Quantitative limit theorems and bootstrap approximations for empirical spectral projectors

TL;DR

This paper develops new quantitative limit theorems and bootstrap methods for approximating spectral projectors of empirical covariance operators in Hilbert spaces, with broad applicability and improved results under mild conditions.

Contribution

It introduces a dimension-free framework based on the relative rank of the covariance operator, providing novel limit theorems and bootstrap techniques for spectral projector approximation.

Findings

Dimension-free bounds using relative rank

Quantitative limit theorems under mild conditions

Bootstrap approximations that improve existing results

Abstract

Given finite i.i.d.~samples in a Hilbert space with zero mean and trace-class covariance operator $\Sigma$, the problem of recovering the spectral projectors of $\Sigma$ naturally arises in many applications. In this paper, we consider the problem of finding distributional approximations of the spectral projectors of the empirical covariance operator $\hat \Sigma$, and offer a dimension-free framework where the complexity is characterized by the so-called relative rank of $\Sigma$. In this setting, novel quantitative limit theorems and bootstrap approximations are presented subject only to mild conditions in terms of moments and spectral decay. In many cases, these even improve upon existing results in a Gaussian setting.