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 instatistical 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.
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