Entrywise Eigenvector Analysis of Random Matrices with Low Expected Rank

Emmanuel Abbe, Jianqing Fan, Kaizheng Wang, Yiqiao Zhong

Abstract:

Recovering low-rank structures via eigenvector perturbation analysis is a common problem in statistical machine learning, such as in factor analysis, community detection, ranking, matrix completion, among others. While a large variety of bounds are available for average errors between empirical and population statistics of eigenvectors, few results are tight for entrywise analyses, which are critical for a number of problems such as community detection.

This paper investigates entrywise behaviors of eigenvectors for a large class of random matrices whose expectations are low-rank, which helps settle the conjecture in \cite{abh_arxiv} that the spectral algorithm achieves exact recovery 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 $\E A$, respectively. The fact that the approximation is both tight and linear in $A$ facilitates sharp comparisons between $u_k$ and $u_k^*$. In particular, 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 to perturbations of eigenspaces, yielding new $\ell_\infty$-type bounds for synchronization ($\mathbb{Z}_2$-spiked Wigner model) and noisy matrix completion.