Capacity of reproducing kernel spaces in learning theory

IEEE Transactions on Information Theory

Volumn 49, Issue 7, 2003, Pages 1743-1752

  Zhou, Ding Xuan ^a  

^a CITY UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Capacity; Covering number; Learning theory; Nodal function; Packing number; Reproducing kernel Hilbert space (RKHS)

Indexed keywords

APPROXIMATION THEORY; LEAST SQUARES APPROXIMATIONS; MATHEMATICAL MODELS; PROBABILITY; REGRESSION ANALYSIS; VECTORS;

KERNEL HILBERT SPACES; LEARNING THEORY;

LEARNING SYSTEMS;

EID: 0038105204     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2003.813564     Document Type: Article

References (26)

1. N. Alon, S. Ben-David, N. Cesa-Bianchi, and D. Haussler, "Scale-sensitive dimensions, uniform convergence, and learnability," J. Assoc. Comput. Mach., vol. 44, pp. 615-631, 1997.
2. N. Aronszajn, "Theory of reproducing kernels," Trans. Amer. Math. Soc., vol. 68, pp. 337-404, 1950.
3. K. Ball, "Eigenvalues of Euclidean distance matrices," J. Approx. Theory, vol. 68, pp. 74-82, 1992.
4. A. Barron, "Approximation and estimation bounds for artificial neural networks," Mach. Learn., vol. 14, pp. 115-133, 1994.
5. F. Cucker and S. Smale, "On the mathematical foundations of learning," Bull. Amer. Math. Soc., vol. 39, pp. 1-49, 2001.
6. R. Dudley, E. Giné, and J. Zinn, "Uniform and universal Glivenko-Cantelli classes," J. Theoret. Probab., vol. 4, pp. 485-510, 1991.
7. D. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators. Cambridge, U.K.: Cambridge Univ. Press, 1996.
8. T. Evgeniou, M. Pontil, and T. Poggio, "Regularization networks and support vector machines," Adv. Comput. Math., vol. 13, pp. 1-50, 2000.
9. F. Girosi and T. Poggio, "Networks and the best approximation property," Biol. Cybern., vol. 63, pp. 169-176, 1990.
10. Y. Guo, P. L. Bartlett, J. Shawe-Taylor, and R. C. Williamson, "Covering numbers for support vector machines," IEEE Trans. Inform. Theory, vol. 48, pp. 239-250, Jan. 2002.
11. S. Mukherjee, R. Rifkin, and T. Poggio, "Regression and classification with regularization," preprint, 2001.
12. F. J. Narcowich, N. Sivakumar, and J. D. Ward, "On condition numbers associated with radial-function interpolation," J. Math. Anal. Appl., vol. 186, pp. 457-485, 1994.
13. F.J. Narcowich and J. D. Ward, "Norms of inverses and condition numbers for matrices associated with scattered data," J. Approx. Theory, vol. 64, pp. 69-94, 1991.
14. ____, "Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices," J. Approx. Theory, vol. 69, pp. 84-109, 1992.
15. P. Niyogi and F. Girosi, "On the relationship between generalization error, hypothesis complexity, and sample complexity for radial basis functions," Neural Comp., vol. 8, pp. 819-842, 1996.
16. M. Pontil, "A note on different covering numbers in learning theory," J. Complexity, to be published.
17. R. Schaback, "Lower bounds for norms of inverses of interpolation matrices for radial basis functions," J. Approx. Theory, vol. 79, pp. 287-306, 1994.
18. S. Smale and D. X. Zhou, "Estimating the approximation error in learning theory," Anal. Appl., vol. 1, pp. 17-41, 2003.
19. A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems. Washington, DC: W. H. Winston, 1977.
20. V. Vapnik, Statistical Learning Theory. New York: Wiley, 1998.
21. V. Vapnik and A. Chervonenkis, "On the uniform convergence of relative frequencies of events to their probabilities," Theory Probab. Appl., vol. 16, pp. 264-280, 1971.
22. G. Wahba, Spline Models for Observational Data. Philadelphia, PA: SIAM, 1990.
23. R. C. Williamson, A. J. Smola, and B. Schölkopf, "Generalization performance of regularization networks and support vector machines via entropy numbers of compact operators," IEEE Trans. Inform. Theory, vol. 47, pp. 2516-2532, Sept. 2001.
24. Z. Wu, "Compactly supported positive definite radial functions," Adv. Comput. Math., vol. 4, pp. 283-292, 1995.
25. D. X. Zhou, "The covering number in learning theory," J. Complexity, vol. 18, pp. 739-767, 2002.
26. ____, "Conditionally reproducing kernel spaces in learning theory," preprint, 2001.