The Empirical Spectral Distribution of i.i.d. Random Matrices with Random Perturbations

TL;DR

This paper studies the spectral properties of large i.i.d. random matrices with random perturbations, showing convergence to the circular law and eigenvalue outlier behavior, and applies these results to eigenvector approximation complexity.

Contribution

It extends spectral analysis from deterministic to random perturbations and establishes the first lower bound for eigenvector approximation of asymmetric matrices.

Findings

Eigenvalue outliers converge to perturbation eigenvalues

ESD converges to the circular law

Eigenvector alignment holds for specific perturbations

Abstract

A large i.i.d. random matrix with deterministic low-rank perturbation has been extensively studied, particularly in the aspects of the ESD (Empirical Spectral Distribution) and the outliers of eigenvalues. In this work, we investigate the analogous scenario where the perturbation is random and extend the previous results from the deterministic perturbation to the random case. Specifically, we consider an i.i.d. matrix with random perturbation, $\mathbf{M}$. Our results show that: (i) the eigenvalue outliers of $\mathbf{M}$ converge to the eigenvalues of its perturbation; (ii) the ESD of $\mathbf{M}$ converges to the circular law; (iii) the eigenvector alignment holds for specific perturbations. As an application of the above random matrices, we present the first optimal query complexity lower bound for approximating the top eigenvector of asymmetric matrices. In the inverse polynomial accuracy regime, the complexity matches the upper bounds that can be obtained via the power method. As far as we know, it is the first lower bound for approximating the eigenvector of an asymmetric matrix.