Spiked separable covariance matrices and principal components

We introduce a class of separable sample covariance matrices of the form $\widetilde{\mathcal{Q}}_1:=\widetilde A^{1/2} X \widetilde B X^* \widetilde A^{1/2}.$ Here $\widetilde{A}$ and $\widetilde{B}$ are positive definite matrices whose spectrums consist of bulk spectrums plus several spikes, i.e. larger eigenvalues that are separated from the bulks. Conceptually, we call $\widetilde{\mathcal{Q}}_1$ a \emph{spiked separable covariance matrix model}. On the one hand, this model includes the spiked covariance matrix as a special case with $\widetilde{B}=I$. On the other hand, it allows for more general correlations of datasets. In particular, for spatio-temporal dataset, $\widetilde{A}$ and $\widetilde{B}$ represent the spatial and temporal correlations, respectively. In this paper, we study the outlier eigenvalues and eigenvectors, i.e. the principal components, of the spiked separable covariance model $\widetilde{\mathcal{Q}}_1$. We prove the convergence of the outlier eigenvalues $\widetilde \lambda_i$ and the generalized components (i.e. $\langle \mathbf v, \widetilde{\mathbf{\xi}}_i \rangle$ for any deterministic vector $\mathbf v$) of the outlier eigenvectors $\widetilde{\mathbf{\xi}}_i$ with optimal convergence rates. Moreover, we also prove the delocalization of the non-outlier eigenvectors. We state our results in full generality, in the sense that they also hold near the so-called BBP transition and for degenerate outliers. Our results highlight both the similarity and difference between the spiked separable covariance matrix model and the spiked covariance model. In particular, we show that the spikes of both $\widetilde{A}$ and $\widetilde{B}$ will cause outliers of the eigenvalue spectrum, and the eigenvectors can help us to select the outliers that correspond to the spikes of $\widetilde{A}$ (or $\widetilde{B}$).