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Recovering low-rank structures via eigenvector perturbation analysis is acommon problem in statistical machine learning, such as in factor analysis,community detection, ranking, matrix completion, among others. While a largevariety of bounds are available for average errors between empirical andpopulation statistics of eigenvectors, few results are tight for entrywiseanalyses, which are critical for a number of problems such as communitydetection.  This paper investigates entrywise behaviors of eigenvectors for a large classof random matrices whose expectations are low-rank, which helps settle theconjecture in Abbe et al. (2014b) that the spectral algorithm achieves exactrecovery in the stochastic block model without any trimming or cleaning steps.The key is a first-order approximation of eigenvectors under the $\ell_\infty$norm: $$u_k \approx \frac{A u_k^*}{\lambda_k^*},$$ where $\{u_k\}$ and$\{u_k^*\}$ are eigenvectors of a random matrix $A$ and its expectation$\mathbb{E} A$, respectively. The fact that the approximation is both tight andlinear in $A$ facilitates sharp comparisons between $u_k$ and $u_k^*$. Inparticular, it allows for comparing the signs of $u_k$ and $u_k^*$ even if $\|u_k - u_k^*\|_{\infty}$ is large. The results are further extended toperturbations of eigenspaces, yielding new $\ell_\infty$-type bounds forsynchronization ($\mathbb{Z}_2$-spiked Wigner model) and noisy matrixcompletion.

Entrywise Eigenvector Analysis of Random Matrices with Low Expected Rank