Non-Euclidean or Non-metric Measures Can Be Informative

  • Elżbieta Pękalska
  • Artsiom Harol
  • Robert P. W. Duin
  • Barbara Spillmann
  • Horst Bunke
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4109)


Statistical learning algorithms often rely on the Euclidean distance. In practice, non-Euclidean or non-metric dissimilarity measures may arise when contours, spectra or shapes are compared by edit distances or as a consequence of robust object matching [1,2]. It is an open issue whether such measures are advantageous for statistical learning or whether they should be constrained to obey the metric axioms.

The k-nearest neighbor (NN) rule is widely applied to general dissimilarity data as the most natural approach. Alternative methods exist that embed such data into suitable representation spaces in which statistical classifiers are constructed [3]. In this paper, we investigate the relation between non-Euclidean aspects of dissimilarity data and the classification performance of the direct NN rule and some classifiers trained in representation spaces. This is evaluated on a parameterized family of edit distances, in which parameter values control the strength of non-Euclidean behavior. Our finding is that the discriminative power of this measure increases with increasing non-Euclidean and non-metric aspects until a certain optimum is reached. The conclusion is that statistical classifiers perform well and the optimal values of the parameters characterize a non-Euclidean and somewhat non-metric measure.


Edit Distance Neighbor Distance Prototype Selection Approximate String Match Dissimilarity Data 
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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  1. 1.
    Dubuisson, M., Jain, A.: Modified Hausdorff distance for object matching. In: ICPR, vol. 1, pp. 566–568 (1994)Google Scholar
  2. 2.
    Jacobs, D., Weinshall, D., Gdalyahu, Y.: Classification with Non-Metric Dist.: Image Retrieval and Class Representation. TPAMI 22, 583–600 (2000)Google Scholar
  3. 3.
    Pękalska, E., Duin, R.: The dissimilarity representation for pattern recognition. Foundations and applications. World Scientific, Singapore (2005)MATHCrossRefGoogle Scholar
  4. 4.
    Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning. Springer, Heidelberg (2001)MATHGoogle Scholar
  5. 5.
    Pękalska, E., Paclík, P., Duin, R.: A Generalized Kernel Approach to Dissimilarity Based Classification. JMLR 2, 175–211 (2002)MATHCrossRefGoogle Scholar
  6. 6.
    Schölkopf, B., Smola, A.: Learning with Kernels. MIT Press, Cambridge (2002)Google Scholar
  7. 7.
    Veltkamp, R., Hagedoorn, M.: State-of-the-art in shape matching. Technical Report UU-CS-1999-27, Utrecht University, The Netherlands (1999)Google Scholar
  8. 8.
    Pękalska, E., Duin, R., Günter, S., Bunke, H.: On not making dissimilarities Euclidean. In: S+SSPR, pp. 1145–1154 (2004)Google Scholar
  9. 9.
    Pękalska, E., Duin, R., Paclík, P.: Prototype selection for dissimilarity-based classifiers. Pattern Recognition 39, 189–208 (2005)CrossRefGoogle Scholar
  10. 10.
    Haasdonk, B.: Feature space interpretation of SVMs with indefinite kernels. TPAMI 25, 482–492 (2005)Google Scholar
  11. 11.
    Laub, J., Müller, K.R.: Feature discovery in non-metric pairwise data. JMLR, 801–818 (2004)Google Scholar
  12. 12.
    Goldfarb, L.: A unified approach to pattern recognition. Pattern Recognition 17, 575–582 (1984)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Graepel, T., Herbrich, R., Schölkopf, B., Smola, A., Bartlett, P., Müller, K.R., Obermayer, K., Williamson, R.: Classification on proximity data with LP-machines. In: ICANN, pp. 304–309 (1999)Google Scholar
  14. 14.
    Tax, D., Veenman, C.: Tuning the hyperparameter of an auc-optimized classifier. In: BNAIC, pp. 224–231 (2005)Google Scholar
  15. 15.
    Andreu, G., Crespo, A., Valiente, J.M.: Selecting the toroidal self-organizing feature maps (TSOFM) best organized to object recogn. In: ICNN, pp. 1341–1346 (1997)Google Scholar
  16. 16.
    Bunke, H., Bühler, U.: Applications of approximate string matching to 2D shape recognition. Pattern recognition 26, 1797–1812 (1993)CrossRefGoogle Scholar
  17. 17.
    Bunke, H., Sanfeliu, A. (eds.): Syntactic and Structural Pattern Recognition Theory and Applications. World Scientific, Singapore (1990)MATHGoogle Scholar
  18. 18.
    Devijver, P., Kittler, J.: Pattern recognition: A statistical approach. Prentice Hall, Englewood Cliffs (1982)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Elżbieta Pękalska
    • 1
    • 2
  • Artsiom Harol
    • 1
  • Robert P. W. Duin
    • 1
  • Barbara Spillmann
    • 3
  • Horst Bunke
    • 3
  1. 1.Faculty of Electrical Engineering, Mathematics and Computer SciencesDelft University of TechnologyThe Netherlands
  2. 2.School of Computer ScienceUniversity of ManchesterUnited Kingdom
  3. 3.Institute of Computer Science and Applied MathematicsUniversity of BernSwitzerland

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