Higher order concentration on Stiefel and Grassmann manifolds

TL;DR

This paper establishes higher order concentration inequalities for functions on Stiefel and Grassmann manifolds, extending previous results on spheres using logarithmic Sobolev techniques, with applications to subspace distances and Hanson–Wright inequalities.

Contribution

It introduces higher order concentration bounds on Stiefel and Grassmann manifolds, expanding the scope of concentration inequalities beyond spheres using advanced mathematical techniques.

Findings

Derived higher order concentration bounds for manifold functions

Extended Hanson–Wright inequalities to Stiefel manifolds

Provided new concentration bounds for subspace distance functions

Abstract

We prove higher order concentration bounds for functions on Stiefel and Grassmann manifolds equipped with the uniform distribution. This partially extends previous work for functions on the unit sphere. Technically, our results are based on logarithmic Sobolev techniques for the uniform measures on the manifolds. Applications include Hanson--Wright type inequalities for Stiefel manifolds and concentration bounds for certain distance functions between subspaces of $\mathbb{R}^n$.