@unpublished{KreitmeierLinder2011, author = {Kreitmeier, Wolfgang and Linder, Tamas}, title = {Entropy Density and Mismatch in High-Rate Scalar Quantization with R{\´e}nyi Entropy Constraint}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-26132}, year = {2011}, abstract = {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results to general values of the entropy order parameter \$\alpha\$}, subject = {Maßtheorie}, language = {de} } @unpublished{Kreitmeier2009, author = {Kreitmeier, Wolfgang}, title = {Error bounds for high-resolution quantization with R{\´e}nyi - \&\#945; - entropy constraints}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-16647}, year = {2009}, abstract = {We consider the problem of optimal quantization with norm exponent r > 0 for Borel probabilities on Rd under constrained R{\´e}nyi-\&\#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exact decay rate is determined for all \&\#945; \&\#8712; [0, \&\#8734;].}, subject = {Maßtheorie}, language = {en} } @unpublished{KreitmeierLinder2011, author = {Kreitmeier, Wolfgang and Linder, Tamas}, title = {High-Resolution Scalar Quantization with R{\´e}nyi Entropy Constraint}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-23787}, year = {2011}, abstract = {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.}, subject = {Maßtheorie}, language = {de} } @unpublished{Kreitmeier2009, author = {Kreitmeier, Wolfgang}, title = {Optimal quantization for the one-dimensional uniform distribution with R{\´e}nyi -α-entropy constraints}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-16983}, year = {2009}, abstract = {We establish the optimal quantization problem for probabilities under constrained R{\´e}nyi-α-entropy of the quantizers. We determine the optimal quantizers and the optimal quantization error of one-dimensional uniform distributions including the known special cases α = 0 (restricted codebook size) and α = 1 (restricted Shannon entropy).}, subject = {Maßtheorie}, language = {de} } @unpublished{Kreitmeier2011, author = {Kreitmeier, Wolfgang}, title = {Optimal vector quantization in terms of Wasserstein distance}, url = {http://nbn-resolving.de/urn:nbn:de:bvb:739-opus-22502}, year = {2011}, abstract = {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.}, subject = {Maßtheorie}, language = {en} }