Matrices with Gaussian noise: optimal estimates for singular subspace perturbation
This addresses a fundamental issue in matrix perturbation theory for researchers in statistics and machine learning, offering a more precise analysis for random noise scenarios.
The paper tackles the problem of how singular subspaces of a matrix change under Gaussian random perturbations, proving a stochastic version of the Davis-Kahan-Wedin theorem that provides an optimal bound, significantly improving upon the classic worst-case result.
The Davis-Kahan-Wedin $\sin Θ$ 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 $\sin Θ$ 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 $\sin Θ$ theorem. One of our key tools is a new perturbation bound for the singular values, which may be of independent interest.