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Showing papers on "Non-linear least squares published in 2017"


Journal ArticleDOI
TL;DR: A decomposition based least squares iterative identification algorithm for multivariate pseudo-linear autoregressive moving average systems using the data filtering to transform the original system to a hierarchical identification model and to decompose this model into three subsystems and to identify each subsystem.
Abstract: This paper develops a decomposition based least squares iterative identification algorithm for multivariate pseudo-linear autoregressive moving average systems using the data filtering. The key is to apply the data filtering technique to transform the original system to a hierarchical identification model, and to decompose this model into three subsystems and to identify each subsystem, respectively. Compared with the least squares based iterative algorithm, the proposed algorithm requires less computational efforts. The simulation results show that the proposed algorithms can work well.

148 citations


Journal ArticleDOI
TL;DR: In this paper, the authors introduced techniques for efficiently preprocessing, sampling, and computing randomized least squares on a dense tensor of arbitrary order, as well as an efficient sampling-based technique for checking the stopping condition.
Abstract: The CANDECOMP/PARAFAC (CP) decomposition is a leading method for the analysis of multiway data. The standard alternating least squares algorithm for the CP decomposition (CP-ALS) involves a series of highly overdetermined linear least squares problems. We extend randomized least squares methods to tensors and show the workload of CP-ALS can be drastically reduced without a sacrifice in quality. We introduce techniques for efficiently preprocessing, sampling, and computing randomized least squares on a dense tensor of arbitrary order, as well as an efficient sampling-based technique for checking the stopping condition. We also show more generally that the Khatri-Rao product (used within the CP-ALS iteration) produces conditions favorable for direct sampling. In numerical results, we see improvements in speed, reductions in memory requirements, and robustness with respect to initialization.

108 citations


Journal ArticleDOI
TL;DR: The Akiake Information Criterion and its related model comparison indices are explained in the context of least squares (ordinary, weighted, iterative weighted or “generalized”, etc.) based inverse problem formulations.

81 citations


Journal ArticleDOI
TL;DR: It is shown that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling, and the least squares formulation is shown to be suitable for collocation-based radial basis function-PUMs.
Abstract: Recently, collocation-based radial basis function (RBF) partition of unity methods (PUMs) for solving partial differential equations have been formulated and investigated numerically and theoretically. When combined with stable evaluation methods such as the RBF-QR method, high order convergence rates can be achieved and sustained under refinement. However, some numerical issues remain. The method is sensitive to the node layout, and condition numbers increase with the refinement level. Here, we propose a modified formulation based on least squares approximation. We show that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling. We derive theoretical error estimates both for the collocation and least squares RBF-PUMs. Numerical experiments are performed for the Poisson equation in two and three space dimensions for regular and irregular geometries. The convergence experiments confirm the theoretical estimates, and the least squares formulation is shown to ...

77 citations


Journal ArticleDOI
TL;DR: A new nonlinear quality-related fault detection method based on kernel partial least squares (KPLS) model that has the advantages of simple diagnosis logic and stable performance is proposed.
Abstract: In this paper, a new nonlinear quality-related fault detection method is proposed based on kernel partial least squares (KPLS) model. To deal with the nonlinear characteristics among process variables, the proposed method maps these original variables into feature space in which the linear relationship between kernel matrix and output matrix is realized by means of KPLS. Then the kernel matrix is decomposed into two orthogonal parts by singular value decomposition (SVD) and the statistics for each part are determined appropriately for the purpose of quality-related fault detection. Compared with relevant existing nonlinear approaches, the proposed method has the advantages of simple diagnosis logic and stable performance. A widely used literature example and an industrial process are used for the performance evaluation for the proposed method.

73 citations


Journal ArticleDOI
TL;DR: A nonlinear quality-related fault detection approach based on kernel least squares (KLS) model that extracts the full correlation information of feature matrix and only uses two statistics to determine the type of fault, which is more stable than the existing approaches.
Abstract: In this paper, a nonlinear quality-related fault detection approach is proposed based on kernel least squares (KLS) model. The major novelty of the proposed method is that it utilizes KLS model to exploit the entire correlation between feature and output matrices. First, it uses a nonlinear projection function to map original process variables into feature space in which the correlation between feature and output matrices is realized by means of KLS. Then, the feature matrix is decomposed into two orthogonal parts by singular value decomposition and the statistics for each part are determined appropriately for the purpose of quality-related fault detection. Compared with existing kernel partial least squares (KPLS) based approaches, the proposed new method has the following obvious advantages. 1) It extracts the full correlation information of feature matrix, while KPLS-based approaches only use the partial correlation of several selected latent variables; therefore, it is more stable than the existing ones. 2) It omits the iterative computation of KPLS model and the determination of the number of latent variables; therefore, it is more efficient in engineering implementation. 3) It only uses two statistics to determine the type of fault, while most of the KPLS-based approaches need four; therefore, it has a more simple diagnosis logic. For simulation verification, a widely used literature example and an industrial benchmark are utilized to evaluate the performance of the proposed method.

