New matrix perturbation bounds with relative norm: Perturbation of eigenspaces

TL;DR

This paper introduces a new relative norm parameter to refine matrix perturbation bounds, especially for eigenspaces, improving analysis in practical scenarios like random noise matrices.

Contribution

The paper proposes a novel relative norm measure and a robust analysis method, enhancing classical perturbation bounds for eigenspaces under practical conditions.

Findings

Improved bounds using the relative norm in certain noise scenarios

Effective application to random noise matrices

New combinatorial analysis method for matrix perturbations

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.