Магистратура
2021/2022
Теория случайных матриц
Лучший по критерию «Полезность курса для расширения кругозора и разностороннего развития»
Статус:
Курс по выбору
Направление:
01.04.02. Прикладная математика и информатика
Кто читает:
Кафедра технологий моделирования сложных систем
Где читается:
Факультет компьютерных наук
Когда читается:
2-й курс, 1 модуль
Формат изучения:
без онлайн-курса
Охват аудитории:
для своего кампуса
Прогр. обучения:
Статистическая теория обучения
Язык:
английский
Кредиты:
6
Контактные часы:
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
- 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 assignmentsHome 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 assignmentsHome 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 module0.2 * 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