New matrix perturbation bounds via combinatorial expansion I: Perturbation of eigenspaces

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.