A comparison principle for functions of a uniformly random subspace

TL;DR

This paper establishes bounds on the expected norm of matrices from the Stiefel manifold by comparing them to Gaussian matrices, providing a useful comparison principle for functions of such matrices.

Contribution

It introduces a comparison principle that relates expectations of functions of Stiefel manifold matrices to those of Gaussian matrices, including reversed inequalities for specific norms.

Findings

Bound on expected norm of Stiefel matrices in terms of Gaussian matrices

Comparison principle for convex functions of Stiefel matrices

Reversed inequalities for certain norms

Abstract

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.