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  • Kreitmeier, Wolfgang (12)
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Optimal quantization for uniform distributions on Cantor-like sets (2008)
Kreitmeier, Wolfgang
In this paper, the problem of optimal quantization is solved for uniform distributions on some higher dimensional, not necessarily self-similar $N-$adic Cantor-like sets. The optimal codebooks are determined and the optimal quantization error is calculated. The existence of the quantization dimension is characterized and it is shown that the quantization coefficient does not exist. The special case of self-similarity is also discussed. The conditions imposed are a separation property of the distribution and strict monotonicity of the first $N$ quantization error differences. Criteria for these conditions are proved and as special examples modified versions of classical fractal distributions are discussed.
Optimal vector quantization in terms of Wasserstein distance (2011)
Kreitmeier, Wolfgang
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.
Optimal Quantization for Dyadic Homogeneous Cantor Distributions (2005)
Kreitmeier, Wolfgang
For a large class of dyadic homogeneous Cantor distributions in \mathbb{R}, which are not necessarily self-similar, we determine the optimal quantizers, give a characterization for the existence of the quantization dimension, and show the non-existence of the quantization coefficient. The class contains all self-similar dyadic Cantor distributions, with contraction factor less than or equal to \frac{1}{3}. For these distributions we calculate the quantization errors explicitly.
High-Resolution Scalar Quantization with Rényi Entropy Constraint (2011)
Kreitmeier, Wolfgang ; Linder, Tamas
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.
Hausdorff measure of uniform self-similar fractals (2009)
Kreitmeier, Wolfgang
Let <i>d</i> &#8805; 1 be an integer and <i>E</i> a self-similar fractal set, which is the attractor of a uniform contracting iterated function system (UIFS) on R<sup>d</sup>. Denote by <i>D</i> the Hausdorff dimension, by <i>H</i><sup>D</sup><i>(E)</i> the Hausdorff measure and by diam <i>(E)</i> the diameter of <i>E</i>. If the UIFS is parametrised by its contracting factor <i>c</i>, while the set &omega; of fixed points of the UIFS does not depend on <i>c</i>, we will show the existence of a positive constant depending only on &omega;, such that the Hausdorff dimension is smaller than one and <i>H</i><sup>D</sup> = <i>(E)</i> <sup>D</sup> if <i>c</i> is smaller than this constant. We apply our result to modified versions of various classical fractals. Moreover we present a parametrised UIFS where &omega; depends on <i>c</i> and <i>H</i><sup>D</sup> < diam<i>(E)</i><sup>D</sup>, if <i>c</i> is small enough.
Optimal quantization for the one-dimensional uniform distribution with Rényi -α-entropy constraints (2009)
Kreitmeier, Wolfgang
We establish the optimal quantization problem for probabilities under constrained Rényi-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).
Error bounds for high-resolution quantization with Rényi - &#945; - entropy constraints (2009)
Kreitmeier, Wolfgang
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;].
Optimale Quantisierung verallgemeinerter Cantor-Verteilungen (2006)
Kreitmeier, Wolfgang
Für verallgemeinerte Cantor-Verteilungen, die im Eindimensionalen mittels klassischer Wischkonstruktion bzw. in höheren Dimensionen über iterierte Funktionensysteme definiert werden, wird das Problem der optimalen Quantisierung unter bestimmten Voraussetzungen vollständig gelöst. Es werden die optimalen Codebücher bestimmt und Formeln für den optimalen Quantisierungsfehler bewiesen. Im eindimensionalen Fall wird eine Existenzcharakterisierung der Quantisierungsdimension gegeben und unter bestimmten Voraussetzungen die Nichtexistenz des Quantisierungskoeffizienten gezeigt. Auch in höheren Dimensionen wird für die betrachteten Verteilungen bewiesen, dass der Quantisierungskoeffizient, bei existenter Quantisierungsdimension, nicht existiert. Die gewonnenen Resultate werden auf die Gleichverteilungen von modifizierten klassischen fraktalen Mengen, wie das Sierpinski-Dreieck, die Cantormenge und den Cantor-Staub angewandt.
Optimal quantization of probabilities concentrated on small balls (2007)
Kreitmeier, Wolfgang
We consider probability distributions which are uniformly distributed on a disjoint union of balls with equal radius. For small enough radius the optimal quantization error is calculated explicitly in terms of the ball centroids. We apply the results to special self-similar measures.
Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure (2007)
Kreitmeier, Wolfgang
For homogeneous one-dimensional Cantor sets, which are not necessarily self-similar, we show under some restrictions that the Euler exponent equals the quantization dimension of the uniform distribution on these Cantor sets. Moreover for a special sub-class of these sets we present a linkage between the Hausdorff and the Packing measure of these sets and the high-rate asymptotics of the quantization error.
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