For unit-row matrices, the maximal average sup-norm over Rademacher vectors is sqrt(2 log(2n)) - log log(2n)/(2 sqrt(2 log(2n))) + o(1/sqrt(log n)) for explicit Hadamard-based matrices and random sign matrices, with extremizers shown to induce nearly equal-volume, isoperimetrically optimal Voronoi…
On the Structure of Bad Science Matrices
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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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The Structure of Extremal Bad Science Matrices
For unit-row matrices, the maximal average sup-norm over Rademacher vectors is sqrt(2 log(2n)) - log log(2n)/(2 sqrt(2 log(2n))) + o(1/sqrt(log n)) for explicit Hadamard-based matrices and random sign matrices, with extremizers shown to induce nearly equal-volume, isoperimetrically optimal Voronoi…