Spectral Concentration in Random Matrices with a Case Study in Clustering

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

Jaron
Jaron Sanders received in 2012 M.Sc. degrees in Mathematics and Physics from the Eindhoven University of Technology, The Netherlands, as well as a PhD degree in Mathematics in 2016. After he obtained his PhD degree, he worked as a post-doctoral researcher at the KTH Royal Institute of Technology in Stockholm, Sweden. Jaron Sanders then worked as an assistant professor at the Delft University of Technology, and now works as an assistant professor at the Eindhoven University of Technology. His research interests are applied probability, queueing theory, stochastic optimization, stochastic networks, wireless networks, and interacting (particle) systems.
https://www.jaronsanders.nl