Resilience of the Rank of Random Matrices

TL;DR

This paper demonstrates that the full rank property of random Rademacher matrices is highly resilient to adversarial sign changes, nearly matching the theoretical limit, and addresses a conjecture by Van Vu.

Contribution

It proves that random Rademacher matrices retain full rank despite significant adversarial sign modifications, advancing understanding of their structural robustness.

Findings

Full rank persists after changing up to $(1-rac{ ext{}\varepsilon}{6})m/2$ entries.

Resilience is asymptotically optimal, matching the minimal changes needed to create proportional rows.

Provides an asymptotic solution to a weakened version of Van Vu's conjecture.

Abstract

Let $M$ be an $n \times m$ matrix of independent Rademacher ($\pm 1$) random variables. It is well known that if $n \leq m$, then $M$ is of full rank with high probability. We show that this property is resilient to adversarial changes to $M$. More precisely, if $m \geq n + n^{1-\varepsilon/6}$, then even after changing the sign of $(1-\varepsilon)m/2$ entries, $M$ is still of full rank with high probability. Note that this is asymptotically best possible as one can easily make any two rows proportional with at most $m/2$ changes. Moreover, this theorem gives an asymptotic solution to a slightly weakened version of a conjecture made by Van Vu.