62H30 Classification and discrimination; cluster analysis [See also 68T10]
Refine
Document Type
- Preprint (5)
- Doctoral Thesis (1)
Has Fulltext
- yes (6)
Is part of the Bibliography
- no (6)
Keywords
- Maßtheorie (6)
- Quantisierung (4)
- optimal quantization (4)
- Iteriertes Funktionensystem (3)
- Entropie (2)
- Fraktale Dimension (2)
- self-similar probabilities (2)
- Asymptotic quantization theory (1)
- Cantor-Menge (1)
- Companding (1)
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 $\alpha$
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.
Optimal quantization for the one-dimensional uniform distribution with Rényi -α-entropy constraints
(2009)
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).
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.
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.