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  <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>
  <doc>
    <id>145</id>
    <completedYear>2011</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>deu</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2011-07-08</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="deu">High-Resolution Scalar Quantization with Rényi Entropy Constraint</title>
    <abstract language="eng">We consider optimal scalar quantization with $r$th power distortion and constrained R\'enyi entropy of order $\alpha$. For sources with absolutely continuous distributions the high rate asymptotics of the quantizer distortion has long been known for $\alpha=0$ (fixed-rate quantization) and $\alpha=1$ (entropy-constrained quantization). These results have recently been extended to quantization with R\'enyi entropy constraint of order $\alpha \ge r+1$. Here we consider the more challenging case $\alpha\in [-\infty,0)\cup (0,1)$ and for a large class of absolutely continuous source distributions we determine the sharp asymptotics of the optimal quantization distortion. The achievability proof is based on finding (asymptotically) optimal quantizers via the companding approach, and is thus constructive.</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-23787</identifier>
    <identifier type="opus3-id">2378</identifier>
    <note>This is a preprint of an article accepted for publication in the IEEE Transactions on Information Theory Journal, ISSN: 0018-9448. The original publication is available at http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18</note>
    <enrichment key="SourceTitle">IEEE Transactions on Information Theory Journal, ISSN: 0018-9448</enrichment>
    <enrichment key="InvalidVerification">wolfgang.kreitmeier@uni-passau.de</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <author>Tamas Linder</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</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>Companding</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>high-resolution asymptotics</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>optimal quantization</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Rényi entropy</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="28D20">Entropy and other invariants</collection>
    <collection role="msc" number="41A46">Approximation by arbitrary nonlinear expressions; widths and entropy</collection>
    <collection role="msc" number="62H30">Classification and discrimination; cluster analysis [See also 68T10]</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/145/renyi_scalar_final_doublecolumn.pdf</file>
  </doc>
  <doc>
    <id>160</id>
    <completedYear>2011</completedYear>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>deu</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>preprint</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation>Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada</contributingCorporation>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>2012-03-27</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="deu">Entropy Density and Mismatch in High-Rate Scalar Quantization with Rényi Entropy Constraint</title>
    <abstract language="eng">Properties of scalar quantization with $r$th power distortion and constrained R\'enyi entropy of order $\alpha\in (0,1)$ are investigated. For an asymptotically (high-rate) optimal sequence of quantizers, the contribution to the R\'enyi entropy due to source values in a fixed interval is identified in terms of the "entropy density" of the quantizer sequence. This extends results related to the well-known point density concept in optimal fixed-rate quantization. A dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion. The distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined. This extends Bucklew's fixed-rate ($\alpha=0$) and Gray \emph{et al.}'s variable-rate ($\alpha=1$)mismatch results to general values of the entropy order parameter $\alpha$</abstract>
    <identifier type="urn">urn:nbn:de:bvb:739-opus-26132</identifier>
    <identifier type="opus3-id">2613</identifier>
    <note>This is a preprint of an article accepted for publication in the IEEE Transactions on Information Theory Journal, ISSN: 0018-9448. The original publication is available at http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=18</note>
    <enrichment key="SourceTitle">IEEE Transactions on Information Theory Journal, ISSN: 0018-9448</enrichment>
    <enrichment key="InvalidVerification">wolfgang.kreitmeier@uni-passau.de</enrichment>
    <licence>Standardbedingung laut Einverständniserklärung</licence>
    <author>Wolfgang Kreitmeier</author>
    <author>Tamas Linder</author>
    <subject>
      <language>deu</language>
      <type>swd</type>
      <value>Maßtheorie</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>Asymptotic quantization theory</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>distortion density</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>entropy density</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>quantizer mismatch</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>Rényi-entropy</value>
    </subject>
    <collection role="ddc" number="510">Mathematik</collection>
    <collection role="msc" number="28D20">Entropy and other invariants</collection>
    <collection role="msc" number="41A46">Approximation by arbitrary nonlinear expressions; widths and entropy</collection>
    <collection role="msc" number="62H30">Classification and discrimination; cluster analysis [See also 68T10]</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/160/renyi_mismatch_final_doublecolumn.pdf</file>
  </doc>
</export-example>
