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Journal of Applied Mathematics
Volume 2014, Article ID 946241, 7 pages
http://dx.doi.org/10.1155/2014/946241
Research Article

Weaker Regularity Conditions and Sparse Recovery in High-Dimensional Regression

1College of Mathematics and Information Sciences, North China University of Water Resources and Electric Power, Zhengzhou 450045, China
2Institute of Environmental and Municipal Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450045, China

Received 27 October 2013; Accepted 7 July 2014; Published 17 July 2014

Academic Editor: Yuesheng Xu

Copyright © 2014 Shiqing Wang 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

Regularity conditions play a pivotal role for sparse recovery in high-dimensional regression. In this paper, we present a weaker regularity condition and further discuss the relationships with other regularity conditions, such as restricted eigenvalue condition. We study the behavior of our new condition for design matrices with independent random columns uniformly drawn on the unit sphere. Moreover, the present paper shows that, under a sparsity scenario, the Lasso estimator and Dantzig selector exhibit similar behavior. Based on both methods, we derive, in parallel, more precise bounds for the estimation loss and the prediction risk in the linear regression model when the number of variables can be much larger than the sample size.