62 citations


Journal ArticleDOI
TL;DR: A data filtering based recursive least squares algorithm is proposed based on the data filtering technique and results show that the proposed algorithm can generate more accurate parameter estimates than the recursive generalized most squares algorithm.
Abstract: Nonlinear systems exist widely in industrial processes. This paper studies the parameter estimation methods of establishing the mathematical models for a class of output nonlinear systems, whose output is nonlinear about the past outputs and linear about the inputs. We use an estimated noise transfer function to filter the input–output data and obtain two identification models, one containing the parameters of the system model, and the other containing the parameters of the noise model. Based on the data filtering technique, a data filtering based recursive least squares algorithm is proposed. The simulation results show that the proposed algorithm can generate more accurate parameter estimates than the recursive generalized least squares algorithm.

60 citations


Journal ArticleDOI
TL;DR: This paper gives the input-output representation of a bilinear system through eliminating the state variables in it, and derives a maximum likelihood least squares based iterative for identifying the parameters of bil inear systems with colored noises by using the maximum likelihood principle.
Abstract: Maximum likelihood methods are significant for parameter estimation and system modeling. This paper gives the input-output representation of a bilinear system through eliminating the state variables in it, and derives a maximum likelihood least squares based iterative for identifying the parameters of bilinear systems with colored noises by using the maximum likelihood principle. A least squares based iterative (LSI) algorithm is presented for comparison. It is proved that the maximum of the likelihood function is equivalent to minimize the least squares cost function. The simulation results indicate that the proposed algorithm is effective for identifying bilinear systems and the maximum likelihood LSI algorithm is more accurate than the LSI algorithm.

59 citations


Journal ArticleDOI
TL;DR: To assess the performance of various least squares and Bayesian modeling approaches to parameter estimation in intravoxel incoherent motion (IVIM) modeling of diffusion‐weighted MRI data, a large number of models were used.
Abstract: Purpose To assess the performance of various least squares and Bayesian modeling approaches to parameter estimation in intravoxel incoherent motion (IVIM) modeling of diffusion-weighted MRI data. Methods Simulated tissue models of different type (breast/liver) and morphology (discrete/continuous) were used to generate noisy data according to the IVIM model at several signal-to-noise ratios. IVIM parameter maps were generated using six different approaches, including full nonlinear least squares (LSQ), segmented least squares (SEG), Bayesian modeling with a Gaussian shrinkage prior (BSP) and Bayesian modeling with a spatial homogeneity prior (FBM), plus two modified approaches. Estimators were compared by calculating the median absolute percentage error and deviation, and median percentage bias. Results The Bayesian modeling approaches consistently outperformed the least squares approaches, with lower relative error and deviation, and provided cleaner parameter maps with reduced erroneous heterogeneity. However, a weakness of the Bayesian approaches was exposed, whereby certain tissue features disappeared completely in regions of high parameter uncertainty. Lower error and deviation were generally afforded by FBM compared with BSP, at the cost of higher bias. Conclusions Bayesian modeling is capable of producing more visually pleasing IVIM parameter maps than least squares approaches, but their potential to mask certain tissue features demands caution during implementation. Magn Reson Med 78:2373–2387, 2017. © 2017 International Society for Magnetic Resonance in Medicine.

59 citations


Journal ArticleDOI
TL;DR: In this article, the first model-based technique to calibrate a magnetic manipulation system by using nonlinear least squares to solve for a scalar potential for each source is presented.
Abstract: Model-based calibration of a magnetic workspace not only provides a smooth representation of the field and its gradient matrix, but also uses physical constraints to smooth the calibration measurements. This paper presents the first model-based technique to calibrate a magnetic manipulation system by using nonlinear least squares to solve for a scalar potential for each source. The performance of the method is verified by comparison to numerical finite element simulation and a case study calibration of a real system, where it is able to achieve an $R^{2}$ value of 0.9997. Furthermore, the analytical representations for the first three spatial derivatives of a spherical multipole expansion are provided for convenience, which correspond to the torque, force, and force-spatial-rate-of-change on a magnetic dipole in the workspace.

