随机矩阵最大kNN相干变化的快速检测
Quickest Detection for Changes in Maximal kNN Coherence of Random Matrices
摘要：
本文讨论了基于具有单个未知变化点的矩阵观测序列的$n\times p$随机矩阵列间最大相干性变化的最快检测问题。假设随机矩阵具有相同分布的行，并且最大相干性被定义为与任何行相关联的最大$p \choose 2$相关系数。同样，$k$最近邻（kNN）相干性被定义为这些相关系数中的第$k$大。假设观测矩阵的变化前和变化后分布的形式属于具有稀疏色散矩阵的椭圆轮廓密度的家族，但在其他方面是未知的。提出了一种非参数停止规则，该规则基于每个观测随机矩阵的列之间的最大k-最近邻样本相干性。这是一个汇总统计量，与样本相关性图中是否存在度至少为$k$的中心顶点的检验相关。在纯高维的制度，其中$p\rightarrow \infty$和$n$是固定的性能界限上的延迟和虚警性能的建议停止规则。当变化前的色散矩阵是对角的，它表明，在所有的功能，建议的汇总统计量，建议的停止规则是渐近最优的最小最快变化检测（QCD）模型下的停止阈值接近无穷大。该理论也适用于序贯假设检验和固定样本量检验。
英文摘要：
This paper addresses the problem of quickest detection of a change in the maximal coherence between columns of a $n\times p$ random matrix based on a sequence of matrix observations having a single unknown change point. The random matrix is assumed to have identically distributed rows and the maximal coherence is defined as the largest of the $p \choose 2$ correlation coefficients associated with any row. Likewise, the $k$ nearest neighbor (kNN) coherence is defined as the $k$-th largest of these correlation coefficients. The forms of the pre- and post-change distributions of the observed matrices are assumed to belong to the family of elliptically contoured densities with sparse dispersion matrices but are otherwise unknown. A non-parametric stopping rule is proposed that is based on the maximal k-nearest neighbor sample coherence between columns of each observed random matrix. This is a summary statistic that is related to a test of the existence of a hub vertex in a sample correlation graph having a degree at least $k$. Performance bounds on the delay and false alarm performance of the proposed stopping rule are obtained in the purely high dimensional regime where $p\rightarrow \infty$ and $n$ is fixed. When the pre-change dispersion matrix is diagonal it is shown that, among all functions of the proposed summary statistic, the proposed stopping rule is asymptotically optimal under a minimax quickest change detection (QCD) model as the stopping threshold approaches infinity. The theory developed also applies to sequential hypothesis testing and fixed sample size tests.
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