## arXiv Analytics

### arXiv:2104.03298 [math.ST]AbstractReferencesReviewsResources

#### Minimax Estimation of Linear Functions of Eigenvectors in the Face of Small Eigen-Gaps

Published 2021-04-07Version 1

Eigenvector perturbation analysis plays a vital role in various statistical data science applications. A large body of prior works, however, focused on establishing $\ell_{2}$ eigenvector perturbation bounds, which are often highly inadequate in addressing tasks that rely on fine-grained behavior of an eigenvector. This paper makes progress on this by studying the perturbation of linear functions of an unknown eigenvector. Focusing on two fundamental problems -- matrix denoising and principal component analysis -- in the presence of Gaussian noise, we develop a suite of statistical theory that characterizes the perturbation of arbitrary linear functions of an unknown eigenvector. In order to mitigate a non-negligible bias issue inherent to the natural "plug-in" estimator, we develop de-biased estimators that (1) achieve minimax lower bounds for a family of scenarios (modulo some logarithmic factor), and (2) can be computed in a data-driven manner without sample splitting. Noteworthily, the proposed estimators are nearly minimax optimal even when the associated eigen-gap is substantially smaller than what is required in prior theory.

Related articles: Most relevant | Search more
arXiv:1705.00807 [math.ST] (Published 2017-05-02)
Minimax Estimation of the $L_1$ Distance
arXiv:1203.3342 [math.ST] (Published 2012-03-15)
Minimax estimation for mixtures of Wishart distributions
arXiv:1409.8150 [math.ST] (Published 2014-09-29)
Minimax estimation of jump activity in semimartingales