Some General Approximation Error and Convergence Rate Estimates in Statistical Learning Theory

  • Saburou Saitoh


In statistical learning theory, reproducing kernel Hilbert spaces are used basically as the hypothese space in the approximation of the regression function. In this paper, in connection with a basic formula by S. Smale and D. X. Zhou which is fundamental in the approximation error estimates, we shall give a general formula based on the general theory of reproducing kernels combined with linear mappings in the framework of Hilbert spaces. We shall give a prototype example.


learning theory convergence rate approximation reproducing kernel Hilbert space linear mapping inequality 


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  1. [1]
    F. Cucker and S. Smale: On the mathematical foundations of learning’, Bull. Amer. Math. Soc. 39 (2001), 1–49.MathSciNetCrossRefGoogle Scholar
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    S. Saitoh: Integral Transforms, Reproducing Kernels and their Applications’, Pitman Res. Notes in Math. Series 369, Addison Wesley Longman, UK, 1997.Google Scholar
  3. [3]
    S. Smale and D. X. Zhou: Estimating the approximation error in learning theory’, Analysis and Applications (to appear).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • Saburou Saitoh
    • 1
  1. 1.Department of MathematicsFaculty of Engineering Gunma UniversityKiryuJapan

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