Metric entropy of homogeneous spaces and Finsler geometry of classical Lie groups

TL;DR

This paper provides asymptotic estimates for the metric entropy of homogeneous spaces of classical Lie groups, with applications to noncommutative probability and operator algebras, and characterizes geodesics in these groups.

Contribution

It generalizes previous results on Grassmann manifolds by estimating covering numbers for a broad class of homogeneous spaces with natural metrics.

Findings

Asymptotic estimates for covering numbers of homogeneous spaces of unitary and orthogonal groups.

Characterization of geodesics in $U(n)$ and $SO(m)$ for non-Riemannian metrics.

Applications to noncommutative probability and operator algebras.

Abstract

For a (compact) subset $K$ of a metric space and $\varepsilon > 0$, the {\em covering number} $N(K , \varepsilon )$ is defined as the smallest number of balls of radius $\varepsilon$ whose union covers $K$. Knowledge of the {\em metric entropy}, i.e., the asymptotic behaviour of covering numbers for (families of) metric spaces is important in many areas of mathematics (geometry, functional analysis, probability, coding theory, to name a few). In this paper we give asymptotically correct estimates for covering numbers for a large class of homogeneous spaces of unitary (or orthogonal) groups with respect to some natural metrics, most notably the one induced by the operator norm. This generalizes earlier author's results concerning covering numbers of Grassmann manifolds; the generalization is motivated by applications to noncommutative probability and operator algebras. In the process we give a characterization of geodesics in $U(n)$ (or $SO(m)$) for a class of non-Riemannian metric structures.