Advertisement

Data Analysis of (Non-)Metric Proximities at Linear Costs

  • Frank-Michael Schleif
  • Andrej Gisbrecht
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7953)

Abstract

Domain specific (dis-)similarity or proximity measures, employed e.g. in alignment algorithms in bio-informatics, are often used to compare complex data objects and to cover domain specific data properties. Lacking an underlying vector space, data are given as pairwise (dis-)similarities. The few available methods for such data do not scale well to very large data sets. Kernel methods easily deal with metric similarity matrices, also at large scale, but costly transformations are necessary starting with non-metric (dis-) similarities. We propose an integrative combination of Nyström approximation, potential double centering and eigenvalue correction to obtain valid kernel matrices at linear costs. Accordingly effective kernel approaches, become accessible for these data. Evaluation at several larger (dis-)similarity data sets shows that the proposed method achieves much better runtime performance than the standard strategy while keeping competitive model accuracy. Our main contribution is an efficient linear technique, to convert (potentially non-metric) large scale dissimilarity matrices into approximated positive semi-definite kernel matrices.

Keywords

Support Vector Machine Similarity Matrix Negative Eigenvalue Kernel Method Dissimilarity Matrix 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chen, Y., Garcia, E.K., Gupta, M.R., Rahimi, A., Cazzanti, L.: Similarity-based classification: Concepts and algorithms. JMLR 10, 747–776 (2009)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Cortes, C., Mohri, M., Talwalkar, A.: On the impact of kernel approximation on learning accuracy. JMLR - Proceedings Track 9, 113–120 (2010)Google Scholar
  3. 3.
    Drineas, P., Mahoney, M.W.: On the nyström method for approximating a gram matrix for improved kernel-based learning. Journal of Machine Learning Research 6, 2153–2175 (2005)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Duin, R.P.: PRTools (March 2012), http://www.prtools.org
  5. 5.
    Duin, R.P.W., Pękalska, E.: Non-euclidean dissimilarities: Causes and informativeness. In: Hancock, E.R., Wilson, R.C., Windeatt, T., Ulusoy, I., Escolano, F. (eds.) SSPR&SPR 2010. LNCS, vol. 6218, pp. 324–333. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Farahat, A.K., Ghodsi, A., Kamel, M.S.: A novel greedy algorithm for nyström approximation. JMLR - Proceedings Track 15, 269–277 (2011)Google Scholar
  7. 7.
    Gisbrecht, A., Mokbel, B., Schleif, F.M., Zhu, X., Hammer, B.: Linear time relational prototype based learning. Journal of Neural Systems 22(5) (2012)Google Scholar
  8. 8.
    Graepel, T., Obermayer, K.: A stochastic self-organizing map for proximity data. Neural Computation 11(1), 139–155 (1999)CrossRefGoogle Scholar
  9. 9.
    Hammer, B., Hasenfuss, A.: Topographic mapping of large dissimilarity data sets. Neural Computation 22(9), 2229–2284 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kohonen, T., Somervuo, P.: How to make large self-organizing maps for nonvectorial data. Neural Networks 15(8-9), 945–952 (2002)CrossRefGoogle Scholar
  11. 11.
    Kumar, S., Mohri, M., Talwalkar, A.: On sampling-based approximate spectral decomposition. In: ICML. ACM International Conference Proceeding Series, vol. 382, p. 70. ACM (2009)Google Scholar
  12. 12.
    Laub, J., Roth, V., Buhmann, J.M., Müller, K.R.: On the information and representation of non-euclidean pairwise data. Pattern Recognition 39(10), 1815–1826 (2006)zbMATHCrossRefGoogle Scholar
  13. 13.
    Li, W.J., Zhang, Z., Yeung, D.Y.: Latent wishart processes for relational kernel learning. JMLR - Proceedings Track 5, 336–343 (2009)Google Scholar
  14. 14.
    Neuhaus, M., Bunke, H.: Edit distance based kernel functions for structural pattern classification. Pattern Recognition 39(10), 1852–1863 (2006)zbMATHCrossRefGoogle Scholar
  15. 15.
    Pekalska, E., Duin, R.: The dissimilarity representation for pattern recognition. World Scientific (2005)Google Scholar
  16. 16.
    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), 729–744 (2008)CrossRefGoogle Scholar
  17. 17.
    Pękalska, E.z., Duin, R.P.W., Günter, S., Bunke, H.: On not making dissimilarities euclidean. In: Fred, A., Caelli, T.M., Duin, R.P.W., Campilho, A.C., de Ridder, D. (eds.) SSPR&SPR 2004. LNCS, vol. 3138, pp. 1145–1154. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Platt, J.: Fastmap, metricmap, and landmark mds are all nyström algorithms (2005)Google Scholar
  19. 19.
    Shawe-Taylor, J., Cristianini, N.: Kernel Methods for Pattern Analysis and Discovery. Cambridge University Press (2004)Google Scholar
  20. 20.
    de Silva, V., Tenenbaum, J.B.: Global versus local methods in nonlinear dimensionality reduction. In: NIPS, pp. 705–712. MIT Press (2002)Google Scholar
  21. 21.
    Tan, J., Kuchibhatla, D., Sirota, F.L.: Tachyon search speeds up retrieval of similar sequences by several orders of magnitude. Bio Informatics (April 23, 2012)Google Scholar
  22. 22.
    Tsang, I.W., Kocsor, A., Kwok, J.T.: Simpler core vector machines with enclosing balls. In: ICML. ACM International Conference Proceeding Series, vol. 227, pp. 911–918. ACM (2007)Google Scholar
  23. 23.
    Williams, C.K.I., Seeger, M.: Using the nyström method to speed up kernel machines. In: NIPS, pp. 682–688. MIT Press (2000)Google Scholar
  24. 24.
    Zhang, K., Kwok, J.T.: Clustered nyström method for large scale manifold learning and dimension reduction. IEEE Transactions on Neural Networks 21(10), 1576–1587 (2010)CrossRefGoogle Scholar
  25. 25.
    Zhang, K., Lan, L., Wang, Z., Moerchen, F.: Scaling up kernel svm on limited resources: A low-rank linearization approach. JMLR - Proceedings Track 22, 1425–1434 (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Frank-Michael Schleif
    • 1
  • Andrej Gisbrecht
    • 1
  1. 1.CITEC Centre of ExcellenceBielefeld UniversityBielefeldGermany

Personalised recommendations