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

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.

arXiv commentAccepted to Linear Algebra and its Applications

References in corpus (4)

• Minimax risk of matrix denoising by singular value thresholding
• Entrywise Eigenvector Analysis of Random Matrices with Low Expected Rank
• An ${\ell }_{\infty }$ Eigenvector Perturbation Bound and Its Application to Robust Covariance Estimation
• Asymmetry Helps: Eigenvalue and Eigenvector Analyses of Asymmetrically Perturbed Low-Rank Matrices

Cited by in corpus (4)

• Entrywise Estimation of Singular Vectors of Low-Rank Matrices with Heteroskedasticity and Dependence
• A Sharp Blockwise Tensor Perturbation Bound for Orthogonal Iteration
• Recursive Importance Sketching for Rank Constrained Least Squares: Algorithms and High-order Convergence
• An exact $\sin \Theta$ formula for matrix perturbation analysis and its applications