Strong robust generalized cross-validation for choosing the regularization parameter

Lukas, Mark A.

doi:10.1088/0266-5611/24/3/034006

Cited by 42 publications

(40 citation statements)

References 20 publications

(80 reference statements)

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“…In fact, for Tikhonov regularization, it is known [103] that as m → ∞, the R 1 GCV function in (34) is consistent with a weighted sum of the prediction risk E Ax δ n − y 2 and the variance E x δ n −Ex δ n 2 , with weights γ and 1−γ, respectively. Since this variance is measured in a stronger norm than for the variance E Ax δ n − EAx δ n 2 , R 1 GCV places more weight than even RGCV on reducing the variability of the regularized solution.…”

Section: Strong Robust Gcvmentioning

confidence: 95%

“…The last term in (33) can also be explained by the fact that, for uncorrelated errors with variance δ 2 , we have δ 2 tr((AA As m → ∞, the RGCV estimate has the same optimal order as the GCV estimate, but it has a different constant resulting in a slightly larger value [102,103]. Consequently, the method does not suffer as badly as GCV from the saturation effect.…”

Section: Robust Generalized Cross-validationmentioning

confidence: 99%

“…The robust GCV (RGCV) method in Section 4.15 is one of a family of robust GCV methods developed in [103] that also includes the strong robust GCV method, denoted R 1 GCV. Like GCV and RGCV, the R 1 GCV method has a good rationale in terms of the influence of data values on the regularized solution [103].…”

Section: Strong Robust Gcvmentioning

confidence: 99%

“…Like GCV and RGCV, the R 1 GCV method has a good rationale in terms of the influence of data values on the regularized solution [103].…”

Section: Strong Robust Gcvmentioning

confidence: 99%

“…As γ is decreased, the method becomes more robust and it is less likely than GCV and (generally) RGCV to choose a large value of n (i.e. to under-smooth) [103].…”

Section: Strong Robust Gcvmentioning

confidence: 99%

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Comparingparameter choice methods for regularization of ill-posed problems

Bauer

Lukas

2011

Mathematics and Computers in Simulation

248

111

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In the literature on regularization, many different parameter choice methods have been proposed in both deterministic and stochastic settings. However, based on the available information, it is not always easy to know how well a particular method will perform in a given situation and how it compares to other methods. This paper reviews most of the existing parameter choice methods, and evaluates and compares them in a large simulation study for spectral cut-off and Tikhonov regularization. The test cases cover a wide range of linear inverse problems with both white and colored stochastic noise. The results show some marked differences between the methods, in particular, in their stability with respect to the noise and its type. We conclude with a table of properties of the methods and a summary of the simulation results, from which we identify the best methods.

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Section: Strong Robust Gcvmentioning

confidence: 95%

Section: Robust Generalized Cross-validationmentioning

confidence: 99%

Section: Strong Robust Gcvmentioning

confidence: 99%

“…Like GCV and RGCV, the R 1 GCV method has a good rationale in terms of the influence of data values on the regularized solution [103].…”

Section: Strong Robust Gcvmentioning

confidence: 99%

“…As γ is decreased, the method becomes more robust and it is less likely than GCV and (generally) RGCV to choose a large value of n (i.e. to under-smooth) [103].…”

Section: Strong Robust Gcvmentioning

confidence: 99%

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