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<export-example>
  <doc>
    <id>140</id>
    <completedYear>2011</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2011-04-18</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Optimal vector quantization in terms of Wasserstein distance</title>
    <abstract language="eng">The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the R\'enyi-$\alpha$-entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained $(\alpha = 1)$ and memory-size constrained $(\alpha = 0)$ quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by Gy\"{o}rgy and Linder.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-22502</identifier>
    <identifier type="opus3-id">2250</identifier>
    <note>This is a preprint of an article accepted for publication in the Journal of Multivariate Analysis ISSN 0047-259X. The original publication is available at http://www.elsevier.com/. The digital object identifier (DOI) of the definitive article is 10.1016/j.jmva.2011.04.005.</note>
    <enrichment key="SourceTitle">Journal of Multivariate Analysis. ISSN 0047-259X</enrichment>
    <enrichment key="InvalidVerification">wolfgang.kreitmeier@uni-passau.de</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Transporttheorie</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Quantisierung</value>
    </subject>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Entropie</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Wasserstein distance</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>optimal quantization error</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>codecell convexity</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>R\'enyi-$\alpha$-entropy</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="60B10">Convergence of probability measures</collection>
    <collection role="msc" number="60E05">Distributions: general theory</collection>
    <collection role="msc" number="62E17">Approximations to distributions (nonasymptotic)</collection>
    <collection role="msc" number="68P30">Coding and information theory (compaction, compression, models of communication, encoding schemes, etc.) [See also 94Axx]</collection>
    <collection role="msc" number="94A17">Measures of information, entropy</collection>
    <collection role="msc" number="94A29">Source coding [See also 68P30]</collection>
    <collection role="open_access" number="">open_access</collection>
    <collection role="institutes" number="">Mitarbeiter Lehrstuhl/Einrichtung der Fakultät für Informatik und Mathematik</collection>
    <thesisPublisher>Universität Passau</thesisPublisher>
    <file>https://opus4.kobv.de/opus4-uni-passau/files/140/quant_wass_rev1.pdf</file>
  </doc>
</export-example>
