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On random matrices with large corank

Zach Hunter, Matthew Kwan, Lisa Sauermann, Mehtaab Sawhney

TL;DR

The paper proves a subgaussian bound for the corank of a random $\{ abla\pm1\}$ matrix by developing a high-dimensional relative anticoncentration inequality that compares hitting a fixed codimension-$k$ subspace under a standard Rademacher vector to hitting it under a sparse, lazy vector. Building on a Kahn–Komlós–Szemerédi framework, the authors partition subspaces into thin and thick classes and control each case via combinatorial bounds and the new anticoncentration lemma, achieving an exponential decay in $nk$ for all $k$ with $1\le k\le n$. The key contributions are the high-dimensional generalization of a Raikov-type doubling inequality and a streamlined, subspace-agnostic proof strategy that avoids unstructuredness considerations. This result extends prior work that covered only small or square-root regimes of $k$ and advances the understanding of randomness-induced linear independence in large-dimension matrices, with potential applications to related distributional models beyond $\pm1$ entries.

Abstract

Let $1\le k\le n$ and $M$ be a random $n\times n$ matrix with independent uniformly random $\{\pm 1\}$-entries. We show that there exists an absolute constant $c > 0$ such that \[\mathbf{P}[\operatorname{rank}(M)\le n-k]\le \exp(-c nk).\]

On random matrices with large corank

TL;DR

The paper proves a subgaussian bound for the corank of a random matrix by developing a high-dimensional relative anticoncentration inequality that compares hitting a fixed codimension- subspace under a standard Rademacher vector to hitting it under a sparse, lazy vector. Building on a Kahn–Komlós–Szemerédi framework, the authors partition subspaces into thin and thick classes and control each case via combinatorial bounds and the new anticoncentration lemma, achieving an exponential decay in for all with . The key contributions are the high-dimensional generalization of a Raikov-type doubling inequality and a streamlined, subspace-agnostic proof strategy that avoids unstructuredness considerations. This result extends prior work that covered only small or square-root regimes of and advances the understanding of randomness-induced linear independence in large-dimension matrices, with potential applications to related distributional models beyond entries.

Abstract

Let and be a random matrix with independent uniformly random -entries. We show that there exists an absolute constant such that \[\mathbf{P}[\operatorname{rank}(M)\le n-k]\le \exp(-c nk).\]
Paper Structure (7 sections, 7 theorems, 44 equations)

This paper contains 7 sections, 7 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. History
  3. Proof ideas
  4. Acknowledgements
  5. Reduction to high-dimensional relative concentration result
  6. Proof of
  7. Proof of \ref{['lem:frieman-check']}

Key Result

Theorem 1.1

There exists an absolute constant $c > 0$ such that the following holds. Take $1\le k\le n$, and let $M$ be a random $n\times n$ matrix with independent entries uniformly random in $\{\pm 1\}$. We have

Theorems & Definitions (18)

  • Theorem 1.1
  • Proposition 1
  • Remark
  • proof : Proof of \ref{['thm:main']}
  • Claim 1
  • Claim 2
  • proof : Proof of \ref{['clm:thin']}
  • proof : Proof of \ref{['clm:thick']}
  • Lemma 1
  • proof : Proof of \ref{['prop:relative']}
  • ...and 8 more