DeepAI AI Chat
Log In Sign Up

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

by   Gen Li, et al.

Eigenvector perturbation analysis plays a vital role in various statistical data science applications. A large body of prior works, however, focused on establishing ℓ_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.


page 1

page 2

page 3

page 4


Minimax Lower Bounds for Noisy Matrix Completion Under Sparse Factor Models

This paper examines fundamental error characteristics for a general clas...

Van Trees inequality, group equivariance, and estimation of principal subspaces

We establish non-asymptotic lower bounds for the estimation of principal...

Optimal Algorithms for Stochastic Multi-Armed Bandits with Heavy Tailed Rewards

In this paper, we consider stochastic multi-armed bandits (MABs) with he...

Minimax Linear Estimation at a Boundary Point

This paper characterizes the minimax linear estimator of the value of an...

A Fourier Analytical Approach to Estimation of Smooth Functions in Gaussian Shift Model

We study the estimation of f() under Gaussian shift model = +, where ∈^d...

Denoising Linear Models with Permuted Data

The multivariate linear regression model with shuffled data and additive...

Theory of functional principal components analysis for discretely observed data

For discretely observed functional data, estimating eigenfunctions with ...