This thesis is concerned with the analysis of the statistical behavior of Gram matrices and with the design and analysis of high-dimensional robust covariance matrix estimators using random matrix theory.

The first and main part of this thesis, divided into two subparts, is devoted to the design and analysis of a large class of robust estimators of covariance matrices known as Maronna’s M-estimators. Under the Big Data paradigm, classical estimators such as the sample covariance matrix typically perform poorly in the recurring scenario where the number of samples is of the same order as the number of variables. Traditional estimators also suffer from possible contamination of the sample by outliers. This motivates the use of more sophisticated, robust covariance matrix estimato...[ Read more ]

This thesis is concerned with the analysis of the statistical behavior of Gram matrices and with the design and analysis of high-dimensional robust covariance matrix estimators using random matrix theory.

The first and main part of this thesis, divided into two subparts, is devoted to the design and analysis of a large class of robust estimators of covariance matrices known as Maronna’s M-estimators. Under the Big Data paradigm, classical estimators such as the sample covariance matrix typically perform poorly in the recurring scenario where the number of samples is of the same order as the number of variables. Traditional estimators also suffer from possible contamination of the sample by outliers. This motivates the use of more sophisticated, robust covariance matrix estimators, the statistical behavior of which shall be investigated.

First, we characterize the asymptotic behavior of regularized robust Maronna’s M-estimators by utilizing random matrix theory tools. We show that, in the large-dimensional regime and in the absence of outliers, these estimators all exhibit the same general behavior, after consistent estimation of their optimal regularization parameters. We then compare the theoretical performance of these estimators in the presence of outliers, providing insight on which estimator is best under different outlier contamination models.

Second, we utilize Maronna’s M-estimators to improve a classical version of linear discriminant analysis. We show that, in a large-dimensional regime, the use of these estimators provides the same asymptotic classification error as non-robust methods when data is free from outliers. Simulations on both synthetic and real data sets attest to the higher performance of our proposed robust method relative to more conventional methods.

In the last part of this thesis, we study the statistical properties of a 2x2 Gram matrix with an arbitrary variance profile. The underpinning random matrix model fundamentally departs from classical Wishart models, and does not readily lend itself to analysis. By making use of finite random matrix theory tools, we derive exact expressions for the distribution of the entries and for the joint eigenvalue distribution of this Gram matrix. Our results are leveraged to study the capacity of a dual-antenna communication system.