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On the Structure of Bad Science Matrices

Alex Albors, Hisham Bhatti, Lukshya Ganjoo, Raymond Guo, Dmitriy Kunisky, Rohan Mukherjee, Alicia Stepin, Tony Zeng

TL;DR

We study the bad science matrix problem: for $A\in\mathbb{R}^{n\times n}$ with unit-$\ell^2$ rows, we maximize $\beta(A)=\mathbb{E}\|AX\|_{\infty}$ over a random Rademacher vector $X$. A central result is a Structure Theorem linking extremizers to normalized sums of cube vertices, which immediately implies that extremal entries are square roots of rationals and guides exact solutions for small $n$. We provide constructive families, notably a Lifting Construction yielding $\beta(A)=\sqrt{\log_2(n)+1}$ for $n=2^k$ and the Unsatisfiable Trees construction, plus a detailed analysis of wide matrices and $\ell^p$ variants with sharp growth rates. Together these results advance understanding of extremizers in Komlós-type problems and yield explicit, scalable constructions with near-optimal behavior across dimensions and norms.

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 $n \leq 4$.

On the Structure of Bad Science Matrices

TL;DR

We study the bad science matrix problem: for with unit- rows, we maximize over a random Rademacher vector . A central result is a Structure Theorem linking extremizers to normalized sums of cube vertices, which immediately implies that extremal entries are square roots of rationals and guides exact solutions for small . We provide constructive families, notably a Lifting Construction yielding for and the Unsatisfiable Trees construction, plus a detailed analysis of wide matrices and variants with sharp growth rates. Together these results advance understanding of extremizers in Komlós-type problems and yield explicit, scalable constructions with near-optimal behavior across dimensions and norms.

Abstract

The bad science matrix problem consists in finding, among all matrices with rows having unit norm, one that maximizes . Our main contribution is an explicit construction of an matrix showing that , 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 .
Paper Structure (22 sections, 13 theorems, 73 equations, 1 figure)

This paper contains 22 sections, 13 theorems, 73 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bad Science.
  3. Rotating the Cube.
  4. The Komlós Conjecture.
  5. Extremizers.
  6. Known Results.
  7. Main Results
  8. Structure Theorem and Consequences.
  9. Construction 1: Lifting Construction.
  10. Construction 2: Unsatisfiable Trees.
  11. Wide Matrices
  12. Lg Variants.
  13. Proofs
  14. Proof of the Structure Theorem
  15. Proof of Corollary 2
  16. ...and 7 more sections

Key Result

Theorem 1

Let $A = ^\top$ be an $m \times n$ matrix maximizing the value of $\beta(A)$ among all such matrices with rows normalized in $\ell^2$. Introducing the set $W_i$ of vertices of the hypercube that are being mapped to a vector whose largest entry is in the $i^{\text{th}}$ coordinate, then the $i^\text{th}$ row of $A$ is given by

Figures (1)

  • Figure 1: An unsatisfiable tree and the corresponding matrix.

Theorems & Definitions (27)

  • Theorem 1: \ref{['proof:structure']}
  • Corollary 1
  • Corollary 2: \ref{['proof:cor2']}
  • Corollary 3
  • Theorem 2: \ref{['proof:thm2']}
  • Corollary 4
  • Definition 1: Highly balanced binary trees
  • Theorem 3: \ref{['proof:thm3']}
  • Lemma 4: \ref{['proof:lemma4']}
  • Conjecture 1
  • ...and 17 more