On the Spielman-Teng Conjecture

TL;DR

This paper proves a near-complete resolution of the Spielman-Teng conjecture by establishing precise probabilistic bounds on the smallest singular value of random matrices with iid subgaussian entries.

Contribution

It provides a sharp probabilistic bound on the least singular value of iid subgaussian matrices, confirming the Spielman-Teng conjecture up to a small error factor.

Findings

Probability that the least singular value is very small is approximately proportional to epsilon.

The bound holds uniformly for all small epsilon.

The result confirms the conjecture up to a 1+o(1) factor.

Abstract

Let $M$ be an $n\times n$ matrix with iid subgaussian entries with mean $0$ and variance $1$ and let $\sigma_n(M)$ denote the least singular value of $M$. We prove that \[\mathbb{P}\big( \sigma_{n}(M) \leq \varepsilon n^{-1/2} \big) = (1+o(1)) \varepsilon + e^{-\Omega(n)}\] for all $0 \leq \varepsilon \ll 1$. This resolves, up to a $1+o(1)$ factor, a seminal conjecture of Spielman and Teng.