Approximate Gács-Körner Common Information

We propose to exploit the structure of the correlation between two random variables X and Y via a relaxation on the Common Information problem of Gács and Körner (GK Common Information). Consider two correlated sources X and Y generated from a joint distribution P
_X,Y
. We study embeddings of X into discrete random variables U, such that H(U|Y) ≤ δ, while maximizing I(X; U). When δ = 0, this reduces to the GK Common Information problem. However, unlike the GK Common Information, which is known to be zero for many pairs of random variables (X, Y), we show that this relaxation allows to capture the structure in the correlation between X and Y for a much broader range of joint distributions, and showcase applications for some problems in multi-terminal information theory.