On Pairwise Kernels: An Efficient Alternative and Generalization Analysis

  • Hisashi Kashima
  • Satoshi Oyama
  • Yoshihiro Yamanishi
  • Koji Tsuda
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5476)


Pairwise classification has many applications including network prediction, entity resolution, and collaborative filtering. The pairwise kernel has been proposed for those purposes by several research groups independently, and become successful in various fields. In this paper, we propose an efficient alternative which we call Cartesian kernel. While the existing pairwise kernel (which we refer to as Kronecker kernel) can be interpreted as the weighted adjacency matrix of the Kronecker product graph of two graphs, the Cartesian kernel can be interpreted as that of the Cartesian graph which is more sparse than the Kronecker product graph. Experimental results show the Cartesian kernel is much faster than the existing pairwise kernel, and at the same time, competitive with the existing pairwise kernel in predictive performance.We discuss the generalization bounds by the two pairwise kernels by using eigenvalue analysis of the kernel matrices.


Kernel Methods Pairwise Kernels Link Prediction 


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  1. 1.
    Shawe-Taylor, J., Cristianini, N.: Kernel Methods for Pattern Analysis. Cambridge University Press, New York (2004)CrossRefzbMATHGoogle Scholar
  2. 2.
    Basilico, J., Hofmann, T.: Unifying collaborative and content-based filtering. In: Proceedings of the 21st International Conference on Machine Learning (ICML) (2004)Google Scholar
  3. 3.
    Ben-Hur, A., Noble, W.S.: Kernel methods for predicting protein-protein interactions. Bioinformatics 21(suppl. 1), i38–i46 (2005)CrossRefGoogle Scholar
  4. 4.
    Oyama, S., Manning, C.D.: Using feature conjunctions across examples for learning pairwise classifiers. In: Boulicaut, J.-F., Esposito, F., Giannotti, F., Pedreschi, D. (eds.) ECML 2004. LNCS, vol. 3201, pp. 322–333. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Schölkopf, B., Shawe-Taylor, J., Smola, A., Williamson, R.: Generalization bounds via eigenvalues of the gram matrix. Technical Report 99-035, NeuroColt (1999)Google Scholar
  6. 6.
    Kroon, R.: Support vector machines, generalization bounds and transduction, masters thesis, stellenbosch university (2003)Google Scholar
  7. 7.
    Laub, A.J.: Matrix Analysis for Scientists and Engineers. Society for Industrial and Applied Mathematics (2005)Google Scholar
  8. 8.
    Imrich, W., Klavzar, S.: Product Graphs: Structure and Recognition. Wiley, Chichester (2000)zbMATHGoogle Scholar
  9. 9.
    Yamanishi, Y., Vert, J.P., Kanehisa, M.: Supervised enzyme network inference from the integration of genomic data and chemical information. Bioinformatics 21, i468–i477 (2005)CrossRefGoogle Scholar
  10. 10.
    Kanehisa, M., Goto, S., Kawashima, S., Okuno, Y., Hattori, M.: The KEGG resources for deciphering the genome. Nucleic Acids Research 32, D277–D280 (2004)CrossRefGoogle Scholar
  11. 11.
    von Mering, C., Krause, R., Snel, B., Cornell, M., Olivier, S., Fields, S., Bork, P.: Comparative assessment of large-scale data sets of protein-protein interactions. Nature 417, 399–403 (2002)CrossRefGoogle Scholar
  12. 12.
    Tsuda, K., Noble, W.S.: Learning kernels from biological networks by maximizing entropy. Bioinformatics 20(suppl. 1), i326–i333 (2004)CrossRefGoogle Scholar
  13. 13.
    Ishibashi, K., Hatano, K., Takeda, M.: Online learning of approximate maximum p-norm margin classifiers with biases. In: Proceedings of the 21st Annual Conference on Learning Theory (COLT 2008) (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Hisashi Kashima
    • 1
  • Satoshi Oyama
    • 2
  • Yoshihiro Yamanishi
    • 3
  • Koji Tsuda
    • 4
  1. 1.Tokyo Research LaboratoryIBM ResearchUSA
  2. 2.Graduate School of InformaticsKyoto UniversityJapan
  3. 3.Centre for Computational BiologyMines ParisTechFrance
  4. 4.Max Planck Institute for Biological CyberneticsGermany

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