New matrix perturbation bounds with relative norm: Perturbation of eigenspaces
Phuc Tran, Van Vu
TL;DR
The paper introduces a relative-norm framework that refines classical matrix perturbation bounds by measuring how noise interacts with the signal’s leading eigenspaces. A novel combinatorial contour-expansion method is developed to bound perturbations of eigenspaces, replacing the reliance on the spectral norm with relative-norm quantities and enabling sharper results, particularly when the noise is random. The authors derive deterministic bounds for general S, low-rank, and rectangular cases, and demonstrate substantial improvements in random-noise settings, with concrete applications to spiked covariance models, matrix completion, and privacy. The work combines contour-integral representations, combinatorial graph profiles, and probabilistic moment-analysis to obtain near-optimal perturbation rates under milder gap conditions, offering a robust, broadly applicable framework for matrix functionals beyond eigenprojections.
Abstract
Matrix perturbation bounds (such as Weyl and Davis-Kahan) are used abundantly in many areas of mathematics and data science. Many bounds (such as the above two) involve the spectral norm of the noise matrix and are sharp in worst case analysis. In order to refine these classical bounds, we introduce a new parameter, which we refer to as the relative norm. This parameter measures the strength of the action of the noise matrix on the relevant eigenvectors of the ground matrix. It has turned out that in a number of situations, we can use the relative norm as a replacement for the spectral norm. This has led to a number of notable improvements under certain sets of assumptions, which are frequently met in practice. For instance, our new results apply very well in the case when the noise matrix is random. For the purpose of our study, we introduce a new method of analysis, which combines the classical contour integral argument with new (combinatorial) ideas. This method is robust and of independent interest. In the current paper, we focus on the perturbation of eigenspaces (Davis-Kahan type results). Perturbation bounds for eigenspaces are essential in statistics and theoretical computer science, and thus deserve a special treatment. Furthermore, this will lay the ground for the more technical treatment of general matrix functionals, which appears in a future paper.