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The optimal transport problem studies how to transport one measure to anotherin the most cost-effective way and has wide range of applications fromeconomics to machine learning. In this paper, we introduce and study aninformation constrained variation of this problem. Our study yields astrengthening and generalