Quickest Detection for Changes in Maximal kNN Coherence of Random Matrices

The problem of quickest detection of a change in the distribution of a 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 . 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 and fixed. The significance is that the purely high dimensional asymptotic regime considered here is asymptotic in and not making it especially well suited to big data regimes. The theory developed also applies to fixed sample size tests.
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