Upper bounds for singular subspaces under unitarily invariant norms with applications to high-dimensional covariance matrix estimation

Abstract

UDC 519.21, 517.98, 512.64

In [J.  Cape, et al., Ann. Statist., 47, Article 2405 (2019)], the authors established perturbation bounds for the $\ell_{2 \to \infty}$-norm of singular subspaces and applied this framework to high-dimensional covariance matrix estimation under multivariate normal distributions. We establish perturbation bounds for unitarily invariant norms and apply these bounds to multivariate normal and sub-Gaussian random vectors. Combined with our upper bounds, these results enable us to get the improved estimation of high-dimensional covariance matrices. Finally, we perform simulation experiments demonstrating the  improved scaling behavior as compared to Cape, et al.'s bounds in the presented experiments.