Perturbation analysis for mixed least squares–total least squares problems

Zheng, Bing; Yang, Zhanshan

doi:10.1002/nla.2239

Cited by 18 publications

(15 citation statements)

References 26 publications

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“…One advantage of our estimator is that it is ready to be extended to location of multiple disjoint targets by concatenating the data matrices in Section 5. Let the number of the disjoint targets be K, and 10 3 Monte Carlo experiments of joint location are performed for each value of K (= 1,2,4,8,16,32,64). Then, the running time of the 10 3 experiments are recorded.…”

Section: Localizing Multiple Disjoint Targetsmentioning

confidence: 99%

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Design and Analysis of a Non-Iterative Estimator for Target Location in Multistatic Sonar Systems with Sensor Position Uncertainties

Wang

Yang

et al. 2020

Mathematics

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Target location is the basic application of a multistatic sonar system. Determining the position/velocity vector of a target from the related sonar observations is a nonlinear estimation problem. The presence of possible sensor position uncertainties turns this problem into a more challenging hybrid parameter estimation problem. Conventional gradient-based iterative estimators suffer from the problems of initialization difficulties and local convergence. Even if there is no problem with initialization and convergence, a large computational cost is required in most cases. In view of these drawbacks, we develop a computationally efficient non-iterative position/velocity estimator. The main numerical computation involved is the weighted least squares optimization, which makes the estimator computationally efficient. Parameter transformation, model linearization and two-stage processing are exploited to prevent the estimator from iterative computation. Through performance analysis and experimental verification, we find that the proposed estimator reaches the hybrid Cramér–Rao bound and has linear computational complexity.

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Section: Localizing Multiple Disjoint Targetsmentioning

confidence: 99%

“…Perturbation analysis of least squares problems is a major topic in numerical linear algebra. Related work has focused on establishing various error bounds [32][33][34]. We combine the basic techniques of perturbation analysis with multivariate statistics [35] to quantitatively evaluate the estimators for a nonlinear estimation problem.…”

Section: Introductionmentioning

confidence: 99%

Design and Analysis of a Non-Iterative Estimator for Target Location in Multistatic Sonar Systems with Sensor Position Uncertainties

Wang

Yang

et al. 2020

Mathematics

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“…The Kronecker product-based formulas for the mixed and componentwise condition numbers to the TLS problem (2) were derived in References [12,13]. The corresponding componentwise perturbation analysis for the multidimensional TLS problem and mixed least squares-TLS problem can be found in References [14,15].…”

Section: Introductionmentioning

confidence: 99%

Condition numbers for the truncated total least squares problem and their estimations

Diao

Bai

2021

Numerical Linear Algebra App

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In this paper, we present explicit expressions for the mixed and componentwise condition numbers of the truncated total least squares (TTLS) solution of Ax ≈ b under the genericity condition, where A is a m × n real data matrix and b is a real m-vector. Moreover, we reveal that normwise, componentwise, and mixed condition numbers for the TTLS problem can recover the previous corresponding counterparts for the total least squares (TLS) problem when the truncated level of the TTLS problem is n. When A is a structured matrix, the structured perturbations for the structured truncated TLS (STTLS) problem are investigated and the corresponding explicit expressions for the structured normwise, componentwise, and mixed condition numbers for the STTLS problem are obtained. Furthermore, the relationships between the structured and unstructured normwise, componentwise, and mixed condition numbers for the STTLS problem are studied. We devise reliable condition estimation algorithms for the TTLS problem by utilizing small-sample statistical condition estimation techniques. The proposed condition estimation algorithms employ the singular value decomposition (SVD) of the augmented matrix [A b] to reduce the computational complexity, where both unstructured and structured normwise, mixed, and componentwise condition estimations are considered. The proposed condition estimation algorithms can be integrated into the SVD-based direct solver for the small and medium size TTLS problem to give the error estimation for the numerical TTLS solution. Numerical experiments are reported to illustrate the reliability of the proposed condition estimation algorithms.

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“…Condition number plays an important role in perturbation theory and error analysis for algorithms; see e.g., [8,9,10]. Recently, the condition numbers of TLS problem, the scaled TLS problem, the multidimensional TLS problem, the mixed LS-TLS problem, truncated-TLS problem and TLSE problem have been considered; see [19,20,21,22,23,24,25,26,27]. Structured TLS problems [11,12,13] had been studied extensively in the past decades.…”

Section: Introductionmentioning

confidence: 99%

Structured condition numbers for the total least squares problem with linear equality constraint and their statistical estimation

Samar¹

2021

Preprint

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In this paper, we derive the mixed and componentwise condition numbers for a linear function of the solution to the total least squares with linear equality constraint (TLSE) problem. The explicit expressions of the mixed and componentwise condition numbers by dual techniques under both unstructured and structured componentwise perturbations is considered. With the intermediate result, i.e. we can recover the both unstructured and structured condition number for the TLS problem. We choose the small-sample statistical condition estimation method to estimate both unstructured and structured condition numbers with high reliability. Numerical experiments are provided to illustrate the obtained results.

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Perturbation analysis for mixed least squares–total least squares problems

Cited by 18 publications

References 26 publications

Design and Analysis of a Non-Iterative Estimator for Target Location in Multistatic Sonar Systems with Sensor Position Uncertainties

Design and Analysis of a Non-Iterative Estimator for Target Location in Multistatic Sonar Systems with Sensor Position Uncertainties

Condition numbers for the truncated total least squares problem and their estimations

Structured condition numbers for the total least squares problem with linear equality constraint and their statistical estimation

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