Spiked separable covariance matrices and principal components
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We introduce a class of separable sample covariance matrices of the form Q_1:= A^1/2 X B X^* A^1/2. Here A and 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 Q_1 a spiked separable covariance matrix model. On the one hand, this model includes the spiked covariance matrix as a special case with B=I. On the other hand, it allows for more general correlations of datasets. In particular, for spatio-temporal dataset, A and B represent the spatial and temporal correlations, respectively. In this paper, we investigate the principal components of spiked separable covariance matrices, i.e., the outlier eigenvalues and eigenvectors of Q_1. We derive precise large deviation estimates on the outlier eigenvalues, and on the generalized components (i.e. 〈v, ξ_i〉 for any deterministic vector v) of the outlier and non-outlier eigenvectors ξ_i. We prove the results in full generality, in the sense that our results 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 A and B will cause outliers of the eigenvalue spectrum, and the eigenvectors can help us to select the outliers that correspond to the spikes of A (or B).
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