Reproducing kernel banach spaces for machine learning

Journal of Machine Learning Research

Volumn 10, Issue , 2009, Pages 2741-2775

  Zhang, Haizhang ^a   Xu, Yuesheng ^a   Zhang, Jun ^b  

^a SYRACUSE UNIVERSITY   (United States)

^b UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Learning theory; Representer theorems; Reproducing kernel banach spaces; Reproducing kernels; Semi innerproducts

Indexed keywords

DUALITY MAPPING; KERNEL PRINCIPAL COMPONENT ANALYSIS; LEARNING SCHEMES; LEARNING THEORY; MACHINE-LEARNING; REGULARIZATION NETWORKS; REPRESENTER THEOREM; REPRODUCING KERNEL; REPRODUCING KERNEL BANACH SPACES;

BANACH SPACES; TOPOLOGY;

PRINCIPAL COMPONENT ANALYSIS;

EID: 75249087221     PISSN: 15324435     EISSN: 15337928     Source Type: Journal    
DOI: None     Document Type: Article

References (40)

1. A.Argyriou, C. A. Micchelli and M. Pontil. When is there a representer theorem? Vector versus matrixregularizers. arXiv:0809.1590vl.
2. N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc, 68: 337-404, 1950.
3. K. P. Bennett and E. J. Bredensteiner. Duality and geometry in SVM classifiers. In Proceeding of the Seventeenth International Conference on Machine Learning, pages 57-64, Morgan Kaufmann, San Francisco, 2000.
4. S. Canu, X. Mary and A. Rakotomamonjy. Functional learning through kernel. In Advances in Learning Theory: Methods, Models and Applications, pages 89-110, NATO Science Series III: Computer and Systems Sciences, Volume 190, IOS Press, Amsterdam, 2003.
5. A. Caponnetto, C. A. Micchelli, M. Pontil and Y. Ying. Universal multi-task kernels. Journal of Machine Learning Research, 9: 1615-1646, 2008.
6. J. B. Conway. A Course in Functional Analysis. 2nd Edition, Springer-Verlag, New York, 1990.
7. F. Cucker and S. Smale. On the mathematical foundations of learning. Bull. Amer. Math. Soc., 39: 1-49, 2002.
8. D. F. Cudia. On the localization and directionalization of uniform convexity. Bull. Amer. Math. Soc., 69: 265-267, 1963.
9. R. Der and D. Lee. Large-margin classification in Banach spaces. JMLR Workshop and Conference Proceedings, 2: AISTATS: 91-98, 2007.
10. T. Evgeniou, M. Pontil and T. Poggio. Regularization networks and support vector machines. Adv. Comput. Math, 13: 1-50, 2000.
11. M. J. Fabian, P Habala, P. Hajek and J. Pelant. Functional Analysis and Infinite-Dimensional Geometry. Springer, New York, 2001.
12. C. Gentile. A new approximate maximal margin classification algorithm. Journal of Machine Learning Research, 2: 213-242, 2001.
13. J. R. Giles. Classes of semi-inner-product spaces. Trans. Amer. Math. Soc., 129: 436-446, 1967.
14. M. Hein, O. Bousquet and B. Schölkopf. Maximal margin classification for metric spaces. J. Comput. System Sci., 71: 333-359, 2005.
15. V. I. Istrǎţescu. Strict Convexity and Complex Strict Convexity: Theory and Applications. Lecture Notes in Pure and Applied Mathematics 89, Marcel Dekker, New York, 1984.
16. D. Kimber and P M. Long. On-line learning of smooth functions of a single variable. Theoret. Comput. Sci., 148: 141-156, 1995.
17. G. Kimeldorf and G. Wahba. Some results on Tchebycheffian spline functions. J. Math. Anal. Appl, 33: 82-95, 1971.
18. G. Lumer. Semi-inner-product spaces. Trans. Amer. Math. Soc., 100: 29-43, 1961.
19. J. Mercer. Functions of positive and negative type and their connection with the theorey of integral equations. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.,- 209: 415-446, 1909.
20. C. A. Micchelli and M. Pontil. A function representation for learning in Banach spaces. In Learning Theory, pages 255-269, Lecture Notes in Computer Science 3120, Springer, Berlin, 2004.
21. C. A. Micchelli and M. Pontil. On learning vector-valued functions. Neural Comput., 17: 177-204, 2005.
22. C. A. Micchelli and M. Pontil. Feature space perspectives for learning the kernel. Machine Learning, 66: 297-319, 2007.
23. C. A. Micchelli, Y Xu and P Ye. Cucker Smale learning theory in Besov spaces. In Advances in Learning Theory: Methods, Models and Applications, pages 47-68, IOS Press, Amsterdam, The Netherlands, 2003.
24. C. A. Micchelli, Y. Xu and H. Zhang. Universal kernels. Journal of Machine Learning Research, 7: 2651-2667, 2006.
25. C. A. Micchelli, Y. Xu and H. Zhang. Optimal learning of bandlimited functions from localized sampling. J. Complexity, 25: 85-114, 2009.
26. W. Rudin. Real and Complex Analysis. 3rd Edition, McGraw-Hill, New York, 1987.
27. S. Saitoh. Integral Transforms, Reproducing Kernels and Their Applications. Pitman Research Notes in Mathematics Series 369, Longman, Harlow, 1997.
28. B. Schölkopf, R. Herbrich and A. J. Smola. A generalized representer theorem. In Proceeding of the 14th Annual Conference on Computational Learning Theory and the 5th European Conference on Computational Learning Theory, pages 416-426, Springer-Verlag, London, UK, 2001.
29. B. Schölkopf and A. J. Smola. Learning with Kernels: Support Vector Machines, Regularizaron, Optimization, and Beyond. MIT Press, Cambridge, Mass, 2002.
30. B. Schölkopf, A. J. Smola and K.-R Müller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Comput., 10: 1299-1319, 1998.
31. J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, Cambridge, 2004.
32. I. Steinwart. On the influence of the kernel on the consistency of support vector machines. Journal of Machine Learning Research, 2: 67-93, 2001.
33. J. A. Tropp. Just relax: convex programming methods for identifying sparse signals in noise. IEEE Trans. Inform. Theory, 52: 1030-1051, 2006.
34. V. N. Vapnik. Statistical Learning Theory. Wiley, New York, 1998.
35. U. von Luxburg and O. Bousquet. Distance-based classification with Lipschitz functions. Journal of Machine Learning Research, 5: 669-695, 2004.
36. G. Wahba. Support vector machines, reproducing kernel Hilbert spaces and the randomized GACV In Advances in Kernel Methods-Support Vector Learning, pages 69-86, MIT Press, Cambridge, Mass, 1999.
37. Y. Xu and H. Zhang. Refinable kernels. Journal of Machine Learning Research, 8: 2083-2120, 2007.
38. Y. Xu and H. Zhang. Refinement of reproducing kernels. Journal of Machine Learning Research, 10: 107-140, 2009.
39. T. Zhang. On the dual formulation of regularized linear systems with convex risks. Machine Learning, 46: 91-129, 2002.
40. D. Zhou, B. Xiao, H. Zhou and R. Dai. Global geometry of SVM classifiers. Technical Report 30-5-02, Institute of Automation, Chinese Academy of Sciences, 2002.