The smallest singular value of inhomogeneous square random matrices

Abstract

We show that, for an $\mathit{n}\times \mathit{n}$ random matrix A with independent uniformly anticoncentrated entries such that $\mathbb{E}\Vert \mathit{A}{\Vert }_{\mathrm{HS}}^2\le \mathit{K}{\mathit{n}}^2$, the smallest singular value ${\mathit{\sigma }}_{\mathit{n}}(\mathit{A})$ of A satisfies

$${\mathbb{P}}_{}\{{\mathit{\sigma }}_{\mathit{n}}(\mathit{A})\le \frac{\mathit{\epsilon }}{\sqrt{\mathit{n}}}\}\le \mathit{C}\mathit{\epsilon }+2{\mathit{e}}^{-\mathit{c}\mathit{n}},\mathit{\epsilon }\ge 0.$$

This extends earlier results (Adv. Math. 218 (2008) 600–633; Israel J. Math. 227 (2018) 507–544) by removing the assumption of mean zero and identical distribution of the entries across the matrix as well as the recent result (Livshyts (2018)) where the matrix was required to have i.i.d. rows. Our model covers inhomogeneous matrices allowing different variances of the entries as long as the sum of the second moments is of order $\mathit{O}({\mathit{n}}^2)$.

In the past advances, the assumption of i.i.d. rows was required due to lack of Littlewood–Offord-type inequalities for weighted sums of non-i.i.d. random variables. Here, we overcome this problem by introducing the Randomized Least Common Denominator (RLCD) which allows to study anti-concentration properties of weighted sums of independent but not identically distributed variables. We construct efficient nets on the sphere with lattice structure and show that the lattice points typically have large RLCD. This allows us to derive strong anticoncentration properties for the distance between a fixed column of A and the linear span of the remaining columns and prove the main result.