Suppose we have a column-orthogonal matrix $\mathbf {U}\in\mathbb{R}^{n\times p}$, satisfying $\mathbf {U}^{\top}\mathbf {U}=\mathbf {I}_p$. We select $m<n$ rows of $\mathbf {U}$ randomly and get $\mathbf {U}_{sub}\in\mathbb{R}^{m\times p}.$ Now I want to investigate the smallest (which can also obtained from the biggest) singular value of $\mathbf {U}_{sub}$.
A simple case: Suppose $\mathbf {U}$ is uniform w.r.t Haar measure, can we get the concentration inequality of the smallest singular value of $\mathbf {U}_{sub}$?
Further: If we don't suggest specific distribution on $\mathbf {U}$, can we get the result?
I appreciate any reference that may help to solve this problem.