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.

翻译：Davis-Kahan-Wedin $\sin \Theta$ 定理描述了矩阵奇异子空间在受到小扰动时的变化方式。这一经典结果在最坏情况下是精确的。本文证明了当扰动为高斯随机矩阵时，Davis-Kahan-Wedin $\sin \Theta$ 定理的一个随机版本。在特定结构假设下，我们得到了一个显著优于经典 Davis-Kahan-Wedin $\sin \Theta$ 定理的最优界。我们的关键工具之一是奇异值的一个新扰动界，该结果可能具有独立的研究价值。