Date of Award
Fall 2023
Document Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Applied Mathematics
First Advisor
Vu, Van
Abstract
Consider a symmetric, additive perturbation $E$ of a symmetric, low rank matrix $A$. Given access to the perturbed data $\tilde{A}$, an important problem in statistics and numerical analysis is to draw conclusions about the matrix $A$. Of particular importance are the eigenvectors of $A$. Classical eigenvector perturbation theory gives conditions under which the eigenvectors and eigenspaces of $A$ and $\tilde{A}$ are close in the $\ell_2$ norm. In many real-world settings, $E$ is a random matrix. I will demonstrate that the classical perturbation theory falls short when $E$ is random. Furthermore, very little is known about the $\ell_{\infty}$ perturbation of the eigenvectors. In this thesis, I will provide a general $\ell_{\infty}$ perturbation framework. This will yield an $\ell_\infty$ perturbation result that substantially improves existing results for random $E$. The resulting bound, while useful, can fall short in a few key applications. To address this, I will subsequently prove a {\it delocalization} bound that is of independent interest. Using this bound, we can improve the $\ell_{\infty}$ bound for these data science applications.
Recommended Citation
Bhardwaj, Abhinav, "Random Matrix Perturbation: Entry-Wise Analysis" (2023). Yale Graduate School of Arts and Sciences Dissertations. 1213.
https://elischolar.library.yale.edu/gsas_dissertations/1213