52 citations


Journal ArticleDOI
TL;DR: In this paper, the disadvantages of the usual linear least-squares analysis of first-and second-order kinetic data are described, and nonlinear least squares fitting is recommended as an alternative.
Abstract: The disadvantages of the usual linear least-squares analysis of first- and second-order kinetic data are described, and nonlinear least-squares fitting is recommended as an alternative.


Journal ArticleDOI
TL;DR: In this article, the authors used a weighted nonlinear least squares (WNLS) algorithm for parameter estimation using measured breakthrough curves (BTCs) for pulse initial conditions and continuous injections.
Abstract: Anomalous transport cannot be adequately described with classical Fickian advection-dispersion equations (ADE) with constant coefficients. Rather, fractional calculus models may be used, which capture salient features of anomalous transport (e.g., skewness and power law tails). FracFit is a parameter estimation tool based on space-fractional and time-fractional models used by the hydrology community. Currently, four fractional models are supported: (1) space-fractional advection-dispersion equation (sFADE), (2) time-fractional dispersion equation with drift (TFDE), (3) fractional mobile-immobile (FMIM) equation, and (4) temporally tempered Levy motion (TTLM). Model solutions using pulse initial conditions and continuous injections are evaluated using stable distributions or subordination integrals. Parameter estimates are extracted from measured breakthrough curves (BTCs) using a weighted nonlinear least squares (WNLS) algorithm. Optimal weights for BTCs for pulse initial conditions and continuous injections are presented. Two sample applications are analyzed: (1) pulse injection BTCs in the Selke River and (2) continuous injection laboratory experiments using natural organic matter. Model parameters are compared across models and goodness-of-fit metrics are presented, facilitating model evaluation.

Journal ArticleDOI
TL;DR: This paper proposes a globally convergent identification algorithm for nonlinear rational systems defined as the ratio of two nonlinear functions of past inputs and outputs, and to the best of the authors' knowledge, this is the first globally convertergent algorithm for the non linear rational systems.

Journal ArticleDOI
TL;DR: Olivieri, Alejandro Cesar, and Cesar as discussed by the authors have published a paper as discussed by the authors, where they present the Consejo Nacional de Investigaciones Cientificas and Tecnicas.
Abstract: Fil: Olivieri, Alejandro Cesar. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Centro Cientifico Tecnologico Conicet - Rosario. Instituto de Quimica Rosario. Universidad Nacional de Rosario. Facultad de Ciencias Bioquimicas y Farmaceuticas. Instituto de Quimica Rosario; Argentina

Journal ArticleDOI
TL;DR: In this paper, the Polynomial Least Squares Method is applied in order to compute analytical approximate polynomial solutions for the fractional Riccati type differential equations.
Abstract: In this paper the Polynomial Least Squares Method is applied in order to compute analytical approximate polynomial solutions for the fractional Riccati type differential equations. The accuracy of the method is tested by means of the comparison with previous results for several applications.

Journal ArticleDOI
TL;DR: A tensor-input/tensor-output Recursive Exponentially Weighted N-Way Partial Least Squares regression algorithm is proposed for high dimension multi-way (tensor) data treatment and adaptive modeling of complex processes in real-time.
Abstract: A tensor-input/tensor-output Recursive Exponentially Weighted N-Way Partial Least Squares (REW-NPLS) regression algorithm is proposed for high dimension multi-way (tensor) data treatment and adaptive modeling of complex processes in real-time. The method unites fast and efficient calculation schemes of the Recursive Exponentially Weighted PLS with the robustness of tensor-based approaches. Moreover, contrary to other multi-way recursive algorithms, no loss of information occurs in the REW-NPLS. In addition, the Recursive-Validation method for recursive estimation of the hyper-parameters is proposed instead of conventional cross-validation procedure. The approach was then compared to state-of-the-art methods. The efficiency of the methods was tested in electrocorticography (ECoG) and magnetoencephalography (MEG) datasets. The algorithms are implemented in software suitable for real-time operation. Although the Brain-Computer Interface applications are used to demonstrate the methods, the proposed approaches could be efficiently used in a wide range of tasks beyond neuroscience uniting complex multi-modal data structures, adaptive modeling, and real-time computational requirements.

