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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.19259 |
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| _version_ | 1866911077448548352 |
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| author | Bhamidi, Shankar Gamarnik, David Gong, Shuyang |
| author_facet | Bhamidi, Shankar Gamarnik, David Gong, Shuyang |
| contents | We consider the problem of finding a dense submatrix of a matrix with i.i.d. Gaussian entries, where density is measured by average value. This problem arose from practical applications in biology and social sciences \cites{madeira-survey,shabalin2009finding} and is known to exhibit a computation-to-optimization gap between the optimal value and best values achievable by existing polynomial time algorithms. In this paper we consider the class of online algorithms, which includes the best known algorithm for this problem, and derive a tight approximation factor ${4\over 3\sqrt{2}}$ for this class. The result is established using a simple implementation of recently developed Branching-Overlap-Gap-Property \cite{huang2025tight}. We further extend our results to $(\mathbb R^n)^{\otimes p}$ tensors with i.i.d. Gaussian entries, for which the approximation factor is proven to be ${2\sqrt{p}/(1+p)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19259 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finding a dense submatrix of a random matrix. Sharp bounds for online algorithms Bhamidi, Shankar Gamarnik, David Gong, Shuyang Probability Statistics Theory 62G32 60G70, 68Q17 We consider the problem of finding a dense submatrix of a matrix with i.i.d. Gaussian entries, where density is measured by average value. This problem arose from practical applications in biology and social sciences \cites{madeira-survey,shabalin2009finding} and is known to exhibit a computation-to-optimization gap between the optimal value and best values achievable by existing polynomial time algorithms. In this paper we consider the class of online algorithms, which includes the best known algorithm for this problem, and derive a tight approximation factor ${4\over 3\sqrt{2}}$ for this class. The result is established using a simple implementation of recently developed Branching-Overlap-Gap-Property \cite{huang2025tight}. We further extend our results to $(\mathbb R^n)^{\otimes p}$ tensors with i.i.d. Gaussian entries, for which the approximation factor is proven to be ${2\sqrt{p}/(1+p)}$. |
| title | Finding a dense submatrix of a random matrix. Sharp bounds for online algorithms |
| topic | Probability Statistics Theory 62G32 60G70, 68Q17 |
| url | https://arxiv.org/abs/2507.19259 |