I think that embracing uncertainty is the key to understand the meaning of our existence and that truth in its absolute sense is nothing but an illusion. With a great passion for statistics, I try to make tiny improvements on the way we quantify uncertainty.

Khalil Elkhalil received the B.Eng. degree (with Honours) in telecommunication engineering from the Higher School of Communications of Tunis (Sup’Com), Ariana, Tunisia, in 2012. He is currently a PhD degree candidate of the Electrical Engineering, King Abdullah University of Science and Technology. His research interests include random matrix theory, statistical learning and high dimensional statistics.

PhD Dissertation Title

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

PhD Dissertation 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.

Education Profile

  • M.Sc. in EE, KAUST, Aug 2013 – Jun 2015, Thuwal, Saudi Arabia. 
  • B.Sc. in Telecommunication, SUP’COM Tunisia, Sep 2009 – Jun 2012, Tunis, Tunisia. 

Areas of Expertise and Research Interests

  • RMT-based Analysis of Supervised Learning Algorithms.
  • Feedback Reduction in Multiuser and Relay Networks.

Awards and Distinctions

  • 1st Place - National entrance examination to the Ecole Normale Superieure de Tunis, 2009
  • KAUST Graduate Fellowship
  • Finalist for the best student paper award in IEEE MLSP conference 2017

Selected Publications

Elkhalil, K., Kammoun, A., Couillet, R., Al-Naffouri, T. Y., Alouini, M.-S. (2017). A Large Dimensional Analysis of Regularized Discriminant Analysis Classifiers, (Preprint). https://hdl.handle.net/10754/626453
Suliman, M., Ballal, T., Kammoun, A., & Al-Naffouri, T. Y. (2016). Constrained Perturbation Regularization Approach for Signal Estimation Using Random Matrix Theory. IEEE Signal Processing Letters, 23(12), 1727–1731. doi:10.1109/lsp.2016.2615683
K. Elkhalil, A. Kammoun, T. Y. Al-Naffouri, and M.-S. Alouini, “Numerically Stable Evaluation of Moments of Random Gram Matrices With Applications,” IEEE Signal Processing Letters, vol. 24, no. 9, pp. 1353–1357, Sep. 2017.
K. Elkhalil, A. Kammoun, T. Y. Al-Naffouri, and M.-S. Alouini, “Fluctuations of the SNR at the output of the MVDR with regularized Tyler estimators,” Signal Processing, vol. 135, pp. 1–8, Jun. 2017.