On the Structure of Bad Science Matrices
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arXiv
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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915111886651392 |
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| author | Albors, Alex Bhatti, Hisham Ganjoo, Lukshya Guo, Raymond Kunisky, Dmitriy Mukherjee, Rohan Stepin, Alicia Zeng, Tony |
| author_facet | Albors, Alex Bhatti, Hisham Ganjoo, Lukshya Guo, Raymond Kunisky, Dmitriy Mukherjee, Rohan Stepin, Alicia Zeng, Tony |
| contents | 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 $β(A) = \frac{1}{2^n} \sum_{x \in \{-1, 1\}^n} \|Ax\|_\infty$. Our main contribution is an explicit construction of an $n \times n$ matrix $A$ showing that $β(A) \geq \sqrt{\log_2(n+1)}$, 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 $n \leq 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00933 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Structure of Bad Science Matrices Albors, Alex Bhatti, Hisham Ganjoo, Lukshya Guo, Raymond Kunisky, Dmitriy Mukherjee, Rohan Stepin, Alicia Zeng, Tony Functional Analysis Discrete Mathematics Combinatorics 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 $β(A) = \frac{1}{2^n} \sum_{x \in \{-1, 1\}^n} \|Ax\|_\infty$. Our main contribution is an explicit construction of an $n \times n$ matrix $A$ showing that $β(A) \geq \sqrt{\log_2(n+1)}$, 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 $n \leq 4$. |
| title | On the Structure of Bad Science Matrices |
| topic | Functional Analysis Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2408.00933 |