Lower bound on Wyner's Common Information

An important notion of common information between two random variables is due to Wyner. In this paper, we derive a lower bound on Wyner's common information for continuous random variables. The new bound improves on the only other general lower bound on Wyner's common information, which is the mutual information. We also show that the new lower bound is tight for the so-called "Gaussian channels" case, namely, when the joint distribution of the random variables can be written as the sum of a single underlying random variable and Gaussian noises. We motivate this work from the recent variations of Wyner's common information and applications to network data compression problems such as the Gray-Wyner network.