Signal-Plus-Noise Matrix Models: Eigenvector Deviations and Fluctuations

• Activity Number: 136 - Recent Advances in Dimension Reduction
• Type: Contributed
• Date/Time: Monday, July 29, 2019 : 8:30 AM to 10:20 AM
• Sponsor: Section on Statistical Learning and Data Science
• Author(s): Joshua Cape* and Minh Tang and Carey E Priebe
• Companies: Johns Hopkins University and Johns Hopkins University and Johns Hopkins University
• Keywords: principal component analysis; eigenvectors; random matrix; signal-plus-noise

Abstract

Estimating eigenvectors and principal subspaces is of central importance for numerous problems in statistics, computer science, and applied mathematics. This paper characterizes the behavior of perturbed eigenvectors for a range of signal-plus-noise matrix models encountered in both statistical and random matrix theoretic settings. We prove both first-order approximation results (i.e., sharp deviations) as well as second-order distributional limit theory (i.e., fluctuations). The concise methodology considered in this paper synthesizes tools rooted in two core concepts, namely (i) deterministic decompositions of matrix perturbations and (ii) probabilistic matrix concentration phenomena. We illustrate our theoretical results via simulation examples involving stochastic block model random graphs.