Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2012, Article ID 902139, 17 pages doi:10.1155/2012/902139 Research Article Approximation Analysis of Learning Algorithms for Support Vector Regression and Quantile Regression Dao-Hong Xiang, 1 Ting Hu, 2 and Ding-Xuan Zhou 3 1 Department of Mathematics, Zhejiang Normal University, Zhejiang 321004, China 2 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 3 Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong Correspondence should be addressed to Ding-Xuan Zhou, mazhou@cityu.edu.hk Received 14 July 2011; Accepted 14 November 2011 Academic Editor: Yuesheng Xu Copyright q 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. We study learning algorithms generated by regularization schemes in reproducing kernel Hilbert spaces associated with an ǫ-insensitive pinball loss. This loss function is motivated by the ǫ-insen- sitive 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 regu- larized regression. 1. Introduction and Motivation In this paper, we study a family of learning algorithms serving both purposes of support vector regression and quantile regression. Approximation analysis and learning rates will be provided, which also helps better understanding of some classical learning methods. Support vector regression is a classical kernel-based algorithm in learning theory introduced in 1. It is a regularization scheme in a reproducing kernel Hilbert space RKHS H K associated with an ǫ-insensitive loss ψ ǫ : R → R  defined for ǫ ≥ 0 by ψ ǫ u max{|u|- ǫ, 0}   |u|- ǫ, if |u|≥ ǫ, 0, otherwise. 1.1