Computational Resolution of Hadamard Product Factorization for $4 \times 4$ Matrices
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915453955211264 |
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| author | Rivin, Igor |
| author_facet | Rivin, Igor |
| contents | We computationally resolve an open problem concerning the expressibility of $4 \times 4$ full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over $\mathbb{F}_2$, we identify 5,304 counterexamples among the 20,160 full-rank binary matrices (26.3\%). We verify that these counterexamples remain valid over $\mathbb{Z}$ through sign enumeration and provide strong numerical evidence for their validity over $\mathbb{R}$.
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 $4 \times 4$ rank-2 matrices). This emergent low-dimensional structure suggests deep algebraic constraints governing Hadamard factorizability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14901 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Computational Resolution of Hadamard Product Factorization for $4 \times 4$ Matrices Rivin, Igor Rings and Algebras Machine Learning Algebraic Geometry 15A23, 15A69, 05B20, 68W30 We computationally resolve an open problem concerning the expressibility of $4 \times 4$ full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over $\mathbb{F}_2$, we identify 5,304 counterexamples among the 20,160 full-rank binary matrices (26.3\%). We verify that these counterexamples remain valid over $\mathbb{Z}$ through sign enumeration and provide strong numerical evidence for their validity over $\mathbb{R}$. 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 $4 \times 4$ rank-2 matrices). This emergent low-dimensional structure suggests deep algebraic constraints governing Hadamard factorizability. |
| title | Computational Resolution of Hadamard Product Factorization for $4 \times 4$ Matrices |
| topic | Rings and Algebras Machine Learning Algebraic Geometry 15A23, 15A69, 05B20, 68W30 |
| url | https://arxiv.org/abs/2508.14901 |