用典型集的几何分析改进割集界
Improving on the Cut-Set Bound via Geometric Analysis of Typical Sets
摘要：
考虑离散无记忆对称原始中继信道，源$X$希望在中继$Z$的帮助下向目的$Y$发送信息，中继可以通过速率为$R_0$的无差错数字链路与目的通信，而$Y$和$Z$在给定$X$的情况下是条件独立同分布的.我们开发了两个新的上限的能力，这个通道，比现有的界限，包括著名的割集界更紧。我们的方法显着偏离了标准的信息理论的方法证明上界的多用户信道的容量。我们建立在爆破引理分析的概率几何关系之间的典型集合的$n$字母随机变量与一个可靠的代码在这个通道上进行通信。这些关系转化为所涉及的$n$字母随机变量之间的新的熵不等式。作为我们的界限的应用，我们研究了一个开放的问题（封面，1987年），即，什么是最小所需的$Z$ - $Y$链路速率$R_0^*$，以使中继信道的容量等于广播切。我们考虑当$X$ - $Y$和$X$ - $Z$链路都是二进制对称信道时的特殊情况。我们对中继信道容量的更严格的限制立即转化为$R_0^*$的更严格的下限。更有趣的是，我们表明，当$p\to 1/2$，$R_0^*\geq 0.1803$，即使广播信道成为完全嘈杂的$p\to 1/2$和它的容量，因此，中继信道的容量，为零，一个严格的正速率$R_0$是必需的中继信道容量等于广播界。
英文摘要：
We consider the discrete memoryless symmetric primitive relay channel, where, a source $X$ wants to send information to a destination $Y$ with the help of a relay $Z$ and the relay can communicate to the destination via an error-free digital link of rate $R_0$, while $Y$ and $Z$ are conditionally independent and identically distributed given $X$. We develop two new upper bounds on the capacity of this channel that are tighter than existing bounds, including the celebrated cut-set bound. Our approach significantly deviates from the standard information-theoretic approach for proving upper bounds on the capacity of multi-user channels. We build on the blowing-up lemma to analyze the probabilistic geometric relations between the typical sets of the $n$-letter random variables associated with a reliable code for communicating over this channel. These relations translate to new entropy inequalities between the $n$-letter random variables involved. As an application of our bounds, we study an open question posed by (Cover, 1987), namely, what is the minimum needed $Z$-$Y$ link rate $R_0^*$ in order for the capacity of the relay channel to be equal to that of the broadcast cut. We consider the special case when the $X$-$Y$ and $X$-$Z$ links are both binary symmetric channels. Our tighter bounds on the capacity of the relay channel immediately translate to tighter lower bounds for $R_0^*$. More interestingly, we show that when $p\to 1/2$, $R_0^*\geq 0.1803$; even though the broadcast channel becomes completely noisy as $p\to 1/2$ and its capacity, and therefore the capacity of the relay channel, goes to zero, a strictly positive rate $R_0$ is required for the relay channel capacity to be equal to the broadcast bound.
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