Davis–Kahan Theorem under a moderate gap condition

The classical Davis–Kahan theorem provides an efficient bound on the perturbation of eigenspaces of a matrix under a large (eigenvalue) gap condition. In this paper, we consider the case when the gap is moderate. By using a bootstrapping argument, we obtain a new bound which is efficient when the perturbation matrix is uncorrelated to the ground matrix. We believe that this bound is sharp up to a logarithmic term.