Optimal covariance change point localization in high dimensions

We study the problem of change point localization for covariance matrices in high dimensions. We assume that we observe a sequence of independent and centered $p$-dimensional sub-Gaussian random vectors whose covariance matrices are piecewise constant, and only change at unknown times. We are concerned with the localization task of estimating the positions of the change points. In our analysis, we allow for all the model parameters to change with the sample size $n$, including the dimension $p$, the minimal spacing between consecutive change points $\Delta $, the maximal Orlicz-$\psi _{2}$ norm $B$ of the sample points and the magnitude $\kappa $ of the smallest distributional change, defined as the minimal operator norm of the difference between the covariance matrix at a change point and the covariance matrix at the previous time point.We introduce two procedures, one based on the binary segmentation algorithm and the other on its popular extension known as wild binary segmentation, and demonstrate that, under suitable conditions, both procedures can consistently estimate the change points. In particular, our second algorithm, called Wild Binary Segmentation through Independent Projection (WBSIP), delivers a localization error of order $B^{4}\kappa ^{-2}\log (n)$, which is shown to be minimax rate optimal, save, possibly, for the $\log (n)$ term. WBSIP requires the model parameters to satisfy the scaling $\Delta \kappa ^{2}\gtrsim pB^{4}\log ^{1+\xi }(n)$, for any $\xi >0$, which we demonstrate to be essentially necessary, in the sense that no algorithm can guarantee consistent localization if $\Delta \kappa ^{2}\lesssim pB^{4}$. This result reveals an interesting phase transition effect separating parameter combinations for which the localization task is feasible from the ones for which it is not.