A Proof of Costa's Conjecture on Gaussian Interference Channel Capacity Region

It is shown that under suitable regularity conditions, differential entropy is $O(sqrt{n})$-Lipschitz on the space of distributions on $n$-dimensional Euclidean space with respect to the quadratic Wasserstein distance. This result allows one to couple the multi-user interference with its i.i.d. approximations where existence of good couplings follows from Talagrand's transportation-information inequality. As an application, a new outer bound for the two-user Gaussian interference channel is proved that improves the state of the art, which, in particular, settles the corner point conjecture of Costa (1985). Extensions of this program to discrete settings will be discussed where Wasserstein distance is replaced by Ornstein's $bar{d}$-distance and Marton's inequality is invoked in place of Talagrand's.

 

This is joint work with Yury Polyanskiy (MIT).