Concentration of measure on spheres and related manifolds

TL;DR

This paper explores advanced concentration of measure phenomena on spheres and manifolds, employing log-Sobolev inequalities to derive bounds for various functions and random matrices, extending classical results to higher orders.

Contribution

It introduces generalized log-Sobolev inequalities for higher order concentration on spheres and measures, providing new bounds for non-Lipschitz functions and random matrices.

Findings

Sudakov-type concentration results established

Higher order concentration bounds derived for $ ext{LS}_q$-measures

Concentration bounds for symmetric functions related to Edgeworth expansions

Abstract

We study various generalizations of concentration of measure on the unit sphere, in particular by means of log-Sobolev inequalities. First, we show Sudakov-type concentration results and local semicircular laws for weighted random matrices. A further branch addresses higher order concentration (i.\,e., concentration for non-Lipschitz functions which have bounded derivatives of higher order) for $\ell_p^n$-spheres. This is based on a type of generalized log-Sobolev inqualities referred to as $\mathrm{LS}_q$-inequalities. More generally, we prove higher order concentration bounds for probability measures on $\mathbb{R}^n$ which satisfy an $\mathrm{LS}_q$-inequality. Finally, we derive concentration bounds for sequences of smooth symmetric functions on the Euclidean sphere which are closely related to Edgeworth-type expansions.