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.

翻译：我们将Tao关于具有固定扰动的独立同分布矩阵最大特征值的结果推广到随机扰动情形。具体而言，我们考虑$\mathbf{M}=\mathbf{W}+\lambda\mathbf{u}\mathbf{u}^*$的设定，其中$\mathbf{W}$取自Ginibre正交系综，扰动向量$\mathbf{u}$均匀取自$\mathcal{S}^{d-1}$。我们给出了该随机矩阵特征值及最大特征向量的若干渐近性质，这些性质无法从确定性扰动情形直接推得。我们还将结果应用于扩展Max Simchowitz的工作，该工作为对称矩阵特征子空间逼近提供了最优下界。针对任意非对称但可对角化矩阵$\mathbf{M}$的最大特征值所对应的特征向量逼近问题，我们提出了\textit{查询复杂度}下界。我们证明：对于任意$\operatorname{gap}\in (0,1/2]$及足够大的维度$d$，存在满足$\operatorname{gap}(\mathbf{M})=\Omega(\operatorname{gap})$的随机矩阵$\mathbf{M}$，使得若矩阵-向量查询乘积算法能识别出满足$\left\|\hat{\mathbf{v}}-\mathbf{v}_1(\mathbf{M}) \right\|_2^2\le \operatorname{const}\times \operatorname{gap}$的向量$\hat{\mathbf{v}}$，则其至少需要$\mathcal{O}\left(\frac{\log d}{\operatorname{gap}}\right)$次矩阵-向量乘积查询。在逆多项式精度体系（$\epsilon \ge \frac{1}{\operatorname{poly}(d)}$）下，该复杂度与通过幂方法可获得的$\mathcal{O}\left(\frac{\log(d/\epsilon)}{\operatorname{gap}}\right)$上界相匹配。据我们所知，这是首次针对非对称矩阵特征向量计算提出的下界，该问题较对称情形更为复杂。