Signal Plus Noise Type Matrix in Random Matrix Theory: A Review

ABSTRACT

Signal‐plus‐noise type matrices provide a natural extension of classical sample covariance models and arise widely in high‐dimensional statistics, signal processing, wireless communications, and machine learning. This review surveys recent advances in the spectral theory of such matrices from both global and local perspectives. We first introduce the basic signal‐plus‐noise and generalized signal‐plus‐noise models, emphasizing the role of deterministic signal components and heterogeneous noise structures. We then review the limiting spectral distribution and its characterization through Stieltjes transform equations, together with related analytic properties such as density regularity and support determination. Building on these first‐order results, we summarize spectral separation and exact separation phenomena, which describe how gaps in the limiting spectrum determine the location and number of sample eigenvalues. We further discuss edge behavior of extreme eigenvalues, including Tracy–Widom law, as well as the asymptotic theory of spiked eigenvalues and eigenvectors under general noise environments. In addition, the review covers recent progress on linear spectral statistics and their central limit theorems, highlighting the impact of deterministic signals on second‐order fluctuations. Overall, this survey provides a unified overview of the main spectral phenomena of signal‐plus‐noise type matrices and illustrates their theoretical significance and practical value in high‐dimensional inference, wireless communications, and modern machine learning problems such as noisy manifold learning and kernel‐based sensor fusion.