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

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 ${\Sigma }_n{\Sigma }_n^{\ast }$ and $n^{-1}{X}_n{D}_n{X}_n^{\ast }$ 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 ${\Sigma }_n{\Sigma }_n^{\ast }$ 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 $\R - \support(μ)$ are studied. The fluctuations of the largest outliers of ${\Sigma }_n{\Sigma }_n^{\ast }$ are also analyzed. The results find applications in the fields of signal processing and radio communications.

35 pages, 2 figures