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

• URL: http://arxiv.org/abs/2104.03298v1
• Date: Wed, 7 Apr 2021 17:55:10 GMT
• Title: Minimax Estimation of Linear Functions of Eigenvectors in the Face of Small Eigen-Gaps
• Authors: Gen Li, Changxiao Cai, Yuantao Gu, H. Vincent Poor, Yuxin Chen
• Abstract summary: Eigenvector perturbation analysis plays a vital role in various statistical data science applications. 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.
• Score: 95.62172085878132
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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