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Dimensions of Metric Spaces

  • Dan A. Simovici
  • Chabane Djeraba
Chapter
Part of the Advanced Information and Knowledge Processing book series (AI&KP)

Abstract

Subsets of \({\mathbb R}^n\) may have “intrinsic” dimensions that are much lower than \(n\). Consider, for example, two distinct vectors \(\mathbf {a},\mathbf {b}\in {\mathbb R}^n\) and the line \(L = \{\mathbf {a}+ t \mathbf {b}\,\mid \,t \in {\mathbb R}\}\). Intuitively, \(L\) has the intrinsic dimensionality \(1\); however, \(L\) is embedded in \({\mathbb R}^n\) and from this point of view is an \(n\)-dimensional object. In this chapter we examine formalisms that lead to the definition of this intrinsic dimensionality.

Keywords

Open Cover Iterative Function System Ultrametric Space Inductive Dimension Outer Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Massachusetts BostonBostonUSA
  2. 2.University Lille 1Laboratoire d’Informatique Fundamentale de LilleVilleneuve d’AscqFrance

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