Generalized Cross-Validation applied to Conjugate Gradient for discrete ill-posed problems

Favati, Paola; Lotti, Grazia; Menchi, Ornella; Romani, Francesco

doi:10.1016/j.amc.2014.05.109

Cited by 10 publications

(10 citation statements)

References 15 publications

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“…e difference between the existing versions of the GCV is due to the estimation of the trace of the matrix in the denominator. To our knowledge, GCV has not yet been used with PLS regression, but it has been used in regularized conjugate gradient method [11]. e above formula of the GCV in the case of PLS regression becomes [11]…”

Section: Generalized Cross Validationmentioning

confidence: 99%

The L-Curve Criterion as a Model Selection Tool in PLS Regression

Kerkri

Allal

Zarrouk

2019

Journal of Probability and Statistics

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Partial least squares (PLS) regression is an alternative to the ordinary least squares (OLS) regression, used in the presence of multicollinearity. As with any other modelling method, PLS regression requires a reliable model selection tool. Cross validation (CV) is the most commonly used tool with many advantages in both preciseness and accuracy, but it also has some drawbacks; therefore, we will use L-curve criterion as an alternative, given that it takes into consideration the shrinking nature of PLS. A theoretical justification for the use of L-curve criterion is presented as well as an application on both simulated and real data. The application shows how this criterion generally outperforms cross validation and generalized cross validation (GCV) in mean squared prediction error and computational efficiency.

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Section: Generalized Cross Validationmentioning

confidence: 99%

The L-Curve Criterion as a Model Selection Tool in PLS Regression

Kerkri

Allal

Zarrouk

2019

Journal of Probability and Statistics

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“…We suggest to use the Generalized Cross Validation method (GCV) described in [10,11]. The performance of GCV has been tested in [12,13], and it resulted that GCV is more effective than other stopping rules when information about the entity of the noise is not available. Given a sequence x ν generated by an iterative method to approximate the solution of the system Ax = b, the GCV functional is defined by…”

Section: Generalized Cross Validation (Gcv)mentioning

confidence: 99%

Regularized Compression of A Noisy Blurred Image

Favati¹,

Lotti²,

Menchi³

et al. 2016

IJCSA

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“…If L = I a solution of (5) having minimal norm is sought and the problem is said to be in standard form. We consider here the problem in standard form with γ = 0, which is equivalent to solving the normal equations (3). Let x (λ) be the solution of (3).…”

mentioning

confidence: 99%

“…3. CG applied to system (3). We examine first how CG applies to solve (3) with different values of the parameter λ > 0.…”

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confidence: 99%

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An inner-outer regularizing method for ill-posed problems

Favati¹,

Lotti²,

Menchi³

et al. 2014

Inverse Problems &Amp; Imaging

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Conjugate Gradient is widely used as a regularizing technique for solving linear systems with ill-conditioned coefficient matrix and right-hand side vector perturbed by noise. It enjoys a good convergence rate and computes quickly an iterate, say xkopt, which minimizes the error with respect to the exact solution. This behavior can be a disadvantage in the regulariza-tion context, because also the high-frequency components of the noise enter quickly the computed solution, leading to a difficult detection of kopt and to a sharp increase of the error after the koptth iteration. In this paper we propose an inner-outer algorithm based on a sequence of restarted Conjugate Gradients, with the aim of overcoming this drawback. A numerical experimentation validates the effectiveness of the proposed algorithm. ©2014 American Institute of Mathematical Sciences

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Generalized Cross-Validation applied to Conjugate Gradient for discrete ill-posed problems

Cited by 10 publications

References 15 publications

The L-Curve Criterion as a Model Selection Tool in PLS Regression

The L-Curve Criterion as a Model Selection Tool in PLS Regression

Regularized Compression of A Noisy Blurred Image

An inner-outer regularizing method for ill-posed problems

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