Master 2019/2020

Random Matrix Theory

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

Category 'Best Course for New Knowledge and Skills'

Type: Elective course (Statistical Learning Theory)

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

Instructors: Leonid Iosipoi, Alexey Naumov

Master’s programme: Statistical Learning Theory

Language: English

ECTS credits: 6

Contact hours: 64

Course 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

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

Course Contents

• Concentration of measure phenomenon
  Tensorization of variance, Sub-Gaussian and subexponential distributions, concentration inequalities for sums of random variables, Bernstein's inequality, Azuma-Hoeffiding inequality
• Random matrices in science and applications
  Random matrices in statistics, physics, telecommunications, numerical analysis, community detection in networks
• Norms of random matrices
  Norm of a random symmetric matrix, norms of rectangular matrices, the moment method, Gaussian processes, Sudakov-Fernique inequality
• Limit theorems for spectra of random matrices
  Stieltjes transform, Wigner’s semicircle law, Marchenko-Pastur law, local limit theorems, Stein’s method
• Sums of random matrices
  Concentration inequalities and moment inequalities for the sample covariance matrices, spectral projectors, principal component analysis
• Sample covariance matrices
  Concentration inequalities and moment inequalities for the sample covariance matrices, spectral projectors, principal component analysis
• Gaussian ensembles of random matrices
  Gaussian Unitary Ensemble (GUE), Gaussian Orthogonal ensemble (GOE), Wishart ensemble, eigenvalues density, eigenvectors, Determinantal structure, Spectral statistics, Wigner-Dyson-Gaudin-Mehta conjecture
• Random vectors in high dimension
  Multivariate Gaussian distribution, distribution of norm of random vector, dimensionality reduction, Johnson-Lindenstrauss lemma
• 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

Interim Assessment

• Interim assessment (1 module)
  0.4 * Fnal exam + 0.4 * Home assignments + 0.2 * Individual project

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