Metric Entropy of Homogeneous Spaces

Stanisław Szarek

Banach Center Publications (1998)

  • Volume: 43, Issue: 1, page 395-410
  • ISSN: 0137-6934

Abstract

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For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the 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 the author's earlier results concerning covering numbers of Grassmann manifolds; the generalization is motivated by applications to noncommutative probability and operator algebras. The argument uses a characterization of geodesics in U(n) (or SO(m)) for a class of non-Riemannian Finsler metric structures.

How to cite

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Szarek, Stanisław. "Metric Entropy of Homogeneous Spaces." Banach Center Publications 43.1 (1998): 395-410. <http://eudml.org/doc/208860>.

@article{Szarek1998,
abstract = {For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the 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 the author's earlier results concerning covering numbers of Grassmann manifolds; the generalization is motivated by applications to noncommutative probability and operator algebras. The argument uses a characterization of geodesics in U(n) (or SO(m)) for a class of non-Riemannian Finsler metric structures.},
author = {Szarek, Stanisław},
journal = {Banach Center Publications},
keywords = {precompact subset; covering number; metric entropy; Grassmann manifolds; noncommutative probability; geodesics; non-Riemannian Finsler metric structures},
language = {eng},
number = {1},
pages = {395-410},
title = {Metric Entropy of Homogeneous Spaces},
url = {http://eudml.org/doc/208860},
volume = {43},
year = {1998},
}

TY - JOUR
AU - Szarek, Stanisław
TI - Metric Entropy of Homogeneous Spaces
JO - Banach Center Publications
PY - 1998
VL - 43
IS - 1
SP - 395
EP - 410
AB - For a precompact subset K of a metric space and ε > 0, the covering number N(K,ε) is defined as the smallest number of balls of radius ε whose union covers K. Knowledge of the 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 the author's earlier results concerning covering numbers of Grassmann manifolds; the generalization is motivated by applications to noncommutative probability and operator algebras. The argument uses a characterization of geodesics in U(n) (or SO(m)) for a class of non-Riemannian Finsler metric structures.
LA - eng
KW - precompact subset; covering number; metric entropy; Grassmann manifolds; noncommutative probability; geodesics; non-Riemannian Finsler metric structures
UR - http://eudml.org/doc/208860
ER -

References

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