Asymptotic limits of spiked eigenvalues and eigenvectors of signal-plus-noise matrices with weak signals and heteroskedastic noise

TL;DR

This paper investigates the asymptotic behavior of spiked eigenvalues and eigenvectors in high-dimensional signal-plus-noise matrices with heteroskedastic noise and potentially infinite-rank signals, providing new theoretical insights and practical clustering methods.

Contribution

It develops the asymptotic limits of spiked eigenvalues and eigenvectors under heteroskedastic noise and high-rank signals, extending existing random matrix theory.

Findings

Derived limits for spiked eigenvalues and eigenvectors in complex noise models

Proposed a new criterion for estimating the number of clusters

Applicable to high-dimensional data with heteroskedastic noise

Abstract

This paper is to study a signal-plus-noise model in high dimensional settings when the dimension and the sample size are comparable. Specifically, we assume that the noise has a general covariance matrix that allows for heteroskedasticity, and that the deterministic signal has the same magnitude as the noise and can have a rank that tends to infinity. We develop the asymptotic limits of the left and right spiked singular vectors of the signal-plusnoise data matrix and the limits of the spiked eigenvalues of the corresponding Gram matrix. As an application, we propose a new criterion to estimate the number of clusters in clustering problems.