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arXiv · 2408.00933

On the Structure of Bad Science Matrices

Alex Albors·Hisham Bhatti·Lukshya Ganjoo·Raymond Guo·Dmitriy Kunisky·Rohan Mukherjee·Alicia Stepin·Tony Zeng
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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 $n \leq 4$.

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Alex Albors, Hisham Bhatti, Lukshya Ganjoo, Raymond Guo, Dmitriy Kunisky, Rohan Mukherjee, Alicia Stepin, Tony Zeng. 2024-08-01. On the Structure of Bad Science Matrices. https://doi.org/10.2140/involve.2026.19.443

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