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.

翻译：本文研究了以单值分解在扰动下进行低位矩阵估计的Schatten-$q$错误。 我们专门通过扰动预测错误对低位矩阵估计进行约束。 然后, 我们设定了较低的界限, 以证明低位矩阵估计错误上层约束的紧性。 我们进一步开发了一个方便用户的sin$\Theta$, 以基于矩阵扰动预测错误进行单次子空间扰动。 最后, 我们通过模拟展示了我们的结果比文献中的结果的优势 。