On universal oracle inequalities related to high-dimensional linear models

This paper deals with recovering an unknown vector $θ$ from the noisy data $Y=Aθ+σξ$, where $A$ is a known $(m\times n)$-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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