The eigenvalues and eigenvectors of nonnormal matrices can be unstable under perturbations of their entries. This renders an obstacle to the analysis of numerical algorithms for non-Hermitian eigenvalue problems. A recent technique to handle this issue is pseudospectral shattering [BGVKS23], showing that adding a random perturbation to any matrix has a regularizing effect on the stability of the eigenvalues and eigenvectors. Prior work has analyzed the regularizing effect of dense Gaussian perturbations, where independent noise is added to every entry of a given matrix [BVKS20, BGVKS23, BKMS21, JSS21]. We show that the same effect can be achieved by adding a sparse random perturbation. In particular, we show that given any $n\times n$ matrix $M$ of polynomially bounded norm: (a) perturbing $O(n\log^2(n))$ random entries of $M$ by adding i.i.d. complex Gaussians yields $\log\kappa_V(A)=O(\text{poly}\log(n))$ and $\log (1/\eta(A))=O(\text{poly}\log(n))$ with high probability; (b) perturbing $O(n^{1+\alpha})$ random entries of $M$ for any constant $\alpha>0$ yields $\log\kappa_V(A)=O_\alpha(\log(n))$ and $\log(1/\eta(A))=O_\alpha(\log(n))$ with high probability. Here, $\kappa_V(A)$ denotes the condition number of the eigenvectors of the perturbed matrix $A$ and $\eta(A)$ denotes its minimum eigenvalue gap. A key mechanism of the proof is to reduce the study of $\kappa_V(A)$ to control of the pseudospectral area and minimum eigenvalue gap of $A$, which are further reduced to estimates on the least two singular values of shifts of $A$. We obtain the required least singular value estimates via a streamlining of an argument of Tao and Vu [TV07] specialized to the case of sparse complex Gaussian perturbations.

翻译：非正规矩阵的特征值和特征向量在其元素受扰动时可能不稳定。这给非厄米特征值问题的数值算法分析带来了障碍。处理此问题的一项近期技术是伪谱破碎[BGVKS23]，该技术表明，对任意矩阵添加随机扰动会对特征值和特征向量的稳定性产生正则化效应。先前的工作分析了稠密高斯扰动的正则化效应，即向给定矩阵的每个元素添加独立噪声[BVKS20, BGVKS23, BKMS21, JSS21]。我们证明，通过添加稀疏随机扰动可实现相同的效应。具体而言，我们证明对于任意范数多项式有界的 $n\times n$ 矩阵 $M$：(a) 通过添加独立同分布的复高斯噪声扰动 $M$ 的 $O(n\log^2(n))$ 个随机元素，以高概率使得 $\log\kappa_V(A)=O(\text{poly}\log(n))$ 且 $\log (1/\eta(A))=O(\text{poly}\log(n))$；(b) 对于任意常数 $\alpha>0$，扰动 $M$ 的 $O(n^{1+\alpha})$ 个随机元素，以高概率使得 $\log\kappa_V(A)=O_\alpha(\log(n))$ 且 $\log(1/\eta(A))=O_\alpha(\log(n))$。此处，$\kappa_V(A)$ 表示扰动后矩阵 $A$ 的特征向量条件数，$\eta(A)$ 表示其最小特征值间隙。证明的一个关键机制是将 $\kappa_V(A)$ 的研究转化为对 $A$ 的伪谱面积和最小特征值间隙的控制，这进一步转化为对 $A$ 的平移矩阵的最小两个奇异值的估计。我们通过简化 Tao 和 Vu [TV07] 的论证（专门针对稀疏复高斯扰动情形）获得了所需的最小奇异值估计。