Abstract
Kernel methods are widely used in statistical learning techniques. We recently introduced a general kernel framework based on weighted transducers or rational relations, rational kernels, to extend kernel methods to the analysis of variable-length sequences or more generally weighted automata. These kernels are efficient to compute and have been successfully used in applications such as spoken-dialog classification. Not all rational kernels are positive definite and symmetric (PDS) however, a sufficient property for guaranteeing the convergence of discriminant classification algorithms such as Support Vector Machines. We present several theoretical results related to PDS rational kernels. We show in particular that under some conditions these kernels are closed under sum, product, or Kleene-closure and give a general method for constructing a PDS rational kernel from an arbitrary transducer defined on some non-idempotent semirings. We also show that some commonly used string kernels or similarity measures such as the edit-distance, the convolution kernels of Haussler, and some string kernels used in the context of computational biology are specific instances of rational kernels. Our results include the proof that the edit-distance over a non-trivial alphabet is not negative definite, which, to the best of our knowledge, was never stated or proved before.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Berg, C., Christensen, J.P.R., Ressel, P.: Harmonic Analysis on Semigroups. Springer, New York (1984)
Boser, B.E., Guyon, I., Vapnik, V.N.: A training algorithm for optimal margin classifiers. In: Proceedings of the 5thAnnual Workshop of Computational Learning Theory, Pittsburg, vol. 5, pp. 144–152. ACM, New York (1992)
Cortes, C., Haffner, P., Mohri, M.: Rational Kernels. In: NIPS 2002, Vancouver, Canada. MIT Press, Cambridge (2002)
Cortes, C., Vapnik, V.N.: Support-Vector Networks. Machine Learning 20(3), 273–297 (1995)
Durbin, R., Eddy, S.R., Krogh, A., Mitchison, G.J.: Biological Sequence Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge University Press, Cambridge (1998)
Haussler, D.: Convolution Kernels on Discrete Structures. Technical Report UCSC-CRL-99-10, University of California at Santa Cruz (1999)
Kuich, W., Salomaa, A.: Semirings, Automata, Languages. ATCS Monographs on Theoretical Computer Science, vol. 5. Springer, Heidelberg (1986)
Leslie, C., Eskin, E., Weston, J., Noble, W.S.: Mismatch String Kernels for SVM Protein Classification. In: NIPS 2002, Vancouver, Canada. MIT Press, Cambridge (2002)
Levenshtein, V.I.: Binary codes capable of correcting deletions, insertions, and reversals. Soviet Physics - Doklady 10, 707–710 (1966)
Lodhi, H., Shawe-Taylor, J., Cristianini, N., Watkins, C.: Text classification using string kernels. In: Leen, T.K., Dietterich, T.G., Tresp, V. (eds.) NIPS 2000, pp. 563–569. MIT Press, Cambridge (2001)
Mohri, M.: Edit-Distance of Weighted Automata: General Definitions and lgorithms. In: International Journal of Foundations of Computer Science (2003)
Schölkopf, B.: The Kernel Trick for Distances. In: Leen, T.K., Ietterich, T.G., Tresp, V. (eds.) NIPS 2001, pp. 301–307. MIT Press, Cambridge (2001)
Schölkopf, B., Smola, A.: Learning with Kernels. MIT Press, Cambridge (2002)
Vapnik, V.N.: Statistical Learning Theory. John Wiley & Sons, Chichester (1998)
Watkins, C.: Dynamic alignment kernels. Technical Report CSD-TR-98-11, Royal Holloway, University of London (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cortes, C., Haffner, P., Mohri, M. (2003). Positive Definite Rational Kernels. In: Schölkopf, B., Warmuth, M.K. (eds) Learning Theory and Kernel Machines. Lecture Notes in Computer Science(), vol 2777. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45167-9_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-45167-9_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40720-1
Online ISBN: 978-3-540-45167-9
eBook Packages: Springer Book Archive