In statistics and machine learning, people are often interested in the
eigenvectors (or singular vectors) of certain matrices (e.g. covariance
matrices, data matrices, etc). However, those matrices are usually perturbed by
noises or statistical errors, either from random sampling or structural
patterns. One usually employs Davis-Kahan $\sin \theta$ theorem to bound the
difference between the eigenvectors of a matrix $A$ and those of a perturbed
matrix $\widetilde{A} = A + E$, in terms of $\ell_2$ norm. In this paper, we
prove that when $A$ is a low-rank and incoherent matrix, the $\ell_{\infty}$
norm perturbation bound of singular vectors (or eigenvectors in the symmetric
case) is smaller by a factor of $\sqrt{d_1}$ or $\sqrt{d_2}$ for left and right
vectors, where $d_1$ and $d_2$ are the matrix dimensions. The power of this new
perturbation result is shown in robust covariance estimation, particularly when
random variables have heavy tails. There, we propose new robust covariance
estimators and establish their asymptotic properties using the newly developed
perturbation bound. Our theoretical results are verified through extensive
numerical experiments.

该论文针对矩阵扰动中的特征向量进行了研究，证明了当矩阵是低秩和不相干的时候，奇异向量的（或对称情况下的特征向量）l∞范数扰动界限比 l2 范数扰动界限更小一个因子。作者在稳健协方差估计方面提出了新的建模方法，并利用所开发的扰动界限确立其渐近性质。