论文标题
通用信息，噪声稳定性及其扩展
Common Information, Noise Stability, and Their Extensions
论文作者
论文摘要
常见信息（CI）在信息理论及相关领域（例如理论计算机科学和离散概率）中无处不在。但是，由于CI有多种概念，因此缺乏对它们之间的深层互连的统一理解。该专着试图通过利用一系列在看似不同的问题的数学技术来填补这一空白。 在第一部分中，我们回顾了与Wyner和Gács-Körner-Witsenhausen（GKW）CI相关的操作任务和属性。在Partii中，我们从分布式源模拟的角度讨论了前者的扩展。这包括RényiCI，它在Wyner的CI和确切的CI之间形成桥梁。通过订单〜$ \ infty $的rényici与确切的CI之间的令人惊讶的等价性，我们证明了存在严格超过Wyner CI的联合来源的存在。第二部分中讨论的其他密切相关的主题包括通道综合问题以及Wyner和确切的CI与矩阵非负等级的联系。 在第三部分中，我们通过噪声稳定性或NICD问题检查了GKW的CI，其中我们量化了从双变量来源提取的位位的一致性概率。然后，我们将其扩展到$ k $ - 用户NICD和$ Q $稳定性问题，并讨论信息理论和离散概率的各种猜想，例如Courtade-Kumar，Li-Médard和Mossell-O'Donnell猜想。最后，我们考虑了超收缩性和Brascamp-lieb不平等，这通过通过非调用函数替换其布尔函数进一步推广噪声稳定性。第三部分中的证明背后的关键思想可以以教学上的连贯方式呈现，并通过信息理论和傅立叶分析方法统一。
Common information (CI) is ubiquitous in information theory and related areas such as theoretical computer science and discrete probability. However, because there are multiple notions of CI, a unified understanding of the deep interconnections between them is lacking. This monograph seeks to fill this gap by leveraging a small set of mathematical techniques that are applicable across seemingly disparate problems. In Part I, we review the operational tasks and properties associated with Wyner's and Gács-Körner-Witsenhausen's (GKW's) CI. In PartII, we discuss extensions of the former from the perspective of distributed source simulation. This includes the Rényi CI which forms a bridge between Wyner's CI and the exact CI. Via a surprising equivalence between the Rényi CI of order~$\infty$ and the exact CI, we demonstrate the existence of a joint source in which the exact CI strictly exceeds Wyner's CI. Other closely related topics discussed in Part II include the channel synthesis problem and the connection of Wyner's and exact CI to the nonnegative rank of matrices. In Part III, we examine GKW's CI with a more refined lens via the noise stability or NICD problem in which we quantify the agreement probability of extracted bits from a bivariate source. We then extend this to the $k$-user NICD and $q$-stability problems, and discuss various conjectures in information theory and discrete probability, such as the Courtade-Kumar, Li-Médard and Mossell-O'Donnell conjectures. Finally, we consider hypercontractivity and Brascamp-Lieb inequalities, which further generalize noise stability via replacing the Boolean functions therein by nonnnegative functions. The key ideas behind the proofs in Part III can be presented in a pedagogically coherent manner and unified via information-theoretic and Fourier-analytic methods.
