Estimating covariance and precision matrices along subspaces

TL;DR

This paper investigates the accuracy of estimating covariance and precision matrices along specific subspaces in high-dimensional data, highlighting the dependence on subspace components and introducing a new estimator with strong guarantees.

Contribution

It provides new theoretical insights into subspace-specific covariance and precision matrix estimation and proposes a novel estimator with improved performance.

Findings

Estimation accuracy depends mainly on the targeted subspace components.

Precision matrix estimation is nearly unaffected by the covariance matrix's condition number.

A new estimator for the single-index model shows strong theoretical and numerical advantages.

Abstract

We study the accuracy of estimating the covariance and the precision matrix of a $D$-variate sub-Gaussian distribution along a prescribed subspace or direction using the finite sample covariance. Our results show that the estimation accuracy depends almost exclusively on the components of the distribution that correspond to desired subspaces or directions. This is relevant and important for problems where the behavior of data along a lower-dimensional space is of specific interest, such as dimension reduction or structured regression problems. We also show that estimation of precision matrices is almost independent of the condition number of the covariance matrix. The presented applications include direction-sensitive eigenspace perturbation bounds, relative bounds for the smallest eigenvalue, and the estimation of the single-index model. For the latter, a new estimator, derived from the analysis, with strong theoretical guarantees and superior numerical performance is proposed.