  A basic problem in operator theory is to estimate how a small perturbation
effects the eigenspaces of a self-adjoint compact operator. In this paper, we
prove upper bounds for the subspace distance, taylored for structured random
perturbations. As a main example, we consider the empirical covariance
operator, and show that a sharp bound can be achieved under a relative gap
condition. The proof is based on a novel contraction phenomenon, contrasting
previous spectral perturbation approaches.
