Wald Statistics in high-dimensional PCA

TL;DR

This paper develops a Berry-Essen bound for Wald statistics in high-dimensional PCA, enabling non-asymptotic confidence sets for spectral projectors under complex covariance structures.

Contribution

It introduces a Berry-Essen type bound for Wald statistics in high-dimensional PCA, applicable even with large effective rank, advancing spectral projector inference.

Findings

Validates the bound for large effective rank scenarios.

Enables construction of non-asymptotic confidence ellipsoids.

Extends perturbation theory to high-dimensional settings.

Abstract

In this note we consider PCA for Gaussian observations $X_1,\dots, X_n$ with covariance $\Sigma=\sum_i \lambda_i P_i$ in the 'effective rank' setting with model complexity governed by $\mathbf{r}(\Sigma):=\text{tr}(\Sigma)/\| \Sigma \|$. We prove a Berry-Essen type bound for a Wald Statistic of the spectral projector $\hat P_r$. This can be used to construct non-asymptotic confidence ellipsoids and tests for spectral projectors $P_r$. Using higher order pertubation theory we are able to show that our Theorem remains valid even when $\mathbf{r}(\Sigma) \gg \sqrt{n}$.