The smallest singular value of sparse discrete random matrices

TL;DR

This paper establishes bounds on the smallest singular value of sparse discrete random matrices, extending previous results to lazy variables and providing a framework for the approximate Spielman-Teng theorem.

Contribution

Develops a simple framework to prove an approximate Spielman-Teng theorem for sparse discrete matrices, extending prior work to lazy random variables.

Findings

Bounds on the smallest singular value with high probability

Extension of results to lazy random variables

Framework applicable to sparse discrete matrices

Abstract

Let $M_n$ be an $n \times n$ random matrix with i.i.d. sparse discrete entries. In this paper, we develop a simple framework to solve the approximate Spielman-Teng theorem for $M_n$, which has the following form: There exist constants $C, c>0$ such that for all $\eta \geq 0$, $\textbf{P}\left(s_n\left(M_n\right) \leq \eta\right) \lesssim n^C \eta+\exp \left(-n^c\right)$. As an application, we give an approximate Spielman-Teng theorem for $M_n$ whose entries are $\mu-$lazy random variables, extending previous work by Tao and Vu.