Empirical risk minimization as parameter choice rule for general linear 
regularization methods.

Li, H.; Werner, F.

doi:10.1214/19-AIHP966

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Empirical risk minimization as parameter choice rule for general linear regularization methods.

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Werner, F.
Research Group of Statistical Inverse-Problems in Biophysics, MPI for Biophysical Chemistry, Max Planck Society;

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Citation

Li, H., & Werner, F. (2020). Empirical risk minimization as parameter choice rule for general linear regularization methods. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 56(1), 405-427. doi:10.1214/19-AIHP966.

Cite as: https://hdl.handle.net/21.11116/0000-0005-AD7A-0

Abstract

We consider the statistical inverse problem to recover f from noisy measurements Y = Tf + sigma xi where xi is Gaussian white noise and T a compact operator between Hilbert spaces. Considering general reconstruction methods of the form (f) over cap (alpha) = q(alpha) (T*T)T*Y with an ordered filter q(alpha), we investigate the choice of the regularization parameter alpha by minimizing an unbiased estiate of the predictive risk E[parallel to T f - T (f) over cap (alpha)parallel to(2)]. The corresponding parameter alpha(pred) and its usage are well-known in the literature, but oracle inequalities and optimality results in this general setting are unknown. We prove a (generalized) oracle inequality, which relates the direct risk E[parallel to f - (f) over cap (alpha pred)parallel to(2)] with the oracle prediction risk inf(alpha>0) E[parallel to T f - T (f) over cap (alpha)parallel to(2)]. From this oracle inequality we are then able to conclude that the investigated parameter choice rule is of optimal order in the minimax sense.

Finally we also present numerical simulations, which support the order optimality of the method and the quality of the parameter choice in finite sample situations.