2025/07/31 by Rivin, Igor
#05B20 #15A23 #15A69 #68W30 #Algebraic Geometry (math.AG) #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2508.14901
We computationally resolve an open problem concerning the expressibility of 4 × 4 full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over \mathbbF2, we identify 5,304 counterexamples among the 20,160 full-rank binary matrices (26.3%). We verify that these counterexamples remain valid over ℤ through sign enumeration and provide strong numerical evidence for their validity over ℝ. Remarkably, our analysis reveals that matrix density (number of ones) is highly predictive of expressibility, achieving 95.7% classification accuracy. Using modern machine learning techniques, we discover that expressible matrices lie on an approximately 10-dimensional variety within the 16-dimensional ambient space, despite the naive parameter count of 24 (12 parameters each for two 4 × 4 rank-2 matrices). This emergent low-dimensional structure suggests deep algebraic constraints governing Hadamard factorizability.