In this paper, we provide the minimax perturbation bounds for low-rank matrices under unitarily invariant norms. Furthermore, we develop a unilateral \(\sin \Theta \) upper bound for singular subspace perturbation based on a matrix perturbation projection error bound, in contrast to Wedin’s uniform upper bound. Additionally, we define a new kind of unitarily invariant norm, \(\Vert \cdot \Vert _{\varvec{\alpha },k,q}\) , which encompasses several commonly used unitarily invariant norms. Finally, we demonstrate through simulations the superiority of our upper bound over classical results in the literature.
