On spiked eigenvalues of a renormalized sample covariance matrix from multi-population

TL;DR

This paper studies the behavior of large eigenvalues in renormalized sample covariance matrices from multiple populations, especially in ultrahigh dimensions, providing theoretical insights and practical criteria for clustering and subgroup detection.

Contribution

It introduces a unified asymptotic framework for spiked eigenvalues in ultrahigh-dimensional multi-population covariance matrices, extending previous results to new regimes.

Findings

Established first- and second-order convergence of spiked eigenvalues.

Applied results to determine the number of subgroups and evaluate clustering without true labels.

Unified framework covering high and ultrahigh dimensional asymptotics.

Abstract

Sample covariance matrices from multi-population typically exhibit several large spiked eigenvalues, which stem from differences between population means and are crucial for inference on the underlying data structure. This paper investigates the asymptotic properties of spiked eigenvalues of a renormalized sample covariance matrices from multi-population in the ultrahigh dimensional context where the dimension-to-sample size ratio p/n go to infinity. The first- and second-order convergence of these spikes are established based on asymptotic properties of three types of sesquilinear forms from multi-population. These findings are further applied to two scenarios,including determination of total number of subgroups and a new criterion for evaluating clustering results in the absence of true labels. Additionally, we provide a unified framework with p/n->c\in (0,\infty] that integrates the asymptotic results in both high and ultrahigh dimensional settings.