On Eigenvector Approximation of Diagonalizable Random Matrices with Random Perturbations: Properties and Applications

We extend the result on the top eigenvalue of the i.i.d.\ matrix with fixed perturbations by Tao to random perturbations. In particular, we consider a setup that $\mathbf{M}=\mathbf{W}+\lambda\mathbf{u}\mathbf{u}^*$ with $\mathbf{W}$ drawn from a Ginibre Orthogonal Ensemble and the perturbation $\mathbf{u}$ drawn uniformly from $\mathcal{S}^{d-1}$. We provide several asymptotic properties about the eigenvalues and the top eigenvector of the random matrix, which can not be obtained trivially from the deterministic perturbation case. We also apply our results to extend the work of Max Simchowitz, which provides an optimal lower bound for approximating the eigenspace of a symmetric matrix. We present a \textit{query complexity} lower bound for approximating the eigenvector of any asymmetric but diagonalizable matrix $\mathbf{M}$ corresponding to the largest eigenvalue. We show that for every $\operatorname{gap}\in (0,1/2]$ and large enough dimension $d$, there exists a random matrix $\mathbf{M}$ with $\operatorname{gap}(\mathbf{M})=\Omega(\operatorname{gap})$, such that if a matrix-vector query product algorithm can identity a vector $\hat{\mathbf{v}}$ which satisfies $\left\|\hat{\mathbf{v}}-\mathbf{v}_1(\mathbf{M}) \right\|_2^2\le \operatorname{const}\times \operatorname{gap}$, it needs at least $\mathcal{O}\left(\frac{\log d}{\operatorname{gap}}\right)$ queries of matrix-vector products. In the inverse polynomial accuracy regime where $\epsilon \ge \frac{1}{\operatorname{poly}(d)}$, the complexity matches the upper bounds $\mathcal{O}\left(\frac{\log(d/\epsilon)}{\operatorname{gap}}\right)$, which can be obtained via the power method. As far as we know, it is the first lower bound for computing the eigenvector of an asymmetric matrix, which is far more complicated than in the symmetric case.