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On universal oracle inequalities related to high-dimensional linear models. (English) Zbl 1200.62074
Summary: This paper deals with recovering an unknown vector \(\theta \) from the noisy data \(Y = A\theta + \sigma \xi \), where \(A\) is a known \((m \times n)\)-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 A. Kneip, Ann. Stat. 22, No. 2, 835–866 (1994; Zbl 0815.62022)], 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.

MSC:
62J05 Linear regression; mixed models
60E15 Inequalities; stochastic orderings
62H99 Multivariate analysis
15A18 Eigenvalues, singular values, and eigenvectors
62C10 Bayesian problems; characterization of Bayes procedures
62G05 Nonparametric estimation
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