On a quadratic eigenproblem occurring in regularized total least squares

Lampe, Jörg; Voß, Heinrich

doi:10.1016/j.csda.2007.05.020

Cited by 27 publications

(27 citation statements)

References 13 publications

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“…In chapter 3 in subsection 3.4.2 it is shown that a similar technique has been successfully applied for accelerating the RTLS solver in [108] which is based on a sequence of quadratic eigenvalue problems, cf. [60,63]. Another method for RTLS problems presented in [95] which is based on a sequence of linear eigenproblems has also been accelerated substantially in [62,66].…”

Section: Nonlinear Arnoldimentioning

confidence: 99%

“…In [60] we studied this quadratic eigenproblem, and we proved that (3.96) always has a rightmost real eigenvalueλ such that λ ≥ real(λ) for every eigenvalue λ of (3.96).…”

Section: Rtls Fixed Point Iterationmentioning

confidence: 99%

“…subsection 3.4.1. Using a maxmin characterization for nonlinear eigenvalue problems [129] the occurring quadratic eigenproblems are analyzed and the existence of a rightmost real eigenvalue is proven in [60]. The difference between Algorithms 3.1 and 3.2 is highlighted by proving the existence of a negative Lagrange multiplier, see also [60] and the end of subsection 3.4.1.…”

Section: Rtlsqepmentioning

confidence: 99%

“…Using a maxmin characterization for nonlinear eigenvalue problems [129] the occurring quadratic eigenproblems are analyzed and the existence of a rightmost real eigenvalue is proven in [60]. The difference between Algorithms 3.1 and 3.2 is highlighted by proving the existence of a negative Lagrange multiplier, see also [60] and the end of subsection 3.4.1. Efficient implementations for solving the sequence of QEPs are discussed in [60,63,66] with the special emphasis on iterative projection methods and updating techniques that accelerates the RTLSQEP method in [108] substantially.…”

Section: Rtlsqepmentioning

confidence: 99%

“…This first-order conditions were used in literature in two ways to solve problem (3.68): In [60,63,66,107,108] for a fixed parameter λ I problem (3.81) is solved for (x, λ L ), which yields a convergent sequence of updates for λ I . A closely related algorithm is presented in [5,6] that solve a similar sequence where in each iteration step the equality constraint in (3.81) is replaced by the original inequality Lx 2 ≤ ∆ 2 .…”

Section: Quadratically Constrained Tlsmentioning

confidence: 99%

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Solving Regularized Total Least Squares Problems Based on Eigenproblems

Lampe

Voß

2010

Taiwanese J. Math.

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Bibliografische Information der Deutschen NationalbibliothekDie Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar. dissertation.de -Verlag im Internet GmbH 2010Alle Rechte, auch das des auszugsweisen Nachdruckes, der auszugsweisen oder vollständigen Wiedergabe, der Speicherung in Datenverarbeitungsanlagen, auf Datenträgern oder im Internet und der Übersetzung, vorbehalten.

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Section: Nonlinear Arnoldimentioning

confidence: 99%

“…In [60] we studied this quadratic eigenproblem, and we proved that (3.96) always has a rightmost real eigenvalueλ such that λ ≥ real(λ) for every eigenvalue λ of (3.96).…”

Section: Rtls Fixed Point Iterationmentioning

confidence: 99%

Section: Rtlsqepmentioning

confidence: 99%

Section: Rtlsqepmentioning

confidence: 99%

Section: Quadratically Constrained Tlsmentioning

confidence: 99%

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Solving Regularized Total Least Squares Problems Based on Eigenproblems

Lampe

Voß

2010

Taiwanese J. Math.

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Total least squares methods

Markovsky

Sima

Huffel

2010

WIREs Computational Stats

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Recent advances in total least squares approaches for solving various errorsin-variables modeling problems are reviewed, with emphasis on the following generalizations:1. the use of weighted norms as a measure of the data perturbation size, capturing prior knowledge about uncertainty in the data;2. the addition of constraints on the perturbation to preserve the structure of the data matrix, motivated by structured data matrices occurring in signal and image processing, systems and control, and computer algebra;3. the use of regularization in the problem formulation, aiming at stabilizing the solution by decreasing the effect because of intrinsic ill-conditioning of certain problems. The focus of this paper is on computational algorithms for solving the generalized TLS problems. The reader is referred to the errors-in-variables literature for the statistical properties of the corresponding estimators, as well as for a wider range of applications. WEIGHTED AND STRUCTURED TOTAL LEAST SQUARES PROBLEMSThe TLS solution where the vector of perturbations vec ([ A b]) is zero mean and has covariance matrix that is equal to the identity up to a scaling factor, i.e.,The noise assumption (3) implies that all elements of the data matrix are measured with equal precision, an assumption that may not be satisfied in practice. A natural generalization of the EIV model (Eq. (2,3)) is to allow the covariance matrix of the vectorized noise to be of the form σ 2 V, where V is a given positive definite matrix. The corresponding estimation problem is the TLS problem (1) with the Frobenius norm · F replaced by the weighted matrix norm

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Regularization of Large Scale Total Least Squares Problems

Voß

Lampe²

2011

Communications in Computer and Information Science

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On a quadratic eigenproblem occurring in regularized total least squares

Cited by 27 publications

References 13 publications

Solving Regularized Total Least Squares Problems Based on Eigenproblems

Solving Regularized Total Least Squares Problems Based on Eigenproblems

Total least squares methods

Regularization of Large Scale Total Least Squares Problems

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