Asymptotic distribution of spiked eigenvalues in the large signal-plus-noise models

TL;DR

This paper derives the asymptotic joint distribution of spiked singular values in large signal-plus-noise matrices with anisotropic noise, revealing non-universality and applications in mixture models and heterogeneity detection.

Contribution

It establishes the joint asymptotic distribution of spiked singular values under general conditions, highlighting non-universality and extending understanding beyond classical models.

Findings

Asymptotic distributions depend on noise entry distributions.

Results apply to mixture models for spiked eigenvalues.

Useful for detecting mean heterogeneity in data matrices.

Abstract

Consider large signal-plus-noise data matrices of the form $S + \Sigma^{1/2} X$, where $S$ is a low-rank deterministic signal matrix and the noise covariance matrix $\Sigma$ can be anisotropic. We establish the asymptotic joint distribution of its spiked singular values when the dimensionality and sample size are comparably large and the signals are supercritical under general assumptions concerning the structure of $(S, \Sigma)$ and the distribution of the random noise $X$. It turns out that the asymptotic distributions exhibit nonuniversality in the sense of dependence on the distributions of the entries of $X$, which contrasts with what has previously been established for the spiked sample eigenvalues in the context of spiked population models. Such a result yields the asymptotic distribution of the sample spiked eigenvalues associated with mixture models. We also explore the application of these findings in detecting mean heterogeneity of data matrices.