Book
09 Aug 2017
TL;DR: In this paper, the Aitken Model and Generalized Least Squares Application: Aggregation Bias Best Estimation in a Constrained Parameter Space Addendum: Variance of Variance Estimator Distributional Theory.
Abstract: Preface Examples of the General Linear Model Introduction One-Sample Problem Simple Linear Regression Multiple Regression One-Way ANOVA First Discussion The Two-Way Nested Model Two-Way Crossed Model Analysis of Covariance Autoregression Discussion The Linear Least Squares Problem The Normal Equations The Geometry of Least Squares Reparameterization Gram-Schmidt Orthonormalization Estimability and Least Squares Estimators Assumptions for the Linear Mean Model Confounding, Identifiability, and Estimability Estimability and Least Squares Estimators First Example: One-Way ANOVA Second Example: Two-Way Crossed without Interaction Two-Way Crossed with Interaction Reparameterization Revisited Imposing Conditions for a Unique Solution to the Normal Equations Constrained Parameter Space Gauss-Markov Model Model Assumptions The Gauss-Markov Theorem Variance Estimation Implications of Model Selection The Aitken Model and Generalized Least Squares Application: Aggregation Bias Best Estimation in a Constrained Parameter Space Addendum: Variance of Variance Estimator Distributional Theory Introduction Multivariate Normal Distribution Chi-Square and Related Distributions Distribution of Quadratic Forms Cochran's Theorem Regression Models with Joint Normality Statistical Inference Introduction Results from Statistical Theory Testing the General Linear Hypothesis The Likelihood Ratio Test and Change in SSE First Principles Test and LRT Confidence Intervals and Multiple Comparisons Identifiability Further Topics in Testing Introduction Reparameterization Applying Cochran's Theorem for Sequential SS Orthogonal Polynomials and Contrasts Pure Error and the Lack-of-Fit Test Heresy: Testing Nontestable Hypotheses Variance Components and Mixed Models Introduction Variance Components: One Way Variance Components: Two-Way Mixed ANOVA Variance Components: General Case The Split Plot Predictions and BLUPs The Multivariate Linear Model Introduction The Multivariate Gauss-Markov Model Inference under Normality Assumptions Testing Repeated Measures Confidence Intervals Appendix A: Review of Linear Algebra Notation and Fundamentals Rank, Column Space, and Nullspace Some Useful Results Solving Equations and Generalized Inverses Projections and Idempotent Matrices Trace, Determinants, and Eigenproblems Definiteness and Factorizations Appendix B: Lagrange Multipliers Main Results Bibliography A Summary, Notes, and Exercises appear at the end of most chapters.

Journal ArticleDOI
TL;DR: Two computationally efficient residual Doppler shift estimation methods based on computing the phase of the root of a low order polynomial and a closed-form least squares estimate given the unwrapped phases of the minimal eigenvector of a small data matrix are proposed.
Abstract: We propose two computationally efficient residual Doppler shift estimation methods for underwater acoustic multicarrier communication. The first method is based on computing the phase of the root of a low order polynomial. The second method is a closed-form least squares estimate given the unwrapped phases of the minimal eigenvector of a small data matrix. The complexities of both estimates are significantly lower compared to the methods commonly used in underwater acoustic multicarrier communication, which result in nonlinear least squares estimators and thus require a fine grid search in the frequency domain. Numerical simulations show that the mean square errors of the proposed methods have similar performance as the common estimation techniques, achieve the Cramer–Rao lower bounds at low noise levels, and agree with their theoretically derived variances. Pool experiments and sea trial results further demonstrate that the suggested estimates yield similar results as the common nonlinear least squares estimates but at a lower complexity.

Journal ArticleDOI
TL;DR: The purpose of this note is to point out that the recently proposed fractional least mean squares (FLMS) algorithm, whose derivation is based on fractional derivative, is not suitable for adaptive signal processing.

