Generalized Common Information: Common Information Extraction and Private Sources Synthesis

In literature, two different common informations were defined by G\'acs and K\"orner and by Wyner, respectively. In this paper, we generalize and unify them, and define a generalized version of common information, information-correlation function, by exploiting maximal correlation as a commonness or privacy measure. The G\'acs-K\"orner common information and Wyner common information are two special and extreme cases of our generalized definition. Besides, we also study the problems of common information extraction and private sources synthesis, and show that information-correlation function is the optimal rate under a given maximal correlation constraint in these problems.