On the Smallest Singular Value of Log-Concave Random Matrices

TL;DR

This paper establishes precise lower bounds for the smallest singular value of log-concave random matrices under various conditions, advancing understanding of their spectral properties in high-dimensional probability.

Contribution

It provides sharp tail estimates for the smallest singular value of log-concave matrices in multiple settings, including dependent entries and tall matrices.

Findings

Sharp lower tail bounds for N=n case with unconditional distribution

Lower bounds for matrices with independent columns when N≥n

Results for sufficiently tall matrices with N≥(1+λ)n

Abstract

Let $A$ be an $N\times n$ random matrix whose entries are coordinates of an isotropic log-concave random vector in $\mathbb{R}^{Nn}$. We prove sharp lower tail estimates for the smallest singular value of $A$ in the following cases: (1) when $N=n$ and $A$ is drawn from an unconditional distribution, with no independence assumption; (2) when the columns of $A$ are independent and $N\geq n$; (3) when $A$ is sufficiently tall, that is $N\geq (1+\lambda)n$ for any positive constant $\lambda$.