Statistics and Data Science 615: Introduction to Random Matrix Theory and Applications

Zhou Fan, Yale University, Spring 2019


***NOTE***: There is no class on Wednesday 1/16. The first class will be held in the second week, on Wednesday 1/23. A make-up lecture will be held on Friday, 1/25.

Description

A seminar-style introduction to random matrix theory. Wigner matrices, sample covariance matrices, spiked models. Applications to principal components analysis, spectral algorithms on graphs and networks, and landscape analysis of non-convex optimization problems. Methods for non-invariant models that commonly arise in applications: moment method, concentration of measure, resolvents and Stieltjes transforms, free probability, Lindeberg exchange.

Prerequisites: Graduate-level probability theory

Lectures

Wednesdays 4:00PM - 6:30PM
24 Hillhouse, Room 107

Office hours: Wednesdays 1 - 2PM
Dunham Laboratory 231

Requirements

A small number of homework problems will be assigned during lectures. You may select any three problems to solve. Solutions must be type-written and are due on the last day of reading period, Wednesday May 1.

A typical homework problem may require you to prove an extension of a result from class or to implement an algorithm discussed in class and to explore its performance in simulation. You are encouraged to consult textbooks and other literature. Any such literature must be properly cited in a References section at the end of your write-up.

Course material

Written lecture notes will be posted on Canvas. Material in the first half of the course will be drawn from:

An Introduction to Random Matrices, Greg W. Anderson, Alice Guionnet, Ofer Zeitouni

Topics in Random Matrix Theory, Terence Tao

Material in the second half of the course will be based on research papers, which will also be posted on Canvas.

Course schedule (tentative)

W 1/16 No class - Make-up lecture on Friday, 1/25
W 1/23 Course introduction, moment method, Wigner semicircle law
F 1/25 Log-Sobolev inequality, Herbst argument, concentration of spectral measure
W 1/30 Stieltjes transforms, resolvents, Schur-complement identities
W 2/6 Gaussian integration by parts, universality, the Lindeberg method
W 2/13 Extremal eigenvalues, local semicircle law
W 2/20 Low-rank deformations, outlier eigenvalues and eigenvectors
W 2/27 Free probability, free convolution, cumulants and R-transforms
W 3/6 Asymptotic freeness, sample covariances, Marcenko-Pastur law
W 3/13 No class - Spring recess
W 3/20 No class - Spring recess
W 3/27 Spiked covariance model, principal components analysis, BBP phase transition
W 4/3 General covariances, spectrum estimation, eigenvalue shrinkage estimates
W 4/10 Dilute random graphs, local weak convergence, adjacency spectrum
W 4/17 Dilute random graphs, non-backtracking spectrum
W 4/24 Kac-Rice formula, complexity of random non-convex functions