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

We observe a $N\times M$ matrix $Y_{ij}=s_{ij}+\xi_{ij}$ with $\xi_{ij}\sim\CN(0,1)$ i.i.d. in $i,j$, and $s_{ij}\in\R$. 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 test procedures and compute asymptotical detection boundary $a$ in order to have maximal test errors tending to 0 as $M\to\infty,\ N\to\infty,\ p=n/N\to 0,\ q=m/M\to 0$. 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 $(n,m)$, which is unknown but belongs to 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, second, to matrices of random variables having a distribution from an exponential family (non Gaussian) and, third, to a two-sided alternative for matrices with Gaussian elements.