Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependence

TL;DR

This paper introduces a new estimator for singular vectors of low-rank matrices affected by heteroskedastic and dependent noise, providing finite-sample bounds and a Berry-Esseen theorem applicable in high-dimensional settings.

Contribution

It develops the first finite-sample $ ext{ell}_{2, ext{infinity}}$ bounds and a Berry-Esseen theorem for singular vector estimation under dependent, heteroskedastic noise.

Findings

Finite-sample $ ext{ell}_{2, ext{infinity}}$ bounds established

Berry-Esseen theorem derived for individual entries

Numerical simulations validate theoretical results

Abstract

We propose an estimator for the singular vectors of high-dimensional low-rank matrices corrupted by additive subgaussian noise, where the noise matrix is allowed to have dependence within rows and heteroskedasticity between them. We prove finite-sample $\ell_{2,\infty}$ bounds and a Berry-Esseen theorem for the individual entries of the estimator, and we apply these results to high-dimensional mixture models. Our Berry-Esseen theorem clearly shows the geometric relationship between the signal matrix, the covariance structure of the noise, and the distribution of the errors in the singular vector estimation task. These results are illustrated in numerical simulations. Unlike previous results of this type, which rely on assumptions of gaussianity or independence between the entries of the additive noise, handling the dependence between entries in the proofs of these results requires careful leave-one-out analysis and conditioning arguments. Our results depend only on the signal-to-noise ratio, the sample size, and the spectral properties of the signal matrix.