A comparison principle for functions of a uniformly random subspace
  This note demonstrates that it is possible to bound the expectation of an arbitrary norm of a random matrix drawn from the Stiefel manifold in terms of the expected norm of a standard Gaussian matrix with the same dimensions. A related comparison holds for any convex function of a random matrix drawn from the Stiefel manifold. For certain norms, a reversed inequality is also valid. 
Joel A. Tropp
math.PR math.MG math.ST stat.TH