Common Information, Noise Stability, and Their Extensions

Abstract

Common information is ubiquitous in information theory and related areas such as theoretical computer science and discrete probability. However, because there are multiple notions of common information, 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) common information. In Part II, we discuss extensions of the former from the perspective of distributed source simulation. This includes the Rènyi common information which forms a bridge between Wyner’s common information and the exact common information. Via a surprising equivalence between the Rènyi common information of order ∞ and the exact common information, we demonstrate the existence of a joint source in which the exact common information strictly exceeds Wyner’s common information. Other closely related topics discussed in Part II include the channel synthesis problem and the connection of Wyner’s and exact common information to the nonnegative rank of matrices. In Part III, recognizing that GKW’s common information is zero for most non-degenerate sources, we examine it with a more refined lens via the Non-Interactive Correlation Distillation (NICD) problem in which we quantify the agreement probability of extracted bits from a bivariate source. We extend this to the noise stability problem which includes as special cases the k-user NICD and q-stability problems. This allows us to seamlessly transition to discussing their connections to various conjectures in information theory and discrete probability, such as the Courtade–Kumar, Li–Mèdard and Mossell–O’Donnell conjectures. Finally, we consider functional inequalities (e.g., the hypercontractivity and Brascamp–Lieb inequalities), which constitute a further generalization of the noise stability problem in which the Boolean functions therein are replaced by nonnnegative functions. We demonstrate that 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.

Topics & Concepts