Covering Numbers for Semicontinuous Functions

Abstract

Considering the metric space of extended real-valued lower semicontinuous functions under the epi-distance, the paper gives an upper bound on the covering numbers of bounded subsets of such functions. No assumptions about continuity, smoothness, variation, and even niteness of the functions are needed. The bound is shown to be nearly sharp through the construction of a set of functions with covering numbers deviating from the upper bound only by a logarithmic factor. The analogy between lower and upper semicontinuous functions implies that identical covering numbers hold for bounded sets of the latter class of functions as well, but now under the hypo-distance metric.

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Document Details

Document Type
Technical Report
Publication Date
Apr 29, 2016
Accession Number
AD1016659

Entities

People

  • Johannes Ø. Røyset

Organizations

  • Naval Postgraduate School

Tags

Communities of Interest

  • Autonomy

DTIC Thesaurus Topics

  • Analytic Functions
  • Convergence
  • Coverings
  • Differential Equations
  • Distribution Functions
  • Equations
  • Hilbert Space
  • Inequalities
  • Information Theory
  • Machine Learning
  • Numbers
  • Operations Research
  • Partial Differential Equations
  • Real Numbers
  • Theorems
  • Topology
  • Weak Convergence

Fields of Study

  • Mathematics

Readers

  • Approximation Theory.
  • Mathematical Modeling and Probability Theory.

Technology Areas

  • Space