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

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 $\rho_n$ through $n\rho_n$, where $n$ is the dimension of the random matrix, and we allow $n\rho_n$ to grow at the rate of $\log n$. 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 $n\rho_n$ is proportional to $\log n$. 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.