高维噪声矩阵稀疏子矩阵的检测
Detection of a sparse submatrix of a high-dimensional noisy matrix
摘要：
我们观察到$N\times M$矩阵$Y_{ij}=s_{ij}+ξ_{ij}$与$ξ_{ij}\sim {\mathcal {N}}(0,1)$ i.i.d.在$i,j$和$s_{ij}\in \mathbb {R}$中。我们测试的零假设$s_{ij}=0$对所有$i,j$的替代方案，存在一些子矩阵的大小$n\times m$的意义上，$s_{ij}\ge a>0$的重要元素。我们提出了一个测试过程，并计算渐近检测边界$a$，使最大测试风险趋于0为$M\to\infty$，$N\to\infty$，$p=n/N\to0$，$q=m/M\to0$。我们证明了这个边界是渐近尖锐的minimax在一些额外的约束。与其他测试问题的关系进行了讨论。我们提出了一个测试程序，适应未知的$(n,m)$在一些给定的集合，并计算自适应锐率。我们的测试程序在合成数据上的实现显示了稀疏矩阵（不一定是平方矩阵）的出色性能。我们扩展我们的尖锐的极大极小结果在不同的方向：首先，高斯矩阵与未知的方差，其次，随机变量的矩阵具有分布从一个指数家庭（非高斯），最后，一个双边替代矩阵与高斯元素。
英文摘要：
We observe a $N\times M$ matrix $Y_{ij}=s_{ij}+ξ_{ij}$ with $ξ_{ij}\sim {\mathcal {N}}(0,1)$ 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.
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