In this paper, we describe the use of Riemannian geometry, and in particular the relationship between the Laplace-Beltrami operator and the graph Laplacian, for the purposes of embedding a graph onto a Riemannian manifold. Using the properties of Jacobi fields, we show how to compute an edge-weight matrix in which the elements reflect the sectional curvatures associated with the geodesic paths between nodes on the manifold. We use the resulting edge-weight matrix to embed the nodes of the graph onto a Riemannian manifold of constant sectional curvature. With the set of embedding coordinates at hand, the graph matching problem is cast as that of aligning pairs of manifolds subject to a geometric transformation. We illustrate the utility of the method on image matching using the COIL database.


Riemannian Manifold Sectional Curvature Curvature Tensor Riemannian Geometry Graph Match 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Antonio Robles-Kelly
    • 1
  • Edwin R. Hancock
    • 2
  1. 1.National ICT Australia, RSISE Bldg 115, ANUAustralia
  2. 2.Dept. of Comp. ScienceThe University of YorkYorkUK

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