Optimal Covariance Change Point Localization in High Dimension

We study the problem of change point detection for covariance matrices in high dimensions. We assume that we observe a sequence {X_i}_{i=1,...,n} of independent and centered p-dimensional sub-Gaussian random vectors whose covariance matrices are piecewise constant. Our task is to recover with high accuracy the number and locations of the change points, which are assumed unknown. Our generic model setting allows for all the model parameters to change with n, including the dimension p, the minimal spacing between consecutive change points, the magnitude of smallest change size and the maximal Orlicz- 2 norm of the covariance matrices of the sample points. Without assuming any additional structural assumption, such as low rank matrices or having sparse principle components, we set up a general framework and a benchmark result for the covariance change point detection problem. We introduce two procedures, one based on the binary segmentation algorithm (e.g. Vostrikova, 1981) and the other on its extension known as wild binary segmentation of Fryzlewicz (2014), and demonstrate that, under suitable conditions, both procedures are able to consistently es- timate the number and locations of change points. Our second algorithm, called Wild Binary Segmentation through Independent Projection (WBSIP), is shown to be optimal in the sense of allowing for the minimax scaling in all the relevant parameters. Our minimax analysis reveals a phase transition effect based on the problem of change point localization. To the best of our knowledge, this type of results has not been established elsewhere in the high-dimensional change point detection literature.