Universal kernels

Journal of Machine Learning Research

Volumn 7, Issue , 2006, Pages 2651-2667

  Micchelli, Charles A ^a   Xu, Yuesheng ^b   Zhang, Haizhang ^b  

^a STATE UNIVERSITY OF NEW YORK   (United States)

^b SYRACUSE UNIVERSITY   (United States)

Author keywords

Density; Radial kernels; Translation invariant kernels

Indexed keywords

APPROXIMATION THEORY; AUTOMATIC TARGET RECOGNITION; FUNCTION EVALUATION; SET THEORY;

DENSITY; RADIAL KERNELS; TRANSLATION INVARIANT KERNELS;

FEATURE EXTRACTION;

EID: 33845518144     PISSN: 15324435     EISSN: 15337928     Source Type: Journal    
DOI: None     Document Type: Article

References (36)

1. A. Argyriou, C. A. Micchelli and M. Pontil. Learning convex combinations of continuously parameterized basic kernels. Proceeding of the 18th Annual Conference on Learning Theory (COLT'05), Bertinoro, Italy, 2005.
2. A. Argyriou, R. Hauser, C. A. Micchelli and M. Pontil. A DC-programming algorithm for kernel selection. Proceeding of the 23rd International Conference on Machine Learning (ICML'06), forthcoming (see also Research Note RN/06/04, Department of Computer Science, UCL, 2006).
3. N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc., 68: 337-404, 1950.
4. F. R. Bach, G. R. G. Lanckriet and M. I. Jordan. Multiple kernel learning, conic duality and the SMO algorithm. Proceeding of the 21st International Conference on Machine learning (ICML'04), 2004.
5. A. Beurling and P. Malliavin. On the closure of characters and the zeros of entire functions. Acta. Math., 118:79-93, 1967.
6. C. M. Bishop. Neural Networks for Pattern Recognition. Clarendon Press, Oxford, 1995.
7. S. Bochner. Lectures on Fourier Integrals With an author's supplement on monotonic functions, Stieltjes integrals, and harmonic analysis. Annals of Mathematics Studies, no. 42, Princeton University Press, New Jersey, 1959.
8. T. Evgeniou, M. Pontil and T. Poggio. Regularization networks and support vector machines. Adv. Comput. Math., 13: 1-50, 2000.
9. C. H. FitzGerald, C. A. Micchelli and A. Pinkus. Functions that preserve families of positive semidefinite matrices. Linear Algebra Appl., 221: 83-102, 1995.
10. T. Hastie, R. Tibshirani and J. Friedman. The Elements of Statistical Learning. Springer, New York, 2001.
11. G. R. G. Lanckriet, N. Cristianini, P. Bartlett, L. El-Ghaoui and M. I. Jordan. Learning the kernel matrix with semi-definite programming. Journal of Machine Learning Research, 5: 27-72, 2004.
12. P. Lax. Functional Analysis. Wiley, New York, 2002.
13. J. Mercer. Functions of positive and negative type and their connection with the theory of integral equations. Philos. Trans. Royal Soc. London, 209: 415-446, 1909.
14. C. A. Micchelli and M. Pontil. A function representation for learning in Banach spaces. Proceeding of the 17th Annual Conference on Learning (COLT'04), 2004.
15. C. A. Micchelli and M. Pontil. Feature space perspectives for learning the kernel. Machine Learning, forthcoming (see also: Research Note RN/05/11, Department of Computer Science, UCL, June, 2005).
16. C. A. Micchelli, M. Pontil, Q. Wu and D. X. Zhou. Error bounds for learning the kernel. Research Note RN/05/04, Department of Computer Science, UCL, 2006.
17. C. A. Micchelli, Y. Xu and P. Ye. Cucker Smale learning theory in Besov spaces. Advances in Learning Theory: Methods, Models and Applications. J. Suykens, G. Horvath, S. Basu, C. A. Micchelli and J. Vandewalle, editors. IOS Press, Amsterdam, The Netherlands, 2003, 47-68.
18. J. Neumann, C. Schnörr and G. Steidl. SVM-based feature selection by direct objective minimization. C.E. Rasmussen, H. H. Bülthoff, B. Schölkopf and M. A. Giese, editors. Lecture Notes in Computer Science, 3175: 212-219, Proceeding of the 26th DAGM Symposium, 2004.
19. C. S. Ong, A. J. Smola and R. C. Williamson. Learning the kernel with hyperkernels. Journal of Machine Learning Research, 6: 1043-1071, 2005.
20. T. Poggio, S. Mukherjee, R. Rifkin, A. Raklin and A. Verri. B. Uncertainty in geometric computations, J. Winkler and M. Niranjan, editors. Kluwer Academic Publishers, 22: 131-141, 2002.
21. R. M. Redheffer. Completeness of sets of complex exponentials. Adv. Math., 24: 1-62, 1977.
22. T. J. Rivlin. Chebyshev Polynomials. 2nd Edition, John Wiley, New York, 1990.
23. H. Royden. Real Analysis. 3rd Edition, Macmillan Publishing Company, New York, 1988.
24. W. Rudin. Functional Analysis. 2nd Edition, McGraw Hill, New York, 1991.
25. I. J. Schoenberg. Metric spaces and completely monotone functions. Ann. of Math. (2), 39: 811-841, 1938.
26. I. J. Schoenberg. Positive definite functions on spheres. Duke. Math. J., 9: 96-108, 1942.
27. B. Schölkopf, C. J. C. Burges and A. Smola. Advances in Kernel Methods: Support Vector Learning. MIT Press, Cambridge, Mass, 1999.
28. B. Schölkopf and A. Smola. Learning with Kernels. MIT Press, Cambridge, Mass, 2002.
29. J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, Cambridge, 2004.
30. S. Sonnenburg, G. Rätsch and C. Schäfer. A general and efficient multiple kernel learning algorithm. Y. Weiss, B. Schölkopf and J. Platt, editors. Advances in Neural Information Processing Systems, 18. MIT Press, Cambridge, Mass, 2006.
31. E. M. Stein and G. Weiss. Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, New Jersey, 1971.
32. I. Steinwart. On the influence of kernel on the consistency of support vector machines. Journal of Machine Learning Research, 2: 67-93, 2001.
33. H. Sun. Mercer theorem for RKHS on noncompact sets. J. Complexity, 21: 337-349, 2005.
34. G. Szegö. Orthogonal Polynomials. American Mathematical Society Colloquium Publications 23. Revised Edition, Providence, RI, 1959.
35. G. Wahba. Splines Models for Observational Data. Series in Applied Mathematics 59. SIAM, Philadelphia, 1990.
36. D. X. Zhou. Density problem and approximation error in learning theory, preprint, 2003.