Website TUEindhoven **TU Eindhoven, Dept. of Mathematics & Computer Science**

Eindhoven University of Technology (TU/e) is a research university specializing in engineering science & technology.

## Description

Random Matrices [7] and their spectrum concentration phenomena are well known (e.g. Wigner’s Semicircle

Law, Marchenko-Pastur, etc) and have appeared in many areas: Applied Probability [6], Machine Learning

[4, 3], number theory [1], etc. In this project we want to understand and learn the methods used to prove

spectral concentration results for random matrices, specially for ensembles of random matrices where entries

are not necessarily independent (e.g., [2, 5]). In particular, depending on the interest, we will look at a case

example used in Unsupervised Learning dealing with Clustering algorithms[6] , where the distribution of the

entries of the random matrix depends on an underlying Markov Chain which possesses a Block structure,

the so called Block Markov Chain (BMC) model.

## Main Objectives

(a) Understand and describe the common methods for analyzing spectrum concentration for Random Ma-

trices (for example, in [7])

(b) Understand the spectral concentration analysis methods which deal with random matrices with depen-

dent entries (for example, in [2] and [5]).

(c) Numerically analyze the spectrum concentration in the Block Markov Chain matrix model (BMC) [6].

(d) Analyse theoretically, if possible, the spectrum of the Block Markov Chain matrix model in some cases.

Note: The reach and content dedicated to each objective can be discussed with the student depending on

her / his motivation and willingness. For example, a student more interested in a bibliographical part of the

topic could focus on objectives (a) and (b). Similarly, a student focused on a research oriented (and thus

riskier) path could focus more on objectives (b), (c) and maybe (d).

## Supervision

You will work with Albert Senen-Cerda, a.senen.cerda@tue.nl.

## References

[1] J. Keating. Random matrices and number theory. In Applications of random matrices in physics, pages

1-32. Springer, 2006.

[2] T. Kemp and D. Zimmermann. Random matrices with log-range correlations, and log-sobolev inequali-

ties. arXiv preprint, arXiv:1405.2581, 2014.

[3] C. Louart, Z. Liao, R. Couillet, et al. A random matrix approach to neural networks. The Annals of

Applied Probability, 28(2):1190-1248, 2018.

[4] C. H. Martin and M. W. Mahoney. Implicit self-regularization in deep neural networks: Evidence from

random matrix theory and implications for learning. arXiv preprint arXiv:1810.01075, 2018.

[5] B. Polaczyk et al. Concentration of the empirical spectral distribution of random matrices with dependent

entries. Electronic Communications in Probability, 24, 2019.

[6] J. Sanders, A. Proutiere, and S.-Y. Yun. Clustering in bloc Markov chains. The Annals of Statistics,

2019.

[7] T. Tao, Topics in random matrix theory, volume 132. American Mathematical Soc., 2012.

To apply for this job **email your details to** jaron.sanders@tue.nl