Minimax optimal submatrix detection: Sharp non-asymptotic rates

TL;DR

This paper establishes sharp non-asymptotic minimax rates for detecting a planted submatrix in noisy observations, providing optimal tests that adapt to unknown sparsity levels.

Contribution

It derives the first non-asymptotic minimax bounds for submatrix detection and constructs adaptive, optimal tests under general conditions.

Findings

Derived sharp minimax lower bounds for submatrix detection

Constructed tests that achieve these bounds, proving optimality

Extended methods to adapt to unknown sparsity levels

Abstract

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 $\mu>0$. We establish a minimax lower bound characterizing how large $\mu$ 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.