Non-asymptotic theory of random matrices: extreme singular values

TL;DR

This paper surveys recent non-asymptotic geometric methods for analyzing the extreme singular values, especially the smallest singular value, of fixed-dimension random matrices with independent entries.

Contribution

It introduces new geometric techniques for estimating the smallest singular value in the non-asymptotic regime, filling a gap in classical asymptotic random matrix theory.

Findings

Development of geometric bounds for the smallest singular value

Application of methods to fixed-dimension matrices

Enhanced understanding of spectral properties in non-asymptotic settings

Abstract

The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to information theory operate with random matrices in fixed dimensions. This survey addresses the non-asymptotic theory of extreme singular values of random matrices with independent entries. We focus on recently developed geometric methods for estimating the hard edge of random matrices (the smallest singular value).