# Approximate Spielman-Teng theorems for the least singular value of random combinatorial matrices

## TL;DR

This paper introduces a new framework for establishing approximate bounds on the smallest singular value of discrete random matrices, with applications to combinatorial matrices with fixed row sums, advancing prior results.

## Contribution

It develops a simple, novel approach for proving approximate Spielman-Teng theorems for discrete matrices, including the first such result for certain combinatorial matrices.

## Key findings

- Established an approximate Spielman-Teng theorem for 0-1 matrices with fixed row sums.
- Improved upon previous bounds for the least singular value in combinatorial matrix settings.
- Provided the first such result in a truly combinatorial context.

## Abstract

An approximate Spielman-Teng theorem for the least singular value $s_n(M_n)$ of a random $n\times n$ square matrix $M_n$ is a statement of the following form: there exist constants $C,c >0$ such that for all $\eta \geq 0$, $\Pr(s_n(M_n) \leq \eta) \lesssim n^{C}\eta + \exp(-n^{c})$. The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for $\{0,1\}$-valued matrices, each of whose rows is an independent vector with exactly $n/2$ zero components. This improves on previous work of Nguyen and Vu, and is the first such result in a `truly combinatorial' setting.