Signal-plus-noise matrix models: eigenvector deviations and fluctuations. (7th January 2019)

• Record Type: Journal Article
• Authors: Cape, J; Tang, M; Priebe, C E
• Is Part Of: Biometrika. Volume 106:Number 1(2019:Mar.)
• Page Start: 243
• Page End: 250
• Publication Date: 2019-01-07
• Subjects: Asymptotic normality -- Eigenvector perturbation -- Principal component analysis -- Random matrix -- Signal-plus-noise; Biometry -- Periodicals; 570.1519505
• DOI: 10.1093/biomet/asy070
• Languages: English
• ISSNs: 0006-3444

Abstract

Summary: Estimating eigenvectors and low-dimensional subspaces is of central importance for numerous problems in statistics, computer science and applied mathematics. In this paper we characterize the behaviour of perturbed eigenvectors for a range of signal-plus-noise matrix models encountered in statistical and random-matrix-theoretic settings. We establish both first-order approximation results, i.e., sharp deviations, and second-order distributional limit theory, i.e., fluctuations. The concise methodology presented in this paper synthesizes tools rooted in two core concepts, namely deterministic decompositions of matrix perturbations and probabilistic matrix concentration phenomena. We illustrate our theoretical results with simulation examples involving stochastic block model random graphs.