- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 694181, 10 pages
The Learning Rates of Regularized Regression Based on Reproducing Kernel Banach Spaces
1Department of Mathematics, Shaoxing University, Shaoxing 312000, China
2School of Mathematics and LPMC, Nankai University, Tianjin 300071, China
Received 13 August 2013; Accepted 16 October 2013
Academic Editor: Qiang Wu
Copyright © 2013 Baohuai Sheng and Peixin Ye. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- S. Loustau, “Aggregation of SVM classifiers using Sobolev spaces,” Journal of Machine Learning Research, vol. 9, pp. 1559–1582, 2008.
- C. A. Micchelli and M. Pontil, “A function representation for learning in Banach spaces,” in Learning Theory, vol. 3120 of Lecture Notes on Computer Science, pp. 255–269, Springer, Berlin, Germany, 2004.
- S. G. Lv and J. D. Zhu, “Error bounds for -norm multiple kernel learning with least square loss,” Abstract and Applied Analysis, vol. 2012, Article ID 915920, 18 pages, 2012.
- H. Zhang, Y. Xu, and J. Zhang, “Reproducing kernel Banach spaces for machine learning,” Journal of Machine Learning Research, vol. 10, pp. 2741–2775, 2009.
- H. Zhang and J. Zhang, “Regularized learning in Banach spaces as an optimization problem: representer theorems,” Journal of Global Optimization, vol. 54, no. 2, pp. 235–250, 2012.
- H. Zhang and J. Zhang, “Generalized semi-inner products with applications to regularized learning,” Journal of Mathematical Analysis and Applications, vol. 372, no. 1, pp. 181–196, 2010.
- N. Aronszajn, “Theory of reproducing kernels,” Transactions of the American Mathematical Society, vol. 68, pp. 337–404, 1950.
- M. Z. Nashed and Q. Sun, “Sampling and reconstruction of signals in a reproducing kernel subspace of Lp( ),” Journal of Functional Analysis, vol. 258, no. 7, pp. 2422–2452, 2010.
- F. H. Clarke, Y. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis And Control Theory, vol. 178 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 1998.
- H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis. Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.
- Z. B. Xu and G. F. Roach, “Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 157, no. 1, pp. 189–210, 1991.
- T. Bonesky, K. S. Kazimierski, P. Maass, F. Schöpfer, and T. Schuster, “Minimization of Tikhonov functionals in Banach spaces,” Abstract and Applied Analysis, vol. 2008, Article ID 192679, 18 pages, 2008.
- Z. B. Xu and Z. S. Zhang, “Another set of characteristic inequalities of Lp Banach spaces,” Acta Mathematica Sinica, vol. 37, no. 4, pp. 433–439, 1994 (Chinese).
- K. S. Kazimierski, “Minimization of the Tikhonov functional in Banach spaces smooth and convex of power type by steepest descent in the dual,” Computational Optimization and Applications, vol. 48, no. 2, pp. 309–324, 2011.
- F. Cucker and D. X. Zhou, Learning Theory: An Approximation Theory Viewpoint, vol. 24 of Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, New York, NY, USA, 2007.
- B. H. Sheng, J. L. Wang, and P. Li, “The covering number for some Mercer kernel Hilbert spaces,” Journal of Complexity, vol. 24, no. 2, pp. 241–258, 2008.
- B. H. Sheng, J. L. Wang, and Z. X. Chen, “The covering number for some Mercer kernel Hilbert spaces on the unit sphere,” Taiwanese Journal of Mathematics, vol. 15, no. 3, pp. 1325–1340, 2011.
- H. W. Sun and D. X. Zhou, “Reproducing kernel Hilbert spaces associated with analytic translation-invariant Mercer kernels,” Journal of Fourier Analysis and Applications, vol. 14, no. 1, pp. 89–101, 2008.
- D. X. Zhou, “The covering number in learning theory,” Journal of Complexity, vol. 18, no. 3, pp. 739–767, 2002.
- C. Ganser, “Modulus of continuity conditions for Jacobi series,” Journal of Mathematical Analysis and Applications, vol. 27, no. 3, pp. 575–600, 1969.
- H. Berens and Y. Xu, “On Bernstein-Durrmeyer polynomials with Jacobi-weights,” in Approximation Theory and Functional Analysis, C. K. Chui, Ed., pp. 25–46, Academic Press, Boston, Mass, USA, 1991.
- E. E. Berdysheva and K. Jetter, “Multivariate Bernstein-Durrmeyer operators with arbitrary weight functions,” Journal of Approximation Theory, vol. 162, no. 3, pp. 576–598, 2010.
- E. E. Berdysheva, “Uniform convergence of Bernstein-Durrmeyer operators with respect to arbitrary measure,” Journal of Mathematical Analysis and Applications, vol. 394, no. 1, pp. 324–336, 2012.
- F. Filbir and H. N. Mhaskar, “Marcinkiewicz-Zygmund measures on manifolds,” Journal of Complexity, vol. 27, no. 6, pp. 568–596, 2011.
- D. X. Zhou and K. Jetter, “Approximation with polynomial kernels and SVM classifiers,” Advances in Computational Mathematics, vol. 25, no. 1–3, pp. 323–344, 2006.
- H. Z. Tong, D. R. Chen, and L. Z. Peng, “Learning rates for regularized classifiers using multivariate polynomial kernels,” Journal of Complexity, vol. 24, no. 5–6, pp. 619–631, 2008.
- B. Z. Li, “Approximation by multivariate Bernstein-Durrmeyer operators and learning rates of least-squares regularized regression with multivariate polynomial kernels,” Journal of Approximation Theory, vol. 173, pp. 33–55, 2013.
- A. Guntuboyina and B. Sen, “Covering numbers for convex functions,” IEEE Transactions on Information Theory, vol. 59, no. 4, pp. 1957–1965, 2013.
- J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research and Financial Engineering, Springer, New York, NY, USA, 2000.
- F. Cucker and S. Smale, “On the mathematical foundations of learning,” Bulletin of the American Mathematical Society, vol. 39, no. 1, pp. 1–49, 2002.
- F. Cucker and S. Smale, “Best choices for regularization parameters in learning theory: on the bias-variance problem,” Foundations of Computational Mathematics, vol. 2, no. 4, pp. 413–428, 2002.