Quickest Detection for Changes in Maximal kNN Coherence of Random Matrices

The problem of quickest detection of a change in the distribution of a $n\times p$ random matrix based on a sequence of observations having a single unknown change point is considered. The forms of the pre- and post-change distributions of the rows of the 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 a novel scalar summary statistic related to the maximal k-nearest neighbor correlation between columns of each observed random matrix, and is related to a test of existence of a vertex in a sample correlation graph having degree at least $k$. Performance bounds on the delay and false alarm performance of the proposed stopping rule are obtained. 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, in the purely high-dimensional regime of $p\rightarrow \infty$ and $n$ fixed. The significance is that the purely high dimensional asymptotic regime considered here is asymptotic in $p$ and not $n$ making it especially well suited to big data regimes. The theory developed also applies to fixed sample size tests.