The Structure of Extremal Bad Science Matrices

Abstract: We study the 'bad science matrix problem': among all matrices $A\in\mathbb{R}^{n\times n}$ whose rows have unit $\ell_2$-norm, determine the maximum of $$\beta(A)=\frac{1}{2<sup>n}\sum_{x\in{\pm1}<sup>n}|Ax|_\infty$$. Steinerberger 1 showed that the optimal asymptotic rate is $(1+o(1))\sqrt{2\log n}$, and that this rate is attained with high probability by matrices with i.i.d. $\pm1$ entries after normalization. More recent explicit constructions 2 achieve $\beta(A)\ge\sqrt{\log_2(n)+1}$, which lies within a constant factor of the asymptotic optimum. In this paper we bridge the gap between the probabilistic and explicit approaches. We give a geometric description of extremizers as (nearly) isoperimetrically extremal partitions of the $n$-dimensional hypercube induced by the rows of $A$. We obtain precise rates for heuristic constructions by recasting the maximization of $\beta(A)$ in the language of high-dimensional central-limit theorems as in Fang, Koike, Liu and Zhao 16. Using these connections, we present a family of explicit deterministic matrices $A_n$ that exist for all $n$ under the assumption of Hadamard's conjecture, and for infinitely many $n$ unconditionally, such that for all $n$ sufficiently large $$\beta(A_n)\ge\bigl(1 - \frac{\log\log(2n)}{4\log(2n)}\bigr)\sqrt{2\log(2n)}.$$