Analysis of singular subspaces under random perturbations

TL;DR

This paper extends classical perturbation theorems to general unitarily invariant norms, providing detailed bounds on singular vectors and subspaces under Gaussian noise, with applications to Gaussian mixture models and submatrix localization.

Contribution

It generalizes the Davis-Kahan-Wedin theorem to all unitarily invariant norms and offers new bounds for singular vectors and subspaces in noisy matrix models.

Findings

Extended Davis-Kahan-Wedin theorem to unitarily invariant norms

Derived $ ext{ell}_ ext{infty}$ bounds for singular vectors

Applied results to Gaussian mixture and submatrix localization

Abstract

We present a comprehensive analysis of singular vector and singular subspace perturbations in the signal-plus-noise matrix model with random Gaussian noise. Assuming a low-rank signal matrix, we extend the Davis-Kahan-Wedin theorem in a fully generalized manner, applicable to any unitarily invariant matrix norm, building on previous results by O'Rourke, Vu, and the author. Our analysis provides fine-grained insights, including $\ell_\infty$ bounds for singular vectors, $\ell_{2, \infty}$ bounds for singular subspaces, and results for linear and bilinear functions of singular vectors. Additionally, we derive $\ell_{2,\infty}$ bounds on perturbed singular vectors, taking into account the weighting by their corresponding singular values. Finally, we explore practical implications of these results in the Gaussian mixture model and the submatrix localization problem.