We study the 'bad science matrix problem': among all matrices A^n n whose rows have unit ₂-norm, determine the maximum of β (A) =12ⁿₗ\₁\ⁿ\|Ax\|_. Steinerberger 1 (arXiv: 2402. 03205) showed that the optimal asymptotic rate is (1+o (1) ) 2 n, and that this rate is attained with high probability by matrices with i. i. d. 1 entries after normalization. More recent explicit constructions 2 (arXiv: 2408. 00933) achieve β (A) 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 β (A) in the language of high-dimensional central-limit theorems as in Fang, Koike, Liu and Zhao 16 (arXiv: 2305. 17365). Using these connections, we present a family of explicit deterministic matrices Aₙ 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 β (Aₙ) (1 - (2n) 4 (2n) ) 2 (2n).
Skand Sinha (Thu,) studied this question.