An upper bound on the smallest singular value of dense random combinatorial matrices

TL;DR

This paper establishes an upper bound on the smallest singular value of dense random combinatorial matrices, confirming it is typically of order $n^{-1/2}$, complementing existing lower bounds.

Contribution

It provides a complementary upper bound for the smallest singular value of such matrices, showing it is usually of order $n^{-1/2}$, which was previously only bounded below.

Findings

The smallest singular value $s_n(M)$ is typically of order $n^{-1/2}$.

The probability that $s_n(M)$ is below a certain threshold is bounded below by a positive constant.

The result confirms the typical scale of the least singular value for dense random combinatorial matrices.

Abstract

Let $M$ be an $n\times n$ random matrix with entries in $\{0, 1\}$, where each row is independently and uniformly sampled from the set of all vectors in $\{0, 1\}^n$ containing exactly $d$ ones, with $d=pn$ for some fixed constant $p\in (0,1/2]$. A recent result of Tran states that the smallest singular value $s_n(M)$ is bounded below by $c_p n^{-1/2}$ with high probability. In this note, we establish a complementary upper bound for $s_n(M)$, proving that \[ \forall \varepsilon >0 \qquad \mathbb{P}\left(s_n(M)\le \frac{\sqrt{d}}{\varepsilon^2 n}\right)\ge 1-C_p\left(\varepsilon+\frac{1}{\sqrt{d}}\right), \]where $C_p$ is a positive constant depending only on $p$. This result confirms that the least singular value $s_n(M)$ of dense random combinatorial matrices is typically of the order $n^{-1/2}$.