A Subpath Kernel for Rooted Unordered Trees

  • Daisuke Kimura
  • Tetsuji Kuboyama
  • Tetsuo Shibuya
  • Hisashi Kashima
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6634)

Abstract

Kernel method is one of the promising approaches to learning with tree-structured data, and various efficient tree kernels have been proposed to capture informative structures in trees. In this paper, we propose a new tree kernel function based on “subpath sets” to capture vertical structures in rooted unordered trees, since such tree-structures are often used to code hierarchical information in data. We also propose a simple and efficient algorithm for computing the kernel by extending the multikey quicksort algorithm used for sorting strings. The time complexity of the algorithm is O((|T1| + |T2|)log(|T1| + |T2|)) time on average, and the space complexity is O(|T1| + |T2|), where |T1| and |T2| are the numbers of nodes in two trees T1 and T2. We apply the proposed kernel to two supervised classification tasks, XML classification in web mining and glycan classification in bioinformatics. The experimental results show that the predictive performance of the proposed kernel is competitive with that of the existing efficient tree kernel for unordered trees proposed by Vishwanathan et al. [1], and is also empirically faster than the existing kernel.

Keywords

Kernel methods tree kernels convolution kernels 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Vishwanathan, S.V.N., Smola, A.: Fast kernels for string and tree matching. In: Advances in Neural Information Processing Systems, vol. 15, pp. 569–576 (2003)Google Scholar
  2. 2.
    Manning, C.D., Schutze, H.: Foundations of Statistical Natural Language Processing. The MIT Press, Cambridge (1999)MATHGoogle Scholar
  3. 3.
    Shawe-Taylor, J., Cristianini, N.: Kernel Methods for Pattern Analysis. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  4. 4.
    Vapnik, V.: The Nature of Statistical Learning Theory. Springer, Heidelberg (1995)CrossRefMATHGoogle Scholar
  5. 5.
    Haussler, D.: Convolution kernels on discrete structures. Technical Report UCSC-CRL-99-10, UC Santa Cruz (1999)Google Scholar
  6. 6.
    Collins, M., Duffy, N.: Convolution kernels for natural language. In: Proceedings of the Fourteenth Annual Conference on Neural Information Processing Systems, pp. 625–632 (2001)Google Scholar
  7. 7.
    Kashima, H., Koyanagi, T.: Kernels for semi-structured data. In: Proceedings of the Nineteenth International Conference on Machine Learning, pp. 291–298 (2002)Google Scholar
  8. 8.
    Kuboyama, T., Hirata, K., Aoki-Kinoshita, K.F., Kashima, H., Yasuda, H.: A gram distribution kernel applied to glycan classification and motif extraction. In: Proceedings of the Seventeenth International Conference on Genome Informatics, pp. 25–34 (2006)Google Scholar
  9. 9.
    Aiolli, F., Martino, G.D.S., Sperduti, A.: Route kernels for trees. In: Proceedings of the Twentie-sixth International Conference on Machine Learning, pp. 17–24 (2009)Google Scholar
  10. 10.
    Daumé III, H., Marcu, D.: A tree-position kernel for document compression. In: Proceedings of the Fourth Document Understanding Conference (2004)Google Scholar
  11. 11.
    Kashima, H.: Machine Learning Approaches for Structured-data. PhD thesis, Kyoto University (2007)Google Scholar
  12. 12.
    Ichikawa, H., Hakodaa, K., Hashimoto, T., Tokunaga, T.: Efficient sentence retrieval based on syntactic structure. In: Proceedings of the COLING/ACL, pp. 407–411 (2006)Google Scholar
  13. 13.
    Bentley, J.L., Sedgewick, R.: Fast algorithms for sorting and searching strings. In: Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 360–369 (1997)Google Scholar
  14. 14.
    Teo, C.H., Vishwanathan, S.V.N.: Fast and space efficient string kernels using suffix arrays. In: Proceedings of the Twentie-third International Conference on Machine Learning, pp. 929–936 (2006)Google Scholar
  15. 15.
    Shibuya, T.: Constructing the suffix tree of a tree with a large alphabet. IEICE Transactions on Fundamentals of Electronics 86(5), 1061–1066 (2003)Google Scholar
  16. 16.
    Kailing, K., Kriegel, H.P., Schönauer, S., Seidl, T.: Efficient similarity search for hierarchical data in large databases. In: Hwang, J., Christodoulakis, S., Plexousakis, D., Christophides, V., Koubarakis, M., Böhm, K. (eds.) EDBT 2004. LNCS, vol. 2992, pp. 676–693. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Teo, C.H., Vishwanathan, S.V.N.: SASK: suffix arrays based string kernels (2006), http://users.cecs.anu.edu.au/~chteo/SASK.html
  18. 18.
    Chang, C.C., Lin, C.J.: LIBSVM: a library for support vector machines (2001), http://www.csie.ntu.edu.tw/~cjlin/libsvm
  19. 19.
    Zaki, M.J., Aggarwal, C.C.: Xrules: An effective structural classifier for xml data. Machine Learning Journal 62(1-2), 137–170 (2006)CrossRefGoogle Scholar
  20. 20.
    Hashimoto, K., Hamajima, M., Goto, S., Masumoto, S., Kawashima, M., Kanehisa, M.: Glycan: The database of carbohydrate structures. Genome Informatics 14, 649–650 (2003)Google Scholar
  21. 21.
    Doubet, S., Albersheim, P.: Carbbank. Glycobiology 2(6), 505 (1992)CrossRefGoogle Scholar
  22. 22.
    Lodhi, H., Saunders, C., Shawe-Taylor, J., Cristianini, N., Watkins, C.: Text classification using string kernels. Journal of Machine Learning Research 2, 419–444 (2002)MATHGoogle Scholar
  23. 23.
    Leslie, C., Eskin, E., Noble, W.: The spectrum kernel: A string kernel for SVM protein classification. In: Proceedings of the Pacific Symposium on Biocomputing, pp. 566–575 (2002)Google Scholar
  24. 24.
    Leslie, C., Eskin, E., Weston, J., Noble, W.S.: Mismatch string kernels for SVM protein classification. Neural Information Processing Systems 15, 1441–1448 (2003)Google Scholar
  25. 25.
    Kashima, H., Tsuda, K., Inokuchi, A.: Marginalized kernels between labeled graphs. In: Proceedings of the Twentieth International Conference on Machine Learning, pp. 321–328 (2003)Google Scholar
  26. 26.
    Gärtner, T., Flach, P., Wrobel, S.: On graph kernels: Hardness results and efficient alternatives. In: Proceedings of the Sixteenth Annual Conference on Computational Learning Theory, pp. 129–143 (2003)Google Scholar
  27. 27.
    Washio, T., Motoda, H.: State of the art of graph-based data mining. SIGKDD Explorations 5(1), 59–68 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Daisuke Kimura
    • 1
  • Tetsuji Kuboyama
    • 2
  • Tetsuo Shibuya
    • 3
  • Hisashi Kashima
    • 1
  1. 1.Graduate School of Information Science and TechnologyThe University of TokyoBunkyo-kuJapan
  2. 2.Computer CentreGakushuin UniversityToyoshima-kuJapan
  3. 3.Human Genome Center, Institute of Medical ScienceThe University of TokyoMinato-kuJapan

Personalised recommendations