Spiked separable covariance matrices and principal components

We study a class of separable sample covariance matrices the form Q˜1:=A˜1/2XB˜X∗A˜1/2. Here, A˜ and B˜ are positive definite whose spectrums consist bulk plus several spikes, that is, larger eigenvalues separated from bulks. Conceptually, we call Q˜1 spiked matrix model. On one hand, this model includes as special case with B˜=I. other it allows for more general correlations datasets. In particular, spatio-temporal dataset, represent spatial temporal correlations, respectively. paper, outlier eigenvectors, principal components, Q˜1. prove convergence λ˜i generalized components (i.e., ⟨v,ξ˜i⟩ any deterministic vector v) eigenvectors ξ˜i optimal rates. Moreover, also delocalization nonoutlier eigenvectors. state our results in full generality, sense they hold near so-called BBP transition degenerate outliers. Our highlight both similarity difference between (Probab. Theory Related Fields 164 (2016) 459–552). show spikes will cause outliers eigenvalue spectrum, can help to select correspond (or B˜).