Matrices with Gaussian noise: optimal estimates for singular subspace perturbation

TL;DR

This paper develops an optimal stochastic version of the Davis-Kahan-Wedin theorem for singular subspace perturbation under Gaussian noise, providing sharper bounds than classical worst-case results.

Contribution

It introduces a new probabilistic bound for singular subspace perturbation with Gaussian noise, improving classical deterministic bounds under certain conditions.

Findings

Achieves sharper bounds for singular subspace changes under Gaussian noise.

Introduces a new perturbation bound for singular values.

Provides theoretical insights with potential applications in statistical analysis.

Abstract

The Davis-Kahan-Wedin $\sin \Theta$ theorem describes how the singular subspaces of a matrix change when subjected to a small perturbation. This classic result is sharp in the worst case scenario. In this paper, we prove a stochastic version of the Davis-Kahan-Wedin $\sin \Theta$ theorem when the perturbation is a Gaussian random matrix. Under certain structural assumptions, we obtain an optimal bound that significantly improves upon the classic Davis-Kahan-Wedin $\sin \Theta$ theorem. One of our key tools is a new perturbation bound for the singular values, which may be of independent interest.