Optimal covariance change point localization in high dimensions

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