Optimal estimation for lower bound of the packing number

Lower bound estimation of the packing number plays an important role in matrix denoising. By using the metric entropy of the Grassmannian manifold, Cai, Ma and Wu provide a lower bound estimation of the packing number with the metric based on Frobenius norm, see Cai et al. (2013). In this paper, we extend their result to all metrics based on unitarily invariant norms and then give an example to show the optimality of our estimation. Finally, we discuss a potential application in low-rank matrix denoising.