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    Regularization Techniques for Linear Least-Squares Problems

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    M Suliman- Final Thesis.pdf
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    Description:
    Final Thesis
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    Type
    Thesis
    Authors
    Suliman, Mohamed Abdalla Elhag cc
    Advisors
    Al-Naffouri, Tareq Y. cc
    Committee members
    Hoteit, Ibrahim cc
    Alouini, Mohamed-Slim cc
    Program
    Electrical and Computer Engineering
    KAUST Department
    Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division
    Date
    2016-04
    Embargo End Date
    2017-05-12
    Permanent link to this record
    http://hdl.handle.net/10754/609160
    
    Metadata
    Show full item record
    Access Restrictions
    At the time of archiving, the student author of this thesis opted to temporarily restrict access to it. The full text of this thesis became available to the public after the expiration of the embargo on 2017-05-12.
    Abstract
    Linear estimation is a fundamental branch of signal processing that deals with estimating the values of parameters from a corrupted measured data. Throughout the years, several optimization criteria have been used to achieve this task. The most astonishing attempt among theses is the linear least-squares. Although this criterion enjoyed a wide popularity in many areas due to its attractive properties, it appeared to suffer from some shortcomings. Alternative optimization criteria, as a result, have been proposed. These new criteria allowed, in one way or another, the incorporation of further prior information to the desired problem. Among theses alternative criteria is the regularized least-squares (RLS). In this thesis, we propose two new algorithms to find the regularization parameter for linear least-squares problems. In the constrained perturbation regularization algorithm (COPRA) for random matrices and COPRA for linear discrete ill-posed problems, an artificial perturbation matrix with a bounded norm is forced into the model matrix. This perturbation is introduced to enhance the singular value structure of the matrix. As a result, the new modified model is expected to provide a better stabilize substantial solution when used to estimate the original signal through minimizing the worst-case residual error function. Unlike many other regularization algorithms that go in search of minimizing the estimated data error, the two new proposed algorithms are developed mainly to select the artifcial perturbation bound and the regularization parameter in a way that approximately minimizes the mean-squared error (MSE) between the original signal and its estimate under various conditions. The first proposed COPRA method is developed mainly to estimate the regularization parameter when the measurement matrix is complex Gaussian, with centered unit variance (standard), and independent and identically distributed (i.i.d.) entries. Furthermore, the second proposed COPRA method deals with discrete ill-posed problems when the singular values of the linear transformation matrix are decaying very fast to a significantly small value. For the both proposed algorithms, the regularization parameter is obtained as a solution of a non-linear characteristic equation. We provide a details study for the general properties of these functions and address the existence and uniqueness of the root. To demonstrate the performance of the derivations, the first proposed COPRA method is applied to estimate different signals with various characteristics, while the second proposed COPRA method is applied to a large set of different real-world discrete ill-posed problems. Simulation results demonstrate that the two proposed methods outperform a set of benchmark regularization algorithms in most cases. In addition, the algorithms are also shown to have the lowest run time.
    Citation
    Suliman, M. A. E. (2016). Regularization Techniques for Linear Least-Squares Problems. KAUST Research Repository. https://doi.org/10.25781/KAUST-KCL5T
    DOI
    10.25781/KAUST-KCL5T
    ae974a485f413a2113503eed53cd6c53
    10.25781/KAUST-KCL5T
    Scopus Count
    Collections
    MS Theses; Electrical and Computer Engineering Program; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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