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

الموضوعات

علوم تطبيقية وتكنولوجية

الملخص

Random matrix theory is an outstanding mathematical tool that has demonstratedits usefulness in many areas ranging from wireless communication to nance andeconomics. The main motivation behind its use comes from the fundamental role thatrandom matrices play in modeling unknown and unpredictable physical quantities. Inmany situations, meaningful metrics expressed as scalar functionals of these randommatrices arise naturally. Along this line, the present work consists in leveraging toolsfrom random matrix theory in an attempt to answer fundamental questions relatedto applications from statistical signal processing and machine learning.In a rst part, this thesis addresses the development of analytical tools for thecomputation of the inverse moments of random Gram matrices with one side correlation.Such a question is mainly driven by applications in signal processing and wirelesscommunications wherein such matrices naturally arise. In particular, we deriveclosed-form expressions for the inverse moments and show that the obtained resultscan help approximate several performance metrics of common estimation techniques.Then, we carry out a large dimensional study of discriminant analysis classi ers.Under mild assumptions, we show that the asymptotic classi cation error approachesa deterministic quantity that depends only on the means and covariances associatedwith each class as well as the problem dimensions. Such result permits a betterunderstanding of the underlying classi ers, in practical large but nite dimensions,and can be used to optimize the performance.Finally, we revisit kernel ridge regression and study a centered version of it that wecall centered kernel ridge regression or CKRR in short. Relying on recent advances onthe asymptotic properties of random kernel matrices, we carry out a large dimensionalanalysis of CKRR under the assumption that both the data dimesion and the trainingsize grow simultaneiusly large at the same rate. We particularly show that both theempirical and prediction risks converge to a limiting risk that relates the performanceto the data statistics and the parameters involved. Such a result is important asit permits a better undertanding of kernel ridge regression and allows to e cientlyoptimize the performance.

روابط وملفات

التعريف والنوع

رقم الوثيقة

0a1be28c-e531-4c64-aa83-c9df3018bf38

رقم العقد

0

نوع الوسائط

Crawler

نوع المحتوى

الرسائل العلمية

صيغة المصدر

رسائل دكتوراة

نوع الملف

pdf text

أسماء الملفات

1698593_1.pdf

بيانات النشر

ألقاب المؤلفين

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اللغة

English

المصدر والدورية

اسم المصدر

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

المحتوى والصفحات

عدد الصفحات

0

كلمات الباحثين

discriminant analysis , High dimensional statistics , kernel regression , Random Matrix Theory

إشراف وإعداد

الإشراف

طارق يوسف النفوري

الإعداد

خليل الخليل

الاقتباسات الببليوغرافية