An ${\ell }_{\infty }$ Eigenvector Perturbation Bound and Its Application to Robust Covariance Estimation

Abstract

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 $\tilde{A}=A+E$, 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.

48 pages, 5 figures

References in corpus (3)

• Asymptotics of Empirical Eigen-structure for Ultra-high Dimensional Spiked Covariance Model
• Robust Estimation of High-Dimensional Mean Regression
• Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics

Cited by in corpus (6)

• Near-optimal bounds for phase synchronization
• Estimation of the covariance structure of heavy-tailed distributions
• Principal component analysis for big data
• Eigenvector Under Random Perturbation: A Nonasymptotic Rayleigh-Schrödinger Theory
• A Schatten-$q$ Low-rank Matrix Perturbation Analysis via Perturbation Projection Error Bound
• Robust Estimation of Covariance Matrices: Adversarial Contamination and Beyond