The outliers among the singular values of large rectangular random matrices with additive fixed rank deformation

TL;DR

This paper analyzes the behavior of outlier eigenvalues in large random matrices with fixed rank deformations, revealing their locations and fluctuations, with implications for signal processing and communications.

Contribution

It provides a detailed study of outlier eigenvalues in large deformed random matrices, including their existence, locations, and fluctuation behavior, extending previous spectral analysis results.

Findings

Finitely many eigenvalues can stay outside the limiting spectral measure.

Locations of outliers are characterized in relation to the spectral measure.

Fluctuations of the largest outliers are analyzed.

Abstract

Consider the matrix $\Sigma_n = n^{-1/2} X_n D_n^{1/2} + P_n$ where the matrix $X_n \in \C^{N\times n}$ has Gaussian standard independent elements, $D_n$ is a deterministic diagonal nonnegative matrix, and $P_n$ is a deterministic matrix with fixed rank. Under some known conditions, the spectral measures of $\Sigma_n \Sigma_n^*$ and $n^{-1} X_n D_n X_n^*$ both converge towards a compactly supported probability measure $\mu$ as $N,n\to\infty$ with $N/n\to c>0$. In this paper, it is proved that finitely many eigenvalues of $\Sigma_n\Sigma_n^*$ may stay away from the support of $\mu$ in the large dimensional regime. The existence and locations of these outliers in any connected component of $\R - \support(\mu)$ are studied. The fluctuations of the largest outliers of $\Sigma_n\Sigma_n^*$ are also analyzed. The results find applications in the fields of signal processing and radio communications.