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.