Asymptotics of Spiked Covariance Model with Random Projection

The spiked covariance model, characterized by a population covariance matrix perturbed by a low-rank matrix, plays a crucial role in data analysis. In this context, the low-rank deformation typically signifies the underlying signal composition, while the extreme eigenvalues and eigenvectors of the sample covariance matrix contain valuable information about the signal. While the spiked covariance model has been extensively studied, its behavior under dimension reduction techniques, such as random projection, remains largely unexplored. These dimension reduction methods are commonly employed to manage the computational complexity associated with high-dimensional data. In this work, we study the behavior of the extreme eigenvalues and eigenvectors of the spiked covariance model with random projection. Specifically, we identify the exact critical threshold for the empirical eigenvalues to be out of the main bulk of the spectrum. Additionally, we determine the asymptotic positions of the isolated eigenvalues, as well as the projections of the isolated eigenvectors. It is quantitatively shown that the signal strength decreases under projection, and the isolated eigenvectors carry the information of the projected signal. Based on the above results, we propose a linear detection method for strong signals and analyze its performance limits. Simulation results validate the accuracy of the theoretical analysis.