Signal-plus-noise matrix models: eigenvector deviations and fluctuations

TL;DR

This paper analyzes how eigenvectors in signal-plus-noise matrix models deviate and fluctuate, providing precise approximations and distributional limits, with applications demonstrated on stochastic block model graphs.

Contribution

It offers new first- and second-order theoretical results on eigenvector behavior in signal-plus-noise models, combining deterministic and probabilistic techniques.

Findings

Sharp deviation bounds for eigenvectors

Distributional limit theorems for fluctuations

Validation through simulations on stochastic block models

Abstract

Estimating eigenvectors and low-dimensional 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.