SPIKED SEPARABLE COVARIANCE MATRICES AND PRINCIPAL COMPONENTS
成果类型：
Article
署名作者：
Ding, Xiucai; Yang, Fan
署名单位：
Duke University; University of Pennsylvania
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
DOI：
10.1214/20-AOS1995
发表日期：
2021
页码：
1113-1138
关键词：
摘要：
We study a class of separable sample covariance matrices of the form (Q) over tilde (1) := (A) over tilde X-1/2 (B) over tilde BX*(B) over tilde (1/2). Here, (A) over tilde and (B) over tilde are positive definite matrices whose spectrums consist of bulk spectrums plus several spikes, that is, larger eigenvalues that are separated from the bulks. Conceptually, we call (Q) over tilde (1) a spiked separable covariance matrix model. On the one hand, this model includes the spiked covariance matrix as a special case with (B) over tilde = I. On the other hand, it allows for more general correlations of datasets. In particular, for spatiotemporal dataset, (A) over tilde and (B) over tilde represent the spatial and temporal correlations, respectively. In this paper, we study the outlier eigenvalues and eigenvectors, that is, the principal components, of the spiked separable covariance model (Q) over tilde (1). We prove the convergence of the outlier eigenvalues (lambda) over tilde (i) and the generalized components (i.e., < v, (xi) over tilde (i)> for any deterministic vector v) of the outlier eigenvectors (xi) over tilde (i) with optimal convergence rates. Moreover, we also prove the delocalization of the nonoutlier 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 matrix model in (Probab. Theory Related Fields 164 (2016) 459-552). In particular, we show that the spikes of both (A) over tilde and (B) over tilde will cause outliers of the eigenvalue spectrum, and the eigenvectors can help to select the outliers that correspond to the spikes of (A) over tilde (or (B) over tilde).