Introduction to the non-asymptotic analysis of random matrices

TL;DR

This tutorial introduces non-asymptotic methods in random matrix theory, focusing on analyzing extreme singular values of matrices with independent rows or columns, with applications in statistics, computer science, and signal processing.

Contribution

It provides a comprehensive introduction to non-asymptotic techniques in random matrix analysis, emphasizing tools from geometric functional analysis and their applications.

Findings

Methods for estimating covariance matrices in statistics

Validation of measurement matrices in compressed sensing

Analysis of extreme singular values of random matrices

Abstract

This is a tutorial on some basic non-asymptotic methods and concepts in random matrix theory. The reader will learn several tools for the analysis of the extreme singular values of random matrices with independent rows or columns. Many of these methods sprung off from the development of geometric functional analysis since the 1970's. They have applications in several fields, most notably in theoretical computer science, statistics and signal processing. A few basic applications are covered in this text, particularly for the problem of estimating covariance matrices in statistics and for validating probabilistic constructions of measurement matrices in compressed sensing. These notes are written particularly for graduate students and beginning researchers in different areas, including functional analysts, probabilists, theoretical statisticians, electrical engineers, and theoretical computer scientists.