the graph, computed by exponentiating the Laplacian eigen-system. By equating the spectral heat kernel and its Gaussian form we are able to approximate the Euclidean distance between nodes on the manifold. The difference between the geodesic and Euclidean distances can be used to compute the sectional curvatures associated with the edges of the graph. To characterise the manifold on which the graph resides, we use the normalised histogram of sectional curvatures. By performing PCA on long-vectors representing the histogram bin-contents, we construct a pattern space for sets of graphs. We apply the technique to images from the COIL database, and demonstrate that it leads to well defined graph clusters.
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References
Alexandrov, A.D., Zalgaller, V.A.: Intrinsic geometry of surfaces. Transl. Math. Monographs 15 (1967)
Ranicki, A.: Algebraic l-theory and topological manifolds. Cambridge University Press, Cambridge (1992)
Wilson, R.C., Luo, B., Hancock, E.R.: Spectral embedding of graphs. Pattern Recognition 36, 2213–2223 (2003)
Chung, F.R.K.: Spectral graph theory. American Mathematical Society (1997)
Gilkey, P.B.: Invariance theory, the heat equation, and the atiyah-singer index theorem. Publish or Perish Inc. (1984)
Harris, C.G., Stephens, M.J.: A combined corner and edge detector. In: Fourth Alvey Vision Conference, pp. 147–151 (1994)
Sachs, H., Cvetkovic, D.M., Doob, M.: Spectra of graphs. Academic Press, London (1980)
Tenenbaum, J.B., Silva, V.D., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 586–591 (2000)
Luo, B., Wilson, R.C., Hancock, E.R.: Pattern vectors from algebraic graph theory. IEEE Transactions on Pattern Analysis and Machine Intelligence 27 (2005) (to appear)
Shokoufandeh, A., Dickinson, S., Siddiqi, K., Zucker, S.: Indexing using a spectral encoding of topological structure. In: CVPR, pp. 491–497 (1999)
Rosenberg, S.: The laplacian on a Riemannian manifold. Cambridge University Press, Cambridge (2002)
Yau, S.T., Schoen, R.M.: Differential geometry. Science Publication (1988)
Bai, X., Hancock, E.R.: Heat kernels, manifolds and graph embedding. Structural, Syntactic, and Statistical Pattern Recognition, 198–206 (2004)
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© 2005 Springer-Verlag Berlin Heidelberg
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Xiao, B., Hancock, E.R. (2005). Geometric Characterisation of Graphs. In: Roli, F., Vitulano, S. (eds) Image Analysis and Processing – ICIAP 2005. ICIAP 2005. Lecture Notes in Computer Science, vol 3617. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11553595_58
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DOI: https://doi.org/10.1007/11553595_58
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28869-5
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