Minimax perturbation bounds of the low-rank matrix under Ky Fan norm

This paper considers the minimax perturbation bounds of the low-rank matrix under Ky Fan norm. We first explore the upper bounds via the best rank- $r$ approximation ${\hat{A}}_r$ of the observation matrix $\hat{A}$. Next, the lower bounds are established by constructing special matrix groups to show the upper bounds are tight on the low-rank matrix estimation error. In addition, we derive the rate-optimal perturbation bounds for the left and right singular subspaces under Ky Fan norm $\sin \mathrm{\Theta }$ distance. Finally, some simulations have been carried out to support our theories.