  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.
