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Bad Science Matrices

Stefan Steinerberger

Abstract

Inspired by the bad scientist who keeps repeating an experiment 20 times to get a single outcome with $p < 0.05$, we consider matrices $A \in \mathbb{R}^{n \times n}$ whose rows are normalized in $\ell^2$ and for which $2^{-n}\sum_{x \in \left\{-1,1\right\}^n} \|Ax\|_{\ell^{\infty}}$ is large. They correspond to affine transformations of the discrete unit cube to points with, on average, at least one large coordinate. Such matrices can be seen as a collection of fair tests on a fair coin where at least one outcome is typically atypical. We prove that, as $n \rightarrow \infty$, the quantity can scale as $$ \max_{A \in \mathbb{R}^{n \times n}} \frac{1}{2^{n}}\sum_{x \in \left\{-1,1\right\}^n} \|Ax\|_{\ell^{\infty}} = (1+o(1)) \cdot \sqrt{2\log{n}}.$$ We also present candidate maximizers up to dimension $n \leq 8$ which appear to be highly structured and have nice closed-form solutions.

Bad Science Matrices

Abstract

Inspired by the bad scientist who keeps repeating an experiment 20 times to get a single outcome with , we consider matrices whose rows are normalized in and for which is large. They correspond to affine transformations of the discrete unit cube to points with, on average, at least one large coordinate. Such matrices can be seen as a collection of fair tests on a fair coin where at least one outcome is typically atypical. We prove that, as , the quantity can scale as We also present candidate maximizers up to dimension which appear to be highly structured and have nice closed-form solutions.
Paper Structure (18 sections, 3 theorems, 55 equations, 4 figures)

This paper contains 18 sections, 3 theorems, 55 equations, 4 figures.

Table of Contents

  1. Introduction and Result
  2. The dishonest scientist.
  3. Results
  4. Related results.
  5. Small values of $n$
  6. Summary.
  7. The case $n=2$
  8. The case $n=3$
  9. The case $n=4$
  10. The case $n=5$
  11. The case $n=6$
  12. The case $n=7$
  13. The case $n=8$
  14. Proofs
  15. Outline.
  16. ...and 3 more sections

Key Result

Theorem 1

Among matrices with rows satisfying $\|a_i\|_{\ell^2} \leq 1$, as $n \rightarrow \infty$,

Figures (4)

  • Figure 1: Lower bounds for $\beta_n = \max \beta(A)$ when $n \leq 8$, see § 2.
  • Figure 2: The extremal case for $n=2$.
  • Figure 3: Four of the eight points $Ax$ in relation to a cube centered in the origin of sidelength 2.7.
  • Figure 4: Left: a near-optimal matrix for $n=8$. Right: visual representation, each entry is $\pm 1/\sqrt{7}$. $\beta(B) = 3 \sqrt{7}/4 \sim 1.984\dots.$

Theorems & Definitions (7)

  • Theorem
  • Proposition
  • proof
  • Lemma
  • proof
  • proof
  • proof