Generalized Common Informations: Measuring Commonness by the Conditional Maximal Correlation
This work provides a theoretical framework for information theory, offering incremental extensions to existing common information measures with applications in privacy and data synthesis.
The paper tackles the problem of measuring common information between random variables by introducing two generalized common information functions based on conditional maximal correlation, which unify existing definitions. It shows these functions characterize the minimum rates needed for private source synthesis and common information extraction under correlation constraints.
In literature, different common informations were defined by Gács and Körner, by Wyner, and by Kumar, Li, and Gamal, respectively. In this paper, we define two generalized versions of common informations, named approximate and exact information-correlation functions, by exploiting the conditional maximal correlation as a commonness or privacy measure. These two generalized common informations encompass the notions of Gács-Körner's, Wyner's, and Kumar-Li-Gamal's common informations as special cases. Furthermore, to give operational characterizations of these two generalized common informations, we also study the problems of private sources synthesis and common information extraction, and show that the information-correlation functions are equal to the minimum rates of commonness needed to ensure that some conditional maximal correlation constraints are satisfied for the centralized setting versions of these problems. As a byproduct, the conditional maximal correlation has been studied as well.