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

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.