Entrywise limit theorems for eigenvectors of signal-plus-noise matrix models with weak signals

Abstract

We establish a finite-sample Berry-Esseen theorem for the entrywise limits of the eigenvectors for a broad collection of signal-plus-noise random matrix models under challenging weak signal regimes. The signal strength is characterized by a scaling factor ${\mathit{\rho }}_n$ through $n{\mathit{\rho }}_n$, where n is the dimension of the random matrix, and we allow $n{\mathit{\rho }}_n$ to grow at the rate of $logn$. The key technical contribution is a sharp finite-sample entrywise eigenvector perturbation bound. The existing error bounds on the two-to-infinity norms of the higher-order remainders are not sufficient when $n{\mathit{\rho }}_n$ is proportional to $logn$. We apply the general entrywise eigenvector analysis results to the symmetric noisy matrix completion problem, random dot product graphs, and two subsequent inference tasks for random graphs: the estimation of pure nodes in mixed membership stochastic block models and the hypothesis testing of the equality of latent positions in random graphs.