论文标题
$ 1D $ $ \ log $ -concave分布的最佳双量化器：唯一性和劳埃德（Lloyd）
Optimal dual quantizers of $1D$ $\log$-concave distributions: uniqueness and Lloyd like algorithm
论文作者
论文摘要
我们为双重量化建立了Kieffer独特性结果的对应物，其紧凑型概率分布具有$ \ log $ $ -concave密度（也称为非常单峰）：对于此类分布，$ l^r $ toptimal-optimal Dual Dual量化器在每个级别上都是独特的$ n $ nsior nortial nortial Irdization kiltiantive kiltians rigitiation suiltiation kilitiative witter Intion juiltiation ride srimation rigity suiltiation suring wigriative。表现出临界点唯一性失败的非巧妙的单型分布的一个例子。 在二次$ r = 2 $ case中，我们提出了一种算法来计算唯一的最佳双量化器。它提供了Voronoi框架中Lloyd方法〜I算法的对应物。最终，建立了$ l^r $ - 最佳双量化器的半关闭形式，以紧凑的间隔和截断的指数分布建立了电源分布。
We establish for dual quantization the counterpart of Kieffer's uniqueness result for compactly supported one dimensional probability distributions having a $\log$-concave density (also called strongly unimodal): for such distributions, $L^r$-optimal dual quantizers are unique at each level $N$, the optimal grid being the unique critical point of the quantization error. An example of non-strongly unimodal distribution for which uniqueness of critical points fails is exhibited. In the quadratic $r=2$ case, we propose an algorithm to compute the unique optimal dual quantizer. It provides a counterpart of Lloyd's method~I algorithm in a Voronoi framework. Finally semi-closed forms of $L^r$-optimal dual quantizers are established for power distributions on compacts intervals and truncated exponential distributions.
