Matrix perturbation bounds (such as Weyl and Davis-Kahan) are frequently used in many branches of mathematics. Most of the classical results in this area are optimal, in the worst case analysis. However, in modern applications, both the ground and the nose matrices frequently have extra structural properties. For instance, it is often assumed that the ground matrix is essentially low rank, and the nose matrix is random or pseudo-random. We aim to rebuild a part of perturbation theory, adapting to these modern assumptions. The key idea is to exploit the skewness between the leading eigenvectors of the ground matrix and the noise matrix. We will do this by combining the classical contour integration method with combinatorial ideas, resulting in a new machinery, which has a wide range of applications. Our new bounds are optimal under mild assumptions, with direct applications to central problems in many different areas. Among others, we derive a sharp result for the perturbation of a low rank matrix with random perturbation, answering an open question in this area. Next, we derive new, optimal, results concerning covariance estimator of the spiked model, an important model in statistics, bridging two different directions of current research. Finally, and somewhat unexpectedly, we can use our results on the perturbation of eigenspaces to derive new results concerning eigenvalues of deterministic and random matrices. In particular, we obtain new results concerning the outliers in the deformed Wigner model and the least singular value of random matrices with non-zero mean.

翻译：矩阵扰动界（如Weyl界和Davis-Kahan界）在数学的众多分支中被广泛应用。该领域大多数经典结果在最坏情况分析下是最优的。然而，在现代应用中，基础矩阵与噪声矩阵往往具有额外的结构特性。例如，通常假设基础矩阵本质上是低秩的，而噪声矩阵是随机或伪随机的。本文旨在重构扰动理论的一部分，以适应这些现代假设。其核心思想是利用基础矩阵与噪声矩阵主导特征向量之间的偏斜性。我们将通过将经典的围道积分方法与组合思想相结合来实现这一目标，从而构建出一种具有广泛应用前景的新工具。在温和假设下，我们的新界是最优的，并能直接应用于多个不同领域的核心问题。特别地，我们针对低秩矩阵在随机扰动下的扰动问题推导出了一个精确结果，解答了该领域的一个开放性问题。其次，我们针对统计学中的重要模型——尖峰模型，提出了关于协方差估计量的新的最优结果，连接了当前研究的两个不同方向。最后，有些出人意料的是，我们能够利用特征空间扰动的结果，推导出关于确定性与随机矩阵特征值的新结论。具体而言，我们在变形Wigner模型的异常值以及非零均值随机矩阵的最小奇异值方面获得了新的结果。