Matrices With Gaussian Noise: Optimal Estimates for Singular Subspace Perturbation
Sean O’Rourke, Van Vu, Ke Wang
Abstract
The Davis–Kahan–Wedin <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sin \Theta $ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sin \Theta $ </tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sin \Theta $ </tex-math></inline-formula> theorem. One of our key tools is a new perturbation bound for the singular values, which may be of independent interest.