Abstract
This paper aims to address two fundamental challenges arising in eigenvector estimation and inference for a low-rank matrix from noisy observations: 1) how to estimate an unknown eigenvector when the eigen-gap (i.e. the spacing between the associated eigenvalue and the rest of the spectrum) is particularly small; 2) how to perform estimation and inference on linear functionals of an eigenvector—a sort of “fine-grained” statistical reasoning that goes far beyond the usual <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell _{2}$ </tex-math></inline-formula> analysis. We investigate how to address these challenges in a setting where the unknown <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\times n$ </tex-math></inline-formula> matrix is symmetric and the additive noise matrix contains independent (and non-symmetric) entries. Based on eigen-decomposition of the asymmetric data matrix, we propose estimation and uncertainty quantification procedures for an unknown eigenvector, which further allow us to reason about linear functionals of an unknown eigenvector. The proposed procedures and the accompanying theory enjoy several important features: 1) distribution-free (i.e. prior knowledge about the noise distributions is not needed); 2) adaptive to heteroscedastic noise; 3) minimax optimal under Gaussian noise. Along the way, we establish valid procedures to construct confidence intervals for the unknown eigenvalues. All this is guaranteed even in the presence of a small eigen-gap (up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(\sqrt {n/\mathrm {poly}\log (n)}\,)$ </tex-math></inline-formula> times smaller than the requirement in prior theory), which goes significantly beyond what generic matrix perturbation theory has to offer.
Highlights
A variety of science and engineering applications ask for spectral analysis of low-rank matrices in high dimension [1]
We propose a new estimator for the linear functional a ul that achieves minimax-optimal statistical accuracy
Two ingredients are needed: (1) a nearly unbiased estimate of a ul, and (2) a valid length of the interval. We describe these ingredients as follows. a) A modified nearly unbiased estimator uma,oldified: While the estimator ua,l (cf. (11)) discussed previously enjoys minimax optimal statistical accuracy, we find it more convenient to work with a modified estimator when conducting uncertainty quantification, in the regime when a ul is very small
Summary
A variety of science and engineering applications ask for spectral analysis of low-rank matrices in high dimension [1]. Imagine that we are interested in a large low-rank matrix r. The aim is to perform reliable estimation and inference on the unseen eigenvectors of M on the basis of noisy data. Motivated by the abundance of applications (e.g. collaborative filtering, harmonic retrieval, sensor network localization, joint shape matching [2]–[6]), research on eigenspace estimation in this context has flourished in the past several years, typically built upon proper exploitation of low-rank structures. We have been equipped with a rich suite of modern statistical theory that delivers statistical performance guarantees for a number of spectral estimators We have been equipped with a rich suite of modern statistical theory that delivers statistical performance guarantees for a number of spectral estimators (e.g. [7]–[16], [16]–[22]); see [23] for a contemporary overview of spectral methods
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