Coherence-free Entrywise Estimation of Eigenvectors in Low-rank Signal-plus-noise Matrix Models

TL;DR

This paper introduces a novel eigenvector estimation method for low-rank signal-plus-noise matrices that is coherence-free, achieving optimal error rates and demonstrating robustness in simulations.

Contribution

The authors develop a coherence-free eigenvector estimation technique with provable guarantees, extending to non-Gaussian noise and higher ranks, and derive new metric entropy bounds.

Findings

Method achieves error bounds independent of coherence in Gaussian noise

Performs well under non-Gaussian noise in simulations

Extends to rank-r matrices with minimal coherence dependence

Abstract

Spectral methods are widely used to estimate eigenvectors of a low-rank signal matrix subject to noise. These methods use the leading eigenspace of an observed matrix to estimate this low-rank signal. Typically, the entrywise estimation error of these methods depends on the coherence of the low-rank signal matrix with respect to the standard basis. In this work, we present a novel method for eigenvector estimation that avoids this dependence on coherence. Assuming a rank-one signal matrix, under mild technical conditions, the entrywise estimation error of our method provably has no dependence on the coherence under Gaussian noise (i.e., in the spiked Wigner model), and achieves the optimal estimation rate up to logarithmic factors. Simulations demonstrate that our method performs well under non-Gaussian noise and that an extension of our method to the case of a rank-$r$ signal matrix has little to no dependence on the coherence. In addition, we derive new metric entropy bounds for rank-$r$ singular subspaces under $\ell_{2,\infty}$ distance, which may be of independent interest. We use these new bounds to improve the best known lower bound for rank-$r$ eigenspace estimation under $\ell_{2,\infty}$ distance.