Constructive Approximation

, Volume 26, Issue 2, pp 153–172 | Cite as

Learning Theory Estimates via Integral Operators and Their Approximations

  • Steve SmaleEmail author
  • Ding-Xuan ZhouEmail author


The regression problem in learning theory is investigated with least square Tikhonov regularization schemes in reproducing kernel Hilbert spaces (RKHS). We follow our previous work and apply the sampling operator to the error analysis in both the RKHS norm and the L2 norm. The tool for estimating the sample error is a Bennet inequality for random variables with values in Hilbert spaces. By taking the Hilbert space to be the one consisting of Hilbert-Schmidt operators in the RKHS, we improve the error bounds in the L2 metric, motivated by an idea of Caponnetto and de Vito. The error bounds we derive in the RKHS norm, together with a Tsybakov function we discuss here, yield interesting applications to the error analysis of the (binary) classification problem, since the RKHS metric controls the one for the uniform convergence.


Hilbert Space Integral Operator Error Bound Reproduce Kernel Hilbert Space Sampling Operator 
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Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Toyota Technological Institute at Chicago, 1427 East 60th StreetChicago, IL 60637USA
  2. 2.Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, KowloonHong KongChina

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