Detection of a sparse submatrix of a high-dimensional noisy matrix

TL;DR

This paper develops a statistical test to detect a sparse, high-value submatrix within a noisy large matrix, establishing the optimal detection boundary and demonstrating the method's effectiveness on synthetic data.

Contribution

It introduces a new test procedure for sparse submatrix detection, derives the asymptotic detection boundary, and extends results to non-Gaussian and unknown variance cases.

Findings

Test procedure achieves asymptotically sharp minimax detection boundary.

Method performs well on synthetic data with sparse, non-square matrices.

Extended results to non-Gaussian matrices and unknown variance scenarios.

Abstract

We observe a $N\times M$ matrix $Y_{ij}=s_{ij}+\xi_{ij}$ with $\xi_{ij}\sim {\mathcal {N}}(0,1)$ i.i.d. in $i,j$, and $s_{ij}\in \mathbb {R}$. We test the null hypothesis $s_{ij}=0$ for all $i,j$ against the alternative that there exists some submatrix of size $n\times m$ with significant elements in the sense that $s_{ij}\ge a>0$. We propose a test procedure and compute the asymptotical detection boundary $a$ so that the maximal testing risk tends to 0 as $M\to\infty$, $N\to\infty$, $p=n/N\to0$, $q=m/M\to0$. We prove that this boundary is asymptotically sharp minimax under some additional constraints. Relations with other testing problems are discussed. We propose a testing procedure which adapts to unknown $(n,m)$ within some given set and compute the adaptive sharp rates. The implementation of our test procedure on synthetic data shows excellent behavior for sparse, not necessarily squared matrices. We extend our sharp minimax results in different directions: first, to Gaussian matrices with unknown variance, next, to matrices of random variables having a distribution from an exponential family (non-Gaussian) and, finally, to a two-sided alternative for matrices with Gaussian elements.