ON UNIVERSAL ORACLE INEQUALITIES RELATED TO HIGH-DIMENSIONAL LINEAR MODELS

• 成果类型: Article
• 署名作者: Golubev, Yuri
• 署名单位: Aix-Marseille Universite; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/10-AOS803
• 发表日期: 2010
• 页码: 2751-2780
• 关键词: INVERSE PROBLEMS; regularization; PRINCIPLE; RISK

摘要

This paper deals with recovering an unknown vector theta from the noisy data Y = A theta + sigma xi, where A is a known (m x n)-matrix and xi is a white Gaussian noise. It is assumed that n is large and A may be severely ill-posed. Therefore, in order to estimate theta, 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.