Journal ArticleDOI
TL;DR: This paper explores the problem of localizing an emitter of radio frequency energy using a network of mobile receiver nodes, each taking measurements of the emitter’s received signal strength (RSS), and proposes two novel estimators to handle these effects as modifications of the nonlinear least squares and the Gaussian particle filter algorithms.
Abstract: This paper explores the problem of localizing an emitter of radio frequency energy using a network of mobile receiver nodes, each taking measurements of the emitter’s received signal strength (RSS). In this paper, we drop the assumption of receiver calibration, leading to biased measurements for each node. We model these bias effects as additive random variables for each receiver in the log-distance path loss model. We propose two novel estimators to handle these effects as modifications of the nonlinear least squares and the Gaussian particle filter algorithms. The estimators are augmented using the principle of variance least squares, in which the biases’ effect on the data covariance is estimated online. These estimates inform subsequent iterations of the nonlinear algorithms. The path loss exponent and emitter power offset are likewise treated as unknowns. Our simulations show the performance improvement evident in various scenarios over the naive approaches, other contemporary algorithms, and with respect to the Cramer-Rao lower bound. We further show the efficacy of our approach through several experiments using real RSS data collected from a mobile network.

Journal ArticleDOI
TL;DR: The proposed method can robustly estimate and then remove image radial distortion with high accuracy and is extremely useful in many applications, particularly those where human-made environments contain abundant lines.
Abstract: This paper presents an approach for estimating and then removing image radial distortion. It works on a single image and does not require a special calibration. The approach is extremely useful in many applications, particularly those where human-made environments contain abundant lines. A division model is applied, in which a straight line in the distorted image is treated as a circular arc. Levenberg–Marquardt (LM) iterative nonlinear least squares method is adopted to calculate the arc’s parameters. Then “Taubin fit” is applied to obtain the initial guess of the arc’s parameters which works as the initial input to the LM iteration. This dramatically improves the convergence rate in the LM process to obtain the required parameters for correcting image radial distortion. Hough entropy, as a measure, has achieved the quantitative evaluation of the estimated distortion based on the probability distribution in one-dimensional θ Hough space. The experimental results on both synthetic and real images have demonstrated that the proposed method can robustly estimate and then remove image radial distortion with high accuracy.

Journal ArticleDOI
TL;DR: P perturbation analysis for the total least squares problems under the genericity condition is presented and three condition numbers proposed respectively by Zhou et al. are reviewed.
Abstract: In this note, we present perturbation analysis for the total least squares (Tls) problems under the genericity condition. We review the three condition numbers proposed respectively by Zhou et al. (Numer. Algorithm, 51 (2009), pp. 381---399), Baboulin and Gratton (SIAM J. Matrix Anal. Appl. 32 (2011), pp. 685---699), Li and Jia (Linear Algebra Appl. 435 (2011), pp. 674---686). We also derive new perturbation bounds.

Journal ArticleDOI
TL;DR: In this paper, the authors proposed a weighted total least squares (WTLS) method to adjust an errors-in-variables (EIV) model containing random errors both in the observation vector and in the coefficient matrix.
Abstract: Weighted total least squares (WTLS) has been widely used as a standard method to optimally adjust an errors-in-variables (EIV) model containing random errors both in the observation vector and in the coefficient matrix. An earlier work provided a simple and flexible formulation forWTLS based on the standard least-squares (SLS) theory. The formulation allows one to directly apply the available SLS theory to the EIV models. Among such applications, this contribution formulates the WTLS problem subject to weighted or hard linear(ized) equality constraints on unknown parameters. The constraints are to be properly incorporated into the system of equations in an EIV model of which a general structure for the (singular) covariance matrix QA of the coefficient matrix is used. The formulation can easily take into consideration any number of weighted linear and nonlinear constraints. Hard constraints turn out to be a special case of the general formulation of the weighted constraints. Because the formulation is based on the SLS theory, the method automatically approximates the covariance matrix of the estimates from which the precision of the constrained estimates can be obtained. Three numerical examples with different scenarios are used to demonstrate the efficacy of the proposed algorithm for geodetic applications. DOI: 10.1061/(ASCE)SU.1943-5428.0000239.© 2017 American Society of Civil Engineers. Author keywords: Weighted total least squares (WTLS); Errors-in-variables (EIV) model; Linear equality constraints; Two-dimensional (2D) affine transformation.

Journal ArticleDOI
TL;DR: By setting the initial iterative value properly and proving that the iterative solution converges to the least squares and minimum-norm solution, the conjugate gradient least squares algorithm for solving the generalized coupled Sylvester matrix equations is proved.
Abstract: This paper discusses the conjugate gradient least squares algorithm for solving the generalized coupled Sylvester matrix equations ∑ j = 1 q A i j X j B i j = F i , i = 1 , 2 , … , p . We prove that if this system is consistent then the iterative solution converges to the exact solution and if this system is inconsistent then the iterative solution converges to the least squares solution within the finite iteration steps in the absence of the roundoff errors. Also by setting the initial iterative value properly we prove that the iterative solution converges to the least squares and minimum-norm solution.

Journal ArticleDOI
TL;DR: In this article, a generalized total least squares prediction (GTLSP) algorithm is proposed to solve the universal 3D similarity transformation problem, where all of the random errors in the original and target coordinates, and their variance-covariance information is considered.

Journal ArticleDOI
TL;DR: In this article, the authors proposed several approximate Gauss-Newton methods for solving underdetermined nonlinear least squares problems, i.e., the truncated, perturbed, and truncated-perturbed GN methods, under the assumption that the Frechet derivatives are Lipschitz continuous and of full row rank.

Journal ArticleDOI
TL;DR: The authors consider the mean-square-deviation (MSD) as the performance metric in the steady-state and derive a theoretical expression for PDRLS algorithm with noisy links and show that under certain statistical assumptions for the measurement data and noise signals, under noisy links the PDR LS algorithm is stable in both mean and mean- square senses.
Abstract: Partial diffusion-based recursive least squares (PDRLS) is an effective way of lowering computational load and power consumption in adaptive network implementation. In this method, every single node distributes a fraction of its intermediate vector estimate with its immediate neighbours at each iteration. In this study, the authors examine the steady-state performance of PDRLS algorithm in the presence of noisy links by means of an energy conservation argument. They consider the mean-square-deviation (MSD) as the performance metric in the steady-state and derive a theoretical expression for PDRLS algorithm with noisy links. The authors' analysis reveals that unlike the established statements on PDRLS scheme under ideal links, the trade-off between MSD performance and the number of selected entries of the intermediate estimate vectors, as a sign of communication cost, is mitigated. They further examine the convergence behaviour of the PDRLS algorithm. The obtained results show that under certain statistical assumptions for the measurement data and noise signals, under noisy links the PDRLS algorithm is stable in both mean and mean-square senses. Finally, they present some simulation results to verify the theoretical findings.

Journal ArticleDOI
TL;DR: In this article, a non-linear least squares (Non-LS) method is proposed for the Weibull modulus estimation of casting properties, which is based on the linear transformation of the weibull function, occurring in the traditional LLS method.
Abstract: The Maximum Likelihood method and the Linear Least Squares (LLS) method have been widely used to estimate Weibull parameters for reliability of brittle and metal materials. In the last 30 years, many researchers focused on the bias of Weibull modulus estimation, and some improvements have been achieved, especially in the case of the LLS method. However, there is a shortcoming in these methods for a specific type of data, where the lower tail deviates dramatically from the well-known linear fit in a classic LLS Weibull analysis. This deviation can be commonly found from the measured properties of materials, and previous applications of the LLS method on this kind of dataset present an unreliable linear regression. This deviation was previously thought to be due to physical flaws (i.e., defects) contained in materials. However, this paper demonstrates that this deviation can also be caused by the linear transformation of the Weibull function, occurring in the traditional LLS method. Accordingly, it may not be appropriate to carry out a Weibull analysis according to the linearized Weibull function, and the Non-linear Least Squares method (Non-LS) is instead recommended for the Weibull modulus estimation of casting properties.

Journal ArticleDOI
TL;DR: In the proposed BPR method, a perturbation with a bounded norm is allowed into the linear transformation matrix to improve the singular-value structure and provides significant improvement over state-of-the-art methods.
Abstract: This paper addresses the problem of selecting the regularization parameter for linear least-squares estimation. We propose a new technique called bounded perturbation regularization (BPR). In the proposed BPR method, a perturbation with a bounded norm is allowed into the linear transformation matrix to improve the singular-value structure. Following this, the problem is formulated as a min–max optimization problem. Next, the min–max problem is converted to an equivalent minimization problem to estimate the unknown vector quantity. The solution of the minimization problem is shown to converge to that of the $\ell _{2}$ -regularized least squares problem, with the unknown regularizer related to the norm bound of the introduced perturbation through a nonlinear constraint. A procedure is proposed that combines the constraint equation with the mean squared error criterion to develop an approximately optimal regularization parameter selection algorithm. Both direct and indirect applications of the proposed method are considered. Comparisons with different Tikhonov regularization parameter selection methods, as well as with other relevant methods, are carried out. Numerical results demonstrate that the proposed method provides significant improvement over state-of-the-art methods.