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Part of the book series: The Springer International Series in Engineering and Computer Science ((SECS,volume 704))

Abstract

In this chapter we summarize density properties of Reproducing Kernel Hilbert Spaces induced by different classes of kernels. They are important to characterize the power of the associated hypothesis spaces. In the process we characterize the role of b, which is the constant in the standard form of the solution provided by the Support Vector Machine technique \(f(x) = \sum\nolimits_{i = 1}^\ell {\alpha _i } K\left( {x,\:x_i } \right) + b,\) which is a special case of Regularization Machines.

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References

  1. N. Aronszajn Theory of reproducing kernels Trans. Amer. Math. Soc., vol. 686, pp. 337–404, 1950.

    Article  MathSciNet  Google Scholar 

  2. C. Berg, J.P.R. Christensen, and P. Ressel Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions Springer, 1984.

    MATH  Google Scholar 

  3. R. Courant and D. Hilbert Methods of Mathematical Physics, Vol 2 Interscience, 1962.

    Google Scholar 

  4. F. Cucker and S. Smale On the Mathematical Foundations of Learning, Bull Amer. Math. Soc., vol. 39, pp. 1–49, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  5. T. Evgeniou, M. Pontil, and T. Poggio Regularization Networks and Support Vector Machines Advances in Computational Mathematics, vol. 13, pp. 1–50, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Friess, N. Cristianini, and C. Campbell The Kernel-Adatron: A fast and simple learning procedure for Support Vector Machines Proceedings of the 15th International Conference in Machine Learning, pages 188–196, 1998.

    Google Scholar 

  7. F. Girosi An equivalence between sparse approximation and support vector machines Neural Computation, vol. 10, pp. 1455–1480, 1998.

    Article  Google Scholar 

  8. F. Girosi and T. Poggio and B. Caprile Extensions of a Theory of Networks for Approximation and Learning: outliers and Negative Examples, Advances in Neural information processings systems 3, R. Lippmann and J. Moody and D. Touretzky, Morgan Kaufmann, 1991, San Mateo, CA.

    Google Scholar 

  9. F. Girosi and T. Poggio Networks and the Best Approximation Property Biological Cybernetics, vol. 63, pp. 169–176, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Girosi, M. Jones, and T. Poggio Regularization theory and neural network architectures Neural Computation, vol. 7, pp. 219–269, 1995.

    Article  Google Scholar 

  11. A.N. Kolmogorov and S.V. Fonim Elements of the Theory of Functions and Functional Analysis Dover, 1957

    Google Scholar 

  12. Y. Lin, Y. Lee, and G. Wahba Support Vector Machines for Classification in Nonstandard Situations TR 1016, March 2000. To appear, Machine Learning

    Google Scholar 

  13. J. Mercer Functions of positive and negative type and their connection with the theory of integral equations Phil.los. Trans. Roy. Soc. London Ser. A, vol. 209, pp. 415–446, 1909.

    Article  MATH  Google Scholar 

  14. C.A. Micchelli Interpolation of scattered data: distance matrices and conditionally positive definite functions Constructive Approximation, vol. 2, pp. 11–22, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  15. T. Poggio, S. Mukherjee, R. Rifkin, A. Rakhlin and A. Verri b CBCL Paper 198/AI Memo 2001-011, Massachusetts Institute of Technology, Cambridge, MA, July 2001

    Google Scholar 

  16. I.J. Schoenberg Metric spaces and completely monotone functions Ann. of Math., vol. 39, pp.811–841, 1938.

    Article  MathSciNet  Google Scholar 

  17. A. N. Tikhonov and V. Y. Arsenin Solutions of Ill-posed Problems W. H. Winston, 1977

    Google Scholar 

  18. V.N. Vapnik The Nature of Statistical Learning Theory Springer, 1995.

    Google Scholar 

  19. V.N. Vapnik Statistical Learning Theory Wiley, 1998.

    Google Scholar 

  20. G. Wahba Spline models for observational data SIAM, 1990.

    Book  Google Scholar 

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Poggio, T., Mukherjee, S., Rifkin, R., Raklin, A., Verri, A. (2002). B. In: Winkler, J., Niranjan, M. (eds) Uncertainty in Geometric Computations. The Springer International Series in Engineering and Computer Science, vol 704. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-0813-7_11

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  • DOI: https://doi.org/10.1007/978-1-4615-0813-7_11

  • Publisher Name: Springer, Boston, MA

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