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 $s in\ Theta$ 理论解说矩阵的单子空间在受到小扰动时是如何变化的。 这个经典结果在最坏的情况下是尖锐的。 在本文中, 当扰动是一个高斯随机矩阵时, 我们证明了 Davis- Kahan-Wedin $sin\ Theta$sem 理论的随机版本。 根据某些结构假设, 我们得到了一个最佳的界限, 大大改进了经典的 Davis- Kahan-Wedin $\ Theta$ 理论。 我们的关键工具之一是对单值的新扰动约束, 这可能具有独立的兴趣 。