联合类型图、非交互式模拟与更强超压缩性
Graphs of Joint Types, Noninteractive Simulation, and Stronger Hypercontractivity
AI总结：
本文研究联合类型诱导二分图的最大边密度与双团速率区域，给出渐近紧界，并用于非交互式模拟、超压缩性及改进二进制加法器信道零误差容量外界。
摘要：
本文研究类型图，即由联合类型诱导的二分图。我们考察该图的诱导二分子图的最大边密度，其中两侧顶点数在类型长度 n 的指数尺度上给定。这可视为一个等周问题。当类型长度趋于无穷时，我们给出了最大边密度指数的渐近紧界。我们还研究类型图的双团速率区域，其定义为所有满足以下条件的 (R1,R2) 构成的集合：类型图中存在一个双团，在两侧分别具有 2^{nR1} 和 2^{nR2} 个顶点。我们同样给出了该双团速率区域的渐近紧界。随后，我们讨论这些结果与非交互式模拟以及超压缩不等式之间的联系。此外，作为结果的一项应用，本文给出了二进制加法器信道零误差容量区域的一个新外界，改进了 Austrin、Kaski、Koivisto 和 Nederlof 此前已知的最佳界。本文证明基于类型方法和线性代数。
英文摘要：
In this paper, we study the type graph, namely, a bipartite graph induced by a joint type. We investigate the maximum edge density of induced bipartite subgraphs of this graph having a number of vertices on each side on an exponential scale in the length $n$ of the type. This can be seen as an isoperimetric problem. We provide asymptotically sharp bounds for the exponent of the maximum edge density as the length of the type goes to infinity. We also study the biclique rate region of the type graph, which is defined as the set of $(R_{1},R_{2})$ such that there exists a biclique of the type graph which has respectively $2^{nR_{1}}$ and $2^{nR_{2}}$ vertices on the two sides. We provide asymptotically sharp bounds for the biclique rate region as well. We then discuss the connections of these results to noninteractive simulation and hypercontractivity inequalities. Furthermore, as an application of our results, a new outer bound for the zero-error capacity region of the binary adder channel is provided, which improves the previously best known bound, due to Austrin, Kaski, Koivisto, and Nederlof. Our proofs in this paper are based on the method of types and linear algebra.
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