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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.
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 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).
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.
Securitization Theory has been applied and advanced continuously since the publication of the seminal work “Security – A New Framework for Analysis” by Buzan et al. in 1998. Various extensions, clarifications and definitions have been added over the years. Ontological and epistemological debates as well as debates about the normativity of the concept have taken place, furthering the approach incrementally and adapting it to new empirical cases. This paper aims at contributing to the improvement of the still useful framework in a more general way by amending it with well-established findings from another discipline: Psychology. The exploratory article will point out what elements of Securitization Theory might benefit most from incorporating insights from Psychology and in which ways they might change our understanding of the phenomenon. Some well-studied phenomena in the field of (Social) Psychology, it is argued here, play an important role for the construction and perception of security threats and the acceptance of the audience to grant the executive branch extraordinary measures to counter these threats: availability heuristic, loss-aversion and social identity theory are central psychological concepts that can help us to better understand how securitization works, and in which situations securitizing moves have great or little chances to reverberate. The empirical cases of the 9/11 and Paris terror attacks will serve to illustrate the potential of this approach, allowing for variances in key factors, among them: (point in) time, system of government and ideological orientation. As a hypotheses-generating pilot study, the paper will conclude by discussing further research possibilities in the field of Securitization.
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.
Let <i>d</i> ≥ 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 ω 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 ω, 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 ω 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.