This paper concerns the analysis of patterns that are specified in terms of non-Euclidean dissimilarity or proximity rather than ordinal values. In prior work we have reported a means of correcting or rectifying the similarities so that the non-Euclidean artifacts are minimized. This is achieved by representing the data using a graph, and evolving the manifold embedding of the graph using Ricci flow. Although the method provides encouraging results, it can prove to be unstable. In this paper we explore how this problem can be overcome using a graph regularisation technique. Specifically, by regularising the curvature of the manifold on which the graph is embedded, then we can improve both the stability and performance of the method. We demonstrate the utility of our method on the standard “Chicken pieces” dataset and show that we can transform the non-Euclidean distances into Euclidean space.


Heat Kernel Gaussian Curvature Negative Eigenvalue Geodesic Distance Dual Graph 
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.


  1. 1.
    Pekalska, E., Duin, R.P.W.: Beyond traditional kernels: classification in two dissimilarity-based representation spaces. IEEE Transactions on Systems Man and Cybernetics-Part C 38(6) (November 2008)Google Scholar
  2. 2.
    Sanfeliu, A., Fu, K.S.: A distance measure between attributed relational graphs for pattern recognition. IEEE transactions on systems, man, and cybernetics 13(3), 353–362 (1983)zbMATHGoogle Scholar
  3. 3.
    Bunke, H.: A graph distance metric based on the maximal common subgraph. Pattern Recognition Letters 19(3-4), 255–259 (1998)zbMATHCrossRefGoogle Scholar
  4. 4.
    Borg, I., Groenen, P.: Modern multidimensional scaling: Theory and applications. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  5. 5.
    Tenenbaum, J., Silva, V., Langford, J.: A global geometric framework for nonlinear dimensionality reduction. Science 290, 2319–2323 (2000)CrossRefGoogle Scholar
  6. 6.
    Roweis, S., Saul, L.: Nonlinear dimensionality reduction by locally linear embedding (2000)Google Scholar
  7. 7.
    Belkin, M., Niyogi, P.: Laplacian eigenmaps and spectral techniques for embedding and clustering. Advances in neural information processing systems 1, 585–592 (2002)Google Scholar
  8. 8.
    Duin, R.P.W., Pekalska, E., Harol, A., Lee, W.J., Bunke, H.: On euclidean corrections for non-euclidean dissimilarities. In: SSPR/SPR, pp. 551–561 (2008)Google Scholar
  9. 9.
    Pekalska, E., Duin, R., Gunter, S., Bunke, H.: On not making dissimilarities euclidean. Lecture notes in computer science, pp. 1145–1154 (2004)Google Scholar
  10. 10.
    Xu, W., Hancock, E.R.W.: Rectifying non-euclidean similarity data using ricci flow embedding. In: To appear ICPR 2010 (August 2010)Google Scholar
  11. 11.
    Pekalska, E., Harol, A., Duin, R., Spillmann, B., Bunke, H.: Non-euclidean or non-metric measures can be informative, pp. 871–880 (2006)Google Scholar
  12. 12.
    Chow, B., Luo, F.: Combinatorial Ricci flows on surfaces. J. Differential Geom. 63(1), 97–129 (2003)zbMATHMathSciNetGoogle Scholar
  13. 13.
    ElGhawalby, H., Hancock, E.R.: Measuring graph similarity using spectral geometry. In: ICIAR, pp. 517–526 (2008)Google Scholar
  14. 14.
    Lindman, H., Caelli, T.: Constant curvature riemannian scaling. Journal of Mathematical Psychology 17, 89–109 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kondor, R., Lafferty, J.: Diffusion kernels on graphs and other discrete structures. In: Proceedings of the ICML, pp. 315–322 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Weiping Xu
    • 1
  • Edwin R. Hancock
    • 1
  • Richard C. Wilson
    • 1
  1. 1.Dept. of Computer ScienceUniversity of YorkUK

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