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