Asymptotically optimal approximate Hadamard matrices

TL;DR

The paper proves that the smallest possible condition number of an $n\times n$ matrix with entries $\pm1$ satisfies $\\kappa(n)\to 1$ as $n\to\infty$, establishing an explicit upper bound $\\kappa(n)\le 1+1/n^{\alpha}$ for large $n$ with $\alpha>0$. It derives this by amplifying flat orthogonal matrices through a submatrix-orthogonalization technique and randomized rounding, supported by matrix Bernstein concentration; it provides concrete $\alpha$ values (e.g., $17/92-\delta$ unconditionally, $1/4-\delta$ conditionally on the Hadamard conjecture) and constructs several explicit infinite families, including Barba and SDS-based matrices, achieving $\\kappa(A)=\sqrt{2}+o(1)$. A Ramsey-theoretic lower bound shows $\\kappa(n)\ge 1+c\log n/n$ for large $n$ not divisible by 4, establishing a gap between upper and lower bounds that narrows with better Hadamard-gap results. The work also documents AI-assisted exploration, with a formally verified submatrix-orthogonalization lemma in Lean and discussion of explosive potential for AI-assisted discovery in mathematical problem-solving. Overall, the results advance the understanding of asymptotically optimal approximate Hadamard matrices and provide explicit constructions approaching Hadamard-ideal conditioning.

Abstract

In this paper, we study approximate Hadamard matrices, that is, well-conditioned $n\times n$ matrices with all entries in $\{\pm1\}$. We show that the smallest-possible condition number goes to $1$ as $n\to\infty$, and we identify some explicit infinite families of approximate Hadamard matrices.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• proof : Proof of Theorem \ref{['thm.main result']}
• Lemma 3: submatrix orthogonalization
• proof
• Lemma 4
• proof
• Lemma 5
• proof : Proof of Lemma \ref{['lem.hadamard gaps']}
• proof : Proof of Theorem \ref{['thm.flat orthogonal matrices']}