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June 2020 Entrywise eigenvector analysis of random matrices with low expected rank
Emmanuel Abbe, Jianqing Fan, Kaizheng Wang, Yiqiao Zhong
Ann. Statist. 48(3): 1452-1474 (June 2020). DOI: 10.1214/19-AOS1854

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 Abbe, Bandeira and Hall (2014) 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: \begin{equation*}u_{k}\approx \frac{Au_{k}^{*}}{\lambda _{k}^{*}},\end{equation*} 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 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.

Citation

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Emmanuel Abbe. Jianqing Fan. Kaizheng Wang. Yiqiao Zhong. "Entrywise eigenvector analysis of random matrices with low expected rank." Ann. Statist. 48 (3) 1452 - 1474, June 2020. https://doi.org/10.1214/19-AOS1854

Information

Received: 1 October 2017; Revised: 1 November 2018; Published: June 2020
First available in Project Euclid: 17 July 2020

zbMATH: 07241598
MathSciNet: MR4124330
Digital Object Identifier: 10.1214/19-AOS1854

Subjects:
Primary: 62H25
Secondary: 60B20, 62H12

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 3 • June 2020
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