An $\ell_{\infty}$ Eigenvector Perturbation Bound and Its Application to Robust Covariance Estimation
Provides a sharper theoretical tool for eigenvector analysis in high-dimensional statistics, with direct application to robust covariance estimation for heavy-tailed data.
The paper proves an ℓ∞ eigenvector perturbation bound for low-rank incoherent matrices, which is tighter than the standard ℓ2 bound by a factor of √d. This bound is applied to robust covariance estimation under heavy-tailed noise, leading to new estimators with established asymptotic properties.
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 $\sin θ$ 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.