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.

提出了一种用于检测具有显著元素的子矩阵的测试方法，并计算了其检测边界和自适应敏锐度。在稀疏矩阵上得到了出色的表现，并扩展了结果以应对不同的矩阵类型和双侧替代假设。