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 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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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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