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arXiv 0904.4541cs.ITmath.IT
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一般广播信道的Marton内界估计

Evaluation of Marton's Inner Bound for the General Broadcast Channel

Amin Aminzadeh Gohari, Venkat Anantharam

摘要:

在没有公共消息的两个接收器通用广播信道上的最著名的内界是由于Marton [3]。这个结果随后在[p.391,问题10(c)2]和[4]中推广到具有公共消息的广播信道。然而,后一个区域是不可计算的(除非在某些特殊情况下),因为其辅助随机变量的基数没有界限。甚至不清楚内界是一个闭集。证明基数界限的主要障碍是传统的Carathéodory定理,证明基数界限的主要已知工具,并不产生有限的基数结果。本文的主要贡献之一是引入了一种新的工具的基础上的身份,涉及到离散随机变量的香农熵的二阶导数(在一定的扰动下)相应的Fisher信息。为了超越传统的Carathéodory型参数,我们确定了某些性质,对应于内界的极值点的辅助随机变量需要满足。这些性质被用来建立内界的辅助随机变量的基数界,从而证明了该区域的可计算性和封闭性。最后,我们建立了\cite{NairZizhou}的一个猜想,即Marton的内界与Nair和El Gamal的最近的外界一般不匹配。

英文摘要:

The best known inner bound on the two-receiver general broadcast channel without a common message is due to Marton [3]. This result was subsequently generalized in [p. 391, Problem 10(c) 2] and [4] to broadcast channels with a common message. However the latter region is not computable (except in certain special cases) as no bounds on the cardinality of its auxiliary random variables exist. Nor is it even clear that the inner bound is a closed set. The main obstacle in proving cardinality bounds is the fact that the traditional use of the Carathéodory theorem, the main known tool for proving cardinality bounds, does not yield a finite cardinality result. One of the main contributions of this paper is the introduction of a new tool based on an identity that relates the second derivative of the Shannon entropy of a discrete random variable (under a certain perturbation) to the corresponding Fisher information. In order to go beyond the traditional Carathéodory type arguments, we identify certain properties that the auxiliary random variables corresponding to the extreme points of the inner bound need to satisfy. These properties are then used to establish cardinality bounds on the auxiliary random variables of the inner bound, thereby proving the computability of the region, and its closedness. Lastly, we establish a conjecture of \cite{NairZizhou} that Marton's inner bound and the recent outer bound of Nair and El Gamal do not match in general.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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