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Journal of Applied Mathematics
Volume 2012, Article ID 902139, 17 pages
http://dx.doi.org/10.1155/2012/902139
Research Article

Approximation Analysis of Learning Algorithms for Support Vector Regression and Quantile Regression

1Department of Mathematics, Zhejiang Normal University, Zhejiang 321004, China
2School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
3Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received 14 July 2011; Accepted 14 November 2011

Academic Editor: Yuesheng Xu

Copyright © 2012 Dao-Hong Xiang et al. 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.

Abstract

We study learning algorithms generated by regularization schemes in reproducing kernel Hilbert spaces associated with an ϵ-insensitive pinball loss. This loss function is motivated by the ϵ-insensitive loss for support vector regression and the pinball loss for quantile regression. Approximation analysis is conducted for these algorithms by means of a variance-expectation bound when a noise condition is satisfied for the underlying probability measure. The rates are explicitly derived under a priori conditions on approximation and capacity of the reproducing kernel Hilbert space. As an application, we get approximation orders for the support vector regression and the quantile regularized regression.