  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.
