Entropy of Convex Functions on Rd View Full Text


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Article Info

DATE

2017-08-17

AUTHORS

Fuchang Gao, Jon A. Wellner

ABSTRACT

Let Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document} be a bounded closed convex set in Rd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^d$$\end{document} with nonempty interior, and let Cr(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal C}_r(\varOmega )$$\end{document} be the class of convex functions on Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document} with Lr\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^r$$\end{document}-norm bounded by 1. We obtain sharp estimates of the ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}-entropy of Cr(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal C}_r(\varOmega )$$\end{document} under Lp(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p(\varOmega )$$\end{document} metrics, 1≤pdrd+(d-1)r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>\frac{dr}{d+(d-1)r}$$\end{document} is attained by the closed unit ball. While a general convex body can be approximated by inscribed polytopes, the entropy rate does not carry over to the limiting body. Our results have applications to questions concerning rates of convergence of nonparametric estimators of high-dimensional shape-constrained functions. More... »

PAGES

565-592

References to SciGraph publications

  • 1996. Weak Convergence in WEAK CONVERGENCE AND EMPIRICAL PROCESSES
  • 1983. Approximation dans les espaces métriques et théorie de l'estimation in PROBABILITY THEORY AND RELATED FIELDS
  • 2009-04-22. Kolmogorov Entropy for Classes of Convex Functions in CONSTRUCTIVE APPROXIMATION
  • 1998-04. Extremal Problems for Geometric Hypergraphs in DISCRETE & COMPUTATIONAL GEOMETRY
  • 1985-06. On triangulations of the convex hull ofn points in COMBINATORICA
  • 1984. A course on empirical processes in ÉCOLE D'ÉTÉ DE PROBABILITÉS DE SAINT-FLOUR XII - 1982
  • 2009-07-05. On the rate of convergence of the maximum likelihood estimator of a k-monotone density in MATHEMATICS
  • 2015-03-07. Covering Numbers of Lp-Balls of Convex Functions and Sets in CONSTRUCTIVE APPROXIMATION
  • 2002. Lectures on Discrete Geometry in NONE
  • 1993-03. Rates of convergence for minimum contrast estimators in PROBABILITY THEORY AND RELATED FIELDS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00365-017-9387-1

    DOI

    http://dx.doi.org/10.1007/s00365-017-9387-1

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1091233467

    PUBMED

    https://www.ncbi.nlm.nih.gov/pubmed/29658960


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