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Numerische Mathematik

, Volume 66, Issue 1, pp 41–66 | Cite as

Asymptotic optimality of generalized cross-validation for choosing the regularization parameter

  • Mark A. Lukas
Article

Summary

Letf n λ be the regularized solution of a general, linear operator equation,Kf 0 =g, from discrete, noisy datay i =g(x) +ɛ i ,i=1,...,n, whereɛ i are uncorrelated random errors. We consider the prominent method of generalized cross-validation (GCV) for choosing the crucial regularization parameter λ. The practical GCV estimate\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\lambda } _V \) and its “expected” counterpart λ V are defined as the minimizers of the GCV functionV(λ) andEV(λ), respectively, whereE denotes expectation. We investigate the asymptotic performance of λ V with respect to each of the following loss functions: the risk, anL2 norm on the output errorKf n λ−g, and a whole class of stronger norms on the input errorf n λ−f0. In the special cases of data smoothing and Fourier differentiation, it is known that asn→∞, λ V is asymptotically optimal (ao) with respect to the risk criterion. We show this to be true in general, and also extend it to theL2 norm criterion. The asymptotic optimality is independent of the error variance, the ill-posedness of the problem and the smoothness index of the solutionf0. For the input error criterion, it is shown that λ V is weakly ao for a certain class off0 if the smoothness off0 relative to the regularization space is not too high, but otherwise λ V is sub-optimal. This result is illustrated in the case of numerical differentiation.

Mathematics Subject Classifications (1991)

65J10 62G05 47A50 

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Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Mark A. Lukas
    • 1
  1. 1.School of Mathematical and Physical SciencesMurdoch UniversityMurdochAustralia

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