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In his famous paper Gersho stressed that the codecells of optimal quantizers asymptotically make an equal contribution to the distortion of the quantizer. Motivated by this fact, we investigate in this paper quantizers in the scalar case, where each codecell contributes with exactly the same portion to the quantization error. We show that such quantizers of Gersho type - or Gersho quantizers for short - exist for non-atomic scalar distributions. As a main result we prove that Gersho quantizers are asymptotically optimal.
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.
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.
We consider the problem of optimal quantization with norm exponent r > 0 for Borel probabilities on R<sup>d</sup> under constrained Rényi-α-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 α = 0 (memory-constrained quantization) and α = 1 (Shannon-entropy-constrained quantization). In this paper we determine sharp asymptotics for the optimal quantization error under large entropy bound with entropy parameter α ∈ [1+r/d, ∞]. For α ∈ [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 α ∈ [0, ∞].
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.
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.
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.