On the Structure of Bad Science Matrices

2024-08-01

arXiv.org

AI Summary

This paper studies the problem of maximizing the expected infinity norm (eta(A) = mathbb{E}_x[|Ax|_infty]), where (x in {-1,1}^n) is a uniformly random sign vector and (A in mathbb{R}^{m imes n}) satisfies unit row norm constraints. Equivalently, it seeks “worst-case” sensing matrices that maximize the largest coordinate magnitude for any sign input. The approach integrates combinatorial optimization, extremal matrix theory, probabilistic methods, and algebraic number theory. The main contributions are: (i) the first asymptotically optimal explicit construction of such matrices; (ii) a proof that entries of any optimal matrix must be square roots of rational numbers; (iii) complete characterization of exact optimal solutions for (n leq 4); and (iv) an explicit family of matrices achieving (eta(A) geq sqrt{log_2(n+1)}), which lies within approximately 18% of the theoretical upper bound—substantially improving upon prior implicit or randomized constructions.

Technology Category

• Search and Optimization: Non-convex Optimization
• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

The bad science matrix problem consists in finding, among all matrices $A in mathbb{R}^{n imes n}$ with rows having unit $ell^2$ norm, one that maximizes $eta(A) = frac{1}{2^n} sum_{x in {-1, 1}^n} |Ax|_infty$. Our main contribution is an explicit construction of an $n imes n$ matrix $A$ showing that $eta(A) geq sqrt{log_2(n+1)}$, 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 $n leq 4$.

Problem

• Optimization
• Matrix Theory
• Educational Accessibility

Innovation

• Optimized Matrix Structure
• Rational Square Root Entries
• High School Accessible Simplification