Matrices with Gaussian noise: optimal estimates for singular subspace perturbation

The Davis--Kahan--Wedin $\sin \Theta$ theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic version of the Davis--Kahan--Wedin $\sin \Theta$ theorem when the perturbation is a Gaussian random matrix. Under certain structural assumptions, we obtain an optimal bound that significantly improves upon the classic Davis--Kahan--Wedin $\sin \Theta$ theorem. One of our key tools is a new perturbation bound for the singular values, which may be of independent interest.