We provide a new framework to lower bound the nonnegative rank of a matrix in terms of common information and information theory. In this framework we can improve on recent results for the correlation polytope. We also compute the exact common information of the unique disjointness patterns and we provide the first family of polytopes that has high approximate extension complexity, both in the average case as well as in the adversarial case. These new results are proven by showing that the UDISJ patterns are extremely robust towards noise and adversarial changes by means of an information theoretic analysis.