We study the hypercontractivity ribbon and the $Φ$-ribbon for joint distributions that obey a given independence structure, obtaining tight bounds in some basic regimes. For general independence structures, modeled as a hypergraph whose hyperedges specify mutually independent subcollections of random variables, we provide an explicit inner bound on the $Φ$-ribbon described by a simple convex hull of incidence vectors. We also provide a new multipartite generalization version and a $Φ$-mutual information analogue of the Zhang--Yeung inequality, which implies nontrivial points in the hypercontractivity ribbon and the $Φ$-ribbon respectively. Finally, we propose the matrix $Φ$-ribbon based on matrix $Φ$-entropy and establish the tensorization and data processing properties, together with the calculation of an exact matrix SDPI constant for the doubly symmetric binary source.

翻译：我们研究了服从给定独立性结构的联合分布的超压缩性带状区域与$Φ$-带状区域，并在若干基本情形下得到了紧界。对于以超图建模的一般独立性结构（其超边指定了随机变量的互独立子集），我们给出了$Φ$-带状区域的一个显式内界，该内界由关联向量的简单凸包描述。我们还提出了一个新的多部推广版本以及张-杨不等式的$Φ$-互信息类比，这分别意味着超压缩性带状区域与$Φ$-带状区域中的非平凡点。最后，我们基于矩阵$Φ$-熵提出了矩阵$Φ$-带状区域，并建立了其张量化与数据处理性质，同时针对双对称二元信源计算了精确的矩阵SDPI常数。