Minimax optimal submatrix detection: Sharp non-asymptotic rates
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916024090099712 |
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| author | Knight, Parker Chhor, Julien |
| author_facet | Knight, Parker Chhor, Julien |
| contents | Given an observation $\mathbf Y \in \mathbb{R}^{d_1\times d_2}$ from the model $\mathbf Y = \mathbf X + \mathbf E$ where $\mathbf X$ is constant and $\mathbf E$ has i.i.d. $N(0,1)$ entries, we consider the problem of detecting a planted submatrix in the mean matrix $\mathbf X$. Specifically, we aim to distinguish the null hypothesis $\mathbf X = 0$ from the alternative hypothesis in which $\mathbf X$ is non-zero only on a submatrix of size $s_1 \times s_2$ with elevated entries bounded below by $μ>0$. We establish a minimax lower bound characterizing how large $μ$ must be to ensure that the two hypotheses are distinguishable with high probability. Furthermore, we derive novel minimax-optimal tests achieving the lower bound, and describe extensions of these tests that are adaptive to unknown sparsity levels $s_1$ and $s_2$. In contrast with previous work, which required restrictive assumptions on $s_1,s_2, d_1$ and $d_2$, our non-asymptotic upper and lower bounds match for any configuration of these parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09569 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimax optimal submatrix detection: Sharp non-asymptotic rates Knight, Parker Chhor, Julien Statistics Theory Information Theory Machine Learning Given an observation $\mathbf Y \in \mathbb{R}^{d_1\times d_2}$ from the model $\mathbf Y = \mathbf X + \mathbf E$ where $\mathbf X$ is constant and $\mathbf E$ has i.i.d. $N(0,1)$ entries, we consider the problem of detecting a planted submatrix in the mean matrix $\mathbf X$. Specifically, we aim to distinguish the null hypothesis $\mathbf X = 0$ from the alternative hypothesis in which $\mathbf X$ is non-zero only on a submatrix of size $s_1 \times s_2$ with elevated entries bounded below by $μ>0$. We establish a minimax lower bound characterizing how large $μ$ must be to ensure that the two hypotheses are distinguishable with high probability. Furthermore, we derive novel minimax-optimal tests achieving the lower bound, and describe extensions of these tests that are adaptive to unknown sparsity levels $s_1$ and $s_2$. In contrast with previous work, which required restrictive assumptions on $s_1,s_2, d_1$ and $d_2$, our non-asymptotic upper and lower bounds match for any configuration of these parameters. |
| title | Minimax optimal submatrix detection: Sharp non-asymptotic rates |
| topic | Statistics Theory Information Theory Machine Learning |
| url | https://arxiv.org/abs/2605.09569 |