On the principal components of sample covariance matrices

• 成果类型: Article
• 署名作者: Bloemendal, Alex; Knowles, Antti; Yau, Horng-Tzer; Yin, Jun
• 署名单位: Harvard University; Swiss Federal Institutes of Technology Domain; ETH Zurich; University of Wisconsin System; University of Wisconsin Madison
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0616-x
• 发表日期: 2016
• 页码: 459-552
• 关键词: generalized wigner matrices; finite rank deformations; spiked population-model; random band matrices; LARGEST EIGENVALUE; airy kernel; UNIVERSALITY; fluctuations; perturbations; ensembles

摘要

We introduce a class of M x M sample covariance matrices Q which subsumes and generalizes several previous models. The associated population covariance matrix Sigma = EQ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of Sigma - I-M may depend on M in an arbitrary fashion. We investigate the principal components, i. e. the top eigenvalues and eigenvectors, of Q. We derive precise large deviation estimates on the generalized components < w, xi(i)> of the outlier and non-outlier eigenvectors xi(i). Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the subcritical spikes of Sigma. The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix Q, satisfying EQ = I-M, as input: the isotropic local Marchenko-Pastur law established in Bloemendal et al. (Electron J Probab 19:1-53, 2014), level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue-eigenvector distribution.