[Paper Review] Probabilistic lower bounds on maximal determinants of binary matrices

Richard P. Brent, Judy-anne H. Osborn+1 more|arXiv (Cornell University)|Jan 26, 2015

graph theory and CDMA systems12 references3 citations

TL;DR

This paper establishes new probabilistic lower bounds on the maximal determinant of $n \times n$ binary matrices with entries $\pm 1$, using a construction that extends a Hadamard matrix of order $h \leq n$ by $d = n - h$ rows and columns. It proves that the ratio $\mathcal{R}(n) = \mathcal{D}(n)/n^{n/2}$ is bounded below by a positive constant depending only on $d$, improving on prior bounds that decay to zero for $d \geq 2$ as $n \to \infty$. The key result is $\mathcal{R}(n) > (2/\pi e)^{d/2}$ for $d \leq 3$, implying a universal lower bound of approximately 0.1133 if the Hadamard conjecture holds.

ABSTRACT

Let ${\mathcal D}(n)$ be the maximal determinant for $n imes n$ $\{\pm 1\}$-matrices, and $\mathcal R(n) = {\mathcal D}(n)/n^{n/2}$ be the ratio of ${\mathcal D}(n)$ to the Hadamard upper bound. Using the probabilistic method, we prove new lower bounds on ${\mathcal D}(n)$ and $\mathcal R(n)$ in terms of $d = n-h$, where $h$ is the order of a Hadamard matrix and $h$ is maximal subject to $h \le n$. For example, $\mathcal R(n) > (πe/2)^{-d/2}$ if $1 \le d \le 3$, and $\mathcal R(n) > (πe/2)^{-d/2}(1 - d^2(π/(2h))^{1/2})$ if $d > 3$. By a recent result of Livinskyi, $d^2/h^{1/2} o 0$ as $n o \infty$, so the second bound is close to $(πe/2)^{-d/2}$ for large $n$. Previous lower bounds tended to zero as $n o \infty$ with $d$ fixed, except in the cases $d \in \{0,1\}$. For $d \ge 2$, our bounds are better for all sufficiently large $n$. If the Hadamard conjecture is true, then $d \le 3$, so the first bound above shows that $\mathcal R(n)$ is bounded below by a positive constant $(πe/2)^{-3/2} > 0.1133$.

Motivation & Objective

• To establish improved lower bounds on the maximal determinant $\mathcal{D}(n)$ of $n \times n$ $\{\pm 1\}$-matrices when $n$ is not a Hadamard order.
• To address the limitation of prior bounds on $\mathcal{R}(n) = \mathcal{D}(n)/n^{n/2}$, which tend to zero as $n \to \infty$ for $d \geq 2$, by introducing a probabilistic method.
• To show that $\mathcal{R}(n)$ is bounded below by a positive constant depending only on $d = n - h$, where $h$ is the largest Hadamard order $\leq n$, thus providing non-vanishing lower bounds for fixed $d \geq 2$.

Proposed method

• A probabilistic construction extends an $h \times h$ Hadamard matrix $A$ by adding $d = n - h$ random $\pm 1$ entries in $d$ new columns, then filling $d$ new rows deterministically to maximize the Schur complement determinant.
• The method relies on computing the expected value $\mu$ and variance $\sigma^2$ of the Schur complement of $A$ in the resulting $n \times n$ matrix $\widetilde{A}$, using tools from random matrix theory.
• Lemmas 2.6 and 2.9 derive the mean $\mu$ and variance $\sigma^2$ of the Schur complement, which are used to bound the determinant of $\widetilde{A}$ from below.
• The determinant of $\widetilde{A}$ is expressed in terms of the Schur complement, enabling the use of concentration inequalities and probabilistic estimates to bound $\mathcal{D}(n)$.
• Theoretical bounds are derived using inequalities involving $\mu$, $\eta$, and $h$, leading to explicit lower bounds on $\mathcal{R}(n)$ in terms of $d$ and $h$, with asymptotic analysis as $n \to \infty$.
• The analysis leverages results from Livinskyi showing $d^2/h^{1/2} \to 0$ as $n \to \infty$, ensuring the bounds remain effective for large $n$.

Experimental results

Research questions

• RQ1 Can probabilistic methods yield non-vanishing lower bounds on $\mathcal{R}(n)$ for $d = n - h \geq 2$ as $n \to \infty$, where $h$ is the largest Hadamard order $\leq n$?
• RQ2 What is the best possible lower bound on $\mathcal{R}(n)$ that can be derived using a probabilistic construction based on Schur complements of bordered Hadamard matrices?
• RQ3 How do the new bounds compare to existing deterministic and probabilistic bounds in the literature, particularly for small $d$ and large $n$?
• RQ4 Does the Hadamard conjecture imply a universal positive lower bound on $\mathcal{R}(n)$, and can this be proven via probabilistic methods?

Key findings

• For $d \leq 3$, the paper proves $\mathcal{R}(n) > (2/\pi e)^{d/2}$, which implies a universal lower bound of approximately 0.1133 if the Hadamard conjecture holds.
• For $d > 3$, the bound $\mathcal{R}(n) > (2/\pi e)^{d/2} \left(1 - d^2 (\pi/(2h))^{1/2}\right)$ is established, which approaches $(\pi e/2)^{-d/2}$ as $n \to \infty$ due to $d^2/h^{1/2} \to 0$.
• The bound $\mathcal{R}(n) > (2/\pi e)^{d/2}$ is sharper than previous bounds by a factor of order $n^{1/2}$ for $d \in \{2,3\}$, and holds for all sufficiently large $n$ when $d$ is fixed.
• Numerical comparisons show that the probabilistic bordering method outperforms both deterministic bordering and minors approaches, with $\mathcal{R}(n) \approx 0.0169$ for $n=668$, $d=4$, compared to $\approx 2.6 \times 10^{-4}$ for the deterministic minors method.
• The bound $\mathcal{R}(n) > (2/\pi e)^{d/2}$ is shown to be sharper than the bound $\mathcal{D}(n) \geq (n+1)^{(n-1)/2}$ from Koukouvinos et al. for $n \geq 135$ when $d=3$.
• The paper confirms that for $d \leq 3$, the lower bound on $\mathcal{R}(n)$ is independent of $n$ and depends only on $d$, providing a significant improvement over previous bounds that decay to zero.