Principal components in linear mixed models with general bulk

TL;DR

This paper analyzes the behavior of principal components in high-dimensional multivariate mixed-effects linear models, revealing biases and aliasing effects, and derives their asymptotic limits using free probability techniques.

Contribution

It extends previous models by deriving the first-order limits of eigenvalues and eigenvectors in high dimensions with general spectral distributions, using novel free probability tools.

Findings

Principal eigenvalues and eigenvectors exhibit bias and aliasing in high dimensions.

Derived asymptotic limits for eigenvalues and eigenvectors in general spectral settings.

Developed new free probability tools for analyzing high-dimensional covariance estimators.

Abstract

We study the principal components of covariance estimators in multivariate mixed-effects linear models. We show that, in high dimensions, the principal eigenvalues and eigenvectors may exhibit bias and aliasing effects that are not present in low-dimensional settings. We derive the first-order limits of the principal eigenvalue locations and eigenvector projections in a high-dimensional asymptotic framework, allowing for general population spectral distributions for the random effects and extending previous results from a more restrictive spiked model. Our analysis uses free probability techniques, and we develop two general tools of independent interest-- strong asymptotic freeness of GOE and deterministic matrices and a free deterministic equivalent approximation for bilinear forms of resolvents.