Work

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

Year: 2024

Type: article

Abstract: Eigenvector perturbation analysis plays a vital role in various 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 issubstantially smallerthan what is required in prior statistical theory.

Source: IEEE Transactions on Information Theory

DOI: 10.1109/tit.2024.3514795

Authors: Gen Li, Changxiao Cai, H. Vincent Poor, Yuxin Chen

Institutions: Chinese University of Hong Kong, University of Michigan, Princeton University, University of Pennsylvania

Language: English

Topic: Random Matrices and Applications

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green