Improved dependence on coherence in eigenvector and eigenvalue estimation error bounds

TL;DR

This paper improves the theoretical bounds on eigenvector and eigenvalue estimation errors in spectral methods by reducing dependence on coherence, especially under subexponential noise, advancing understanding in matrix analysis.

Contribution

The authors introduce a new matrix concentration inequality that reduces coherence dependence in spectral estimator error bounds, applicable to various low-rank matrix problems.

Findings

Coherence dependence in error bounds is significantly improved.

Coherence-free bounds are achievable under certain noise conditions.

Results match minimax lower bounds up to logarithmic factors.

Abstract

Spectral estimators are fundamental in lowrank matrix models and arise throughout machine learning and statistics, with applications including network analysis, matrix completion and PCA. These estimators aim to recover the leading eigenvalues and eigenvectors of an unknown signal matrix observed subject to noise. While extensive research has addressed the statistical accuracy of spectral estimators under a variety of conditions, most previous work has assumed that the signal eigenvectors are incoherent with respect to the standard basis. This assumption typically arises because of suboptimal dependence on coherence in one or more concentration inequalities. Using a new matrix concentration result that may be of independent interest, we establish estimation error bounds for eigenvector and eigenvalue recovery whose dependence on coherence significantly improves upon prior work. Our results imply that coherence-free bounds can be achieved when the standard deviation of the noise is comparable to its Orlicz 1-norm (i.e., its subexponential norm). This matches known minimax lower bounds under Gaussian noise up to logarithmic factors.