A Schatten-$q$ Low-rank Matrix Perturbation Analysis via Perturbation Projection Error Bound

TL;DR

This paper provides a new perturbation projection error bound for Schatten-$q$ low-rank matrix estimation, establishing tight bounds and a sin$ heta$ bound for singular subspace perturbation, with validation through simulations.

Contribution

It introduces a novel perturbation projection error bound for Schatten-$q$ low-rank matrices and develops a user-friendly sin$ heta$ bound, enhancing understanding of matrix perturbation effects.

Findings

Established a tight perturbation bound for low-rank matrix estimation.

Developed a new sin$ heta$ bound for singular subspace perturbation.

Validated the theoretical results through simulation comparisons.

Abstract

This paper studies the Schatten-$q$ error of low-rank matrix estimation by singular value decomposition under perturbation. We specifically establish a perturbation bound on the low-rank matrix estimation via a perturbation projection error bound. Then, we establish lower bounds to justify the tightness of the upper bound on the low-rank matrix estimation error. We further develop a user-friendly sin$\Theta$ bound for singular subspace perturbation based on the matrix perturbation projection error bound. Finally, we demonstrate the advantage of our results over the ones in the literature by simulation.