Non-asymptotic properties of spectral decomposition of large Gram-type matrices and applications

Gram-type matrices and their spectral decomposition are of central importance for numerous problems in statistics, applied mathematics, physics, and machine learning. In this paper, we carefully study the non-asymptotic properties of spectral decomposition of large Gram-type matrices when data are not necessarily independent. Specifically, we derive the exponential tail bounds for the deviation between eigenvectors of the right Gram matrix to their population counterparts as well as the Berry-Esseen type bound for these deviations. We also obtain the non-asymptotic tail bound of the ratio between eigenvalues of the left Gram matrix, namely the sample covariance matrix, and their population counterparts regardless of the size of the data matrix. The documented non-asymptotic properties are further demonstrated in a suite of applications, including the non-asymptotic characterization of the estimated number of latent factors in factor models and relate machine learning problems, the estimation and forecasting of high-dimensional time series, the spectral properties of large sample covariance matrix such as perturbation bounds and inference on the spectral projectors, and low-rank matrix denoising using dependent data.