On the Spielman-Teng Conjecture

Abstract

Let M be an n×n matrix with iid subgaussian entries with mean 0 and variance 1 and let σ_n(M) denote the least singular value of M. We prove that

$$ \mathbb{P}\big( \sigma _{n}(M) \leqslant \varepsilon n^{-1/2} \big) = (1+o(1)) \varepsilon + e^{- \Omega (n)} $$

for all 0⩽ε≪1. This resolves, up to a 1+o(1) factor, a seminal conjecture of Spielman and Teng.