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    Random Matrix Theory: Selected Applications from Statistical Signal Processing and Machine Learning

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    Final Thesis.pdf
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    Final thesis
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    Type
    Dissertation
    Authors
    Elkhalil, Khalil cc
    Advisors
    Al-Naffouri, Tareq Y. cc
    Committee members
    Kammoun,Alba
    Hero, Alfred
    Zhang, Xiangliang cc
    Alouini, Mohamed-Slim cc
    Program
    Electrical Engineering
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2019-06
    Permanent link to this record
    http://hdl.handle.net/10754/655682
    
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    Abstract
    Random matrix theory is an outstanding mathematical tool that has demonstrated its usefulness in many areas ranging from wireless communication to finance and economics. The main motivation behind its use comes from the fundamental role that random matrices play in modeling unknown and unpredictable physical quantities. In many situations, meaningful metrics expressed as scalar functionals of these random matrices arise naturally. Along this line, the present work consists in leveraging tools from random matrix theory in an attempt to answer fundamental questions related to applications from statistical signal processing and machine learning. In a first part, this thesis addresses the development of analytical tools for the computation of the inverse moments of random Gram matrices with one side correlation. Such a question is mainly driven by applications in signal processing and wireless communications wherein such matrices naturally arise. In particular, we derive closed-form expressions for the inverse moments and show that the obtained results can help approximate several performance metrics of common estimation techniques. Then, we carry out a large dimensional study of discriminant analysis classifiers. Under mild assumptions, we show that the asymptotic classification error approaches a deterministic quantity that depends only on the means and covariances associated with each class as well as the problem dimensions. Such result permits a better understanding of the underlying classifiers, in practical large but finite dimensions, and can be used to optimize the performance. Finally, we revisit kernel ridge regression and study a centered version of it that we call centered kernel ridge regression or CKRR in short. Relying on recent advances on the asymptotic properties of random kernel matrices, we carry out a large dimensional analysis of CKRR under the assumption that both the data dimesion and the training size grow simultaneiusly large at the same rate. We particularly show that both the empirical and prediction risks converge to a limiting risk that relates the performance to the data statistics and the parameters involved. Such a result is important as it permits a better undertanding of kernel ridge regression and allows to efficiently optimize the performance.
    Citation
    Elkhalil, K. (2019). Random Matrix Theory: Selected Applications from Statistical Signal Processing and Machine Learning. KAUST Research Repository. https://doi.org/10.25781/KAUST-Q7W78
    DOI
    10.25781/KAUST-Q7W78
    ae974a485f413a2113503eed53cd6c53
    10.25781/KAUST-Q7W78
    Scopus Count
    Collections
    Dissertations; Electrical Engineering Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

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