Random Matrix Theory

Master 2021/2022

Category 'Best Course for Broadening Horizons and Diversity of Knowledge and Skills'

Type: Elective course

Area of studies: Applied Mathematics and Informatics

Delivered by: Department of Complex System Modelling Technologies

Where: Faculty of Computer Science

When: 2 year, 1 module

Mode of studies: offline

Open to: students of one campus

Instructors: Alexey Naumov, Nikita Puchkin

Master’s programme: Statistical Learning Theory

Language: English

ECTS credits: 6

Contact hours: 64

Full Syllabus

Abstract

The aim of this course is to provide an introduction to asymptotic and non-asymptotic methods for the study of random structures in high dimension that arise in probability, statistics, computer science, and mathematics. One of the emphases is on the development of a common set of tools that has proved to be useful in a wide range of applications in different areas. Topics will include concentration of measure, Stein’s methods, suprema of random processes and etc. Another main emphasis is on the application of these tools for the study of spectral statistics of random matrices, which are remarkable examples of random structures in high dimension and may be used as models for data, physical phenomena or within randomised computer algorithms. The topics of this course form an essential basis for work in the area of high dimensional data.

Learning Objectives

Students will study how to apply the main modern probabilistic methods in practice and learn important topics from the random matrix theory

Expected Learning Outcomes

Be able ability to make an oral and written presentation
Be able ability to solve practical problems with methods from modern probability and random matrix theory
Be able ability to work with research literature on the modern probability theory
Be able compute and estimate spectral statistics of random matrices from different random matrix ensembles
Be able select the most efficient probability methods to solve problems in science and practice
Know how to apply the main measure concentration inequalities in science and practice
Know interrelation between different directions of modern high-dimensional probability theory
Know understand random matrix theory and its applications in science and practice
Know аcquaintance with the main aspects of the measure concentration phenomenon

Course Contents

Concentration of measure phenomenon
Random matrices in science and applications
Norms of random matrices
Limit theorems for spectra of random matrices
Sums of random matrices
Sample covariance matrices
Gaussian ensembles of random matrices
Random vectors in high dimension
Individual projects

Assessment Elements

Home assignments
Home assignment: should be done in the form of a written report. The sample of the task structure: • title page • A4 format • Task solution
Individual project
Fnal exam
Home assignments
Home assignment: should be done in the form of a written report. The sample of the task structure: • title page • A4 format • Task solution
Individual project
Fnal exam

Interim Assessment

2021/2022 1st module
0.2 * Individual project + 0.4 * Fnal exam + 0.4 * Home assignments

Bibliography

Recommended Core Bibliography

Bai, Z., & Silverstein, J. W. (2010). Spectral Analysis of Large Dimensional Random Matrices (Vol. 2nd ed). New York: Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=341481
van Handel, R. (2016). Structured Random Matrices. https://doi.org/10.1007/978-1-4939-7005-6_4

Recommended Additional Bibliography

Götze, F., Naumov, A., Tikhomirov, A., & Timushev, D. (2016). On the Local Semicircular Law for Wigner Ensembles. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsbas&AN=edsbas.1C0BB6C9

Course Syllabus