On the structure of bad science matrices
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 $β(A) = \frac{1}{2^n} \sum_{x \in \{-1, 1\}^n} \|Ax\|_\infty$. Our main contribution is an explicit construction of an $n \times n$ matrix $A$ showing that $β(A) \geq \sqrt{\log_2(n+1)}$, 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 $n \leq 4$.
Fields of Study & Subjects
Computability
Logic
AI Algorithms
Publication details
- Journal / source
- Involve a Journal of Mathematics
- Volume
- 19
- Issue
- 3
- Pages
- 443–460
- Publication date
- 12 June 2026
- Language
- EN