On the Structure of Bad Science Matrices

TL;DR

This paper investigates the problem of maximizing a specific matrix norm over all matrices with unit norm rows, providing explicit constructions and properties of near-optimal matrices, advancing understanding of the bad science matrix problem.

Contribution

The paper offers an explicit construction of matrices nearly maximizing the bad science matrix functional and characterizes entries of optimal matrices, including exact solutions for small sizes.

Findings

Constructed matrices achieve at least rac{}( +1), close to the asymptotic rate.

Every entry of an optimal matrix is a square root of a rational number.

Found provably optimal matrices for n  4.

Abstract

The bad science matrix problem consists in finding, among all matrices $A \in \mathbb{R}^{n \times n}$ with rows having unit $\ell^2$ norm, one that maximizes $\beta(A) = \frac{1}{2^n} \sum_{x \in \{-1, 1\}^n} \|Ax\|_\infty$. Our main contribution is an explicit construction of an $n \times n$ matrix $A$ showing that $\beta(A) \geq \sqrt{\log_2(n+1)}$, which is only 18% smaller than the asymptotic rate. We prove that every entry of any optimal matrix is a square root of a rational number, and we find provably optimal matrices for $n \leq 4$.