Error bounds for high-resolution quantization with Rényi - α - entropy constraints

  • We consider the problem of optimal quantization with norm exponent r > 0 for Borel probabilities on R<sup>d</sup> under constrained Rényi-&#945;-entropy of the quantizers. If the bound on the entropy becomes large, then sharp asymptotics for the optimal quantization error are well-known in the special cases &#945; = 0 (memory-constrained quantization) and &#945; = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter &#945; &#8712; [1+r/d, &#8734;]. For &#945; &#8712; [0,1+r/d[ we specify the asymptotical order of the optimal quantization error under large entropy bound. The optimal quantization error decays exponentially fast with the entropy bound and the exact decay rate is determined for all &#945; &#8712; [0, &#8734;].

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Metadaten
Author:Wolfgang Kreitmeier
URN:urn:nbn:de:bvb:739-opus-16647
Document Type:Preprint
Language:English
Year of Completion:2009
Date of Publication (online):2009/10/09
Publishing Institution:Universität Passau
Release Date:2009/10/09
Tag:Rényi-&#945;-entropy; Vector quantization; approximation of probabilities; high-resolution quantization
GND Keyword:Entropie <Informationstheorie>; Maßtheorie; Quantisierung <Nachrichtentechnik>; Vektorquantisierung
Source:This is a preprint of an article accepted for publication in Acta Mathematica Hungarica ISSN: 0236-5294 (print version) ISSN: 1588-2632 (electronic version) Copyright (c) by Springer. The original publication is available at www.springerlink.com, see also http://dx.doi.org/10.1007/s10474-010-9079-9
Institutes:Fakultät für Informatik und Mathematik / Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik
MSC-Classification:60-XX PROBABILITY THEORY AND STOCHASTIC PROCESSES (For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX) / 60Exx Distribution theory [See also 62Exx, 62Hxx]
62-XX STATISTICS / 62Exx Distribution theory [See also 60Exx] / 62E17 Approximations to distributions (nonasymptotic)
68-XX COMPUTER SCIENCE (For papers involving machine computations and programs in a specific mathematical area, see Section {04 in that areag 68-00 General reference works (handbooks, dictionaries, bibliographies, etc.) / 68Pxx Theory of data / 68P30 Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) [See also 94Axx]
94-XX INFORMATION AND COMMUNICATION, CIRCUITS / 94Axx Communication, information / 94A17 Measures of information, entropy
94-XX INFORMATION AND COMMUNICATION, CIRCUITS / 94Axx Communication, information / 94A29 Source coding [See also 68P30]
open_access (DINI-Set):open_access
Licence (German):License LogoStandardbedingung laut Einverständniserklärung

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