We revisit the Wyner-Ahlswede-Körner network, focusing especially on the converse part of the dispersion analysis, which is known to be challenging. Using the functional-entropic duality and the reverse hypercontractivity of the transposition semigroup, we lower bound the error probability for each joint type. Then by averaging the error probability over types, we lower bound the c-dispersion (which characterizes the second-order behavior of the weighted sum of the rates of the two compressors when a nonvanishing error probability is small) as the variance of the gradient of $\inf _{ {P}_{ {U}| {X}}}\{{ { cH}}( {Y}| {U})+ {I}( {U}; {X})\}$ with respect to $ {Q}_{{{ XY}}}$ , the per-letter side information and source distribution. In comparison, using standard achievability arguments based on the method of types, we upper-bound the c-dispersion as the variance of $ {c}\imath _{ {Y}| {U}}( {Y}| {U})+\imath _{ {U}; {X}}( {U}; {X})$ , which improves the existing upper bounds but has a gap to the aforementioned lower bound. Our converse analysis should be immediately extendable to other distributed source-type problems, such as distributed source coding, common randomness generation, and hypothesis testing with communication constraints. We further present improved bounds for the general image-size problem via our semigroup technique.

中文翻译：

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Wyner-Ahlswede-Körner 网络的色散界通过类型的半群方法

我们重新审视了 Wyner-Ahlswede-Körner 网络，特别关注分散分析的逆向部分，这是众所周知的具有挑战性的部分。使用转置半群的功能-熵对偶性和反向超收缩性，我们降低了每个关节类型的错误概率。然后通过对类型的错误概率求平均，我们将 c 离散度（它表征了当非零错误概率很小时两个压缩器的加权和的二阶行为）作为梯度的方差 $\inf _{ {P}_{ {U}| {X}}}\{{ { cH}}( {Y}| {U})+ {I}( {U}; {X})\}$ 关于 $ {Q}_{{{ XY}}}$ ，每个字母的辅助信息和源分布。相比之下，使用基于类型方法的标准可实现性参数，我们将 c 离散度上限设置为 $ {c}\imath _{ {Y}| {U}}( {Y}| {U})+\imath _{ {U}; {X}}( {U}; {X})$，这改进了现有的上限，但与上述下限有差距。我们的逆向分析应该可以立即扩展到其他分布式源类型问题，例如分布式源编码、公共随机性生成和具有通信约束的假设检验。我们通过我们的半群技术进一步提出了一般图像大小问题的改进边界。