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October 2010 On universal oracle inequalities related to high-dimensional linear models
Yuri Golubev
Ann. Statist. 38(5): 2751-2780 (October 2010). DOI: 10.1214/10-AOS803


This paper deals with recovering an unknown vector θ from the noisy data Y =  + σξ, where A is a known (m × n)-matrix and ξ is a white Gaussian noise. It is assumed that n is large and A may be severely ill-posed. Therefore, in order to estimate θ, a spectral regularization method is used, and our goal is to choose its regularization parameter with the help of the data Y. For spectral regularization methods related to the so-called ordered smoothers [see Kneip Ann. Statist. 22 (1994) 835–866], we propose new penalties in the principle of empirical risk minimization. The heuristical idea behind these penalties is related to balancing excess risks. Based on this approach, we derive a sharp oracle inequality controlling the mean square risks of data-driven spectral regularization methods.


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Yuri Golubev. "On universal oracle inequalities related to high-dimensional linear models." Ann. Statist. 38 (5) 2751 - 2780, October 2010.


Published: October 2010
First available in Project Euclid: 20 July 2010

zbMATH: 1200.62074
MathSciNet: MR2722455
Digital Object Identifier: 10.1214/10-AOS803

Primary: 62C10
Secondary: 62C10 , 62G05

Keywords: empirical risk minimization , excess risk , Oracle inequality , ordered smoother , Spectral regularization

Rights: Copyright © 2010 Institute of Mathematical Statistics


Vol.38 • No. 5 • October 2010
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