Information Constrained Optimal Transport: From Talagrand, to Marton, to Cover

The optimal transport problem studies how to one measure another in the most cost-effective way and has wide range of applications from economics machine learning. In this paper, we introduce study an information constrained variation problem. Our yields a strengthening generalization Talagrand’s celebrated transportation cost inequality. Following Marton’s approach, show that new inequality can be used recover old concentration results. Finally, provide application network theory. We it almost immediately recent solution long-standing open posed by Cover regarding capacity relay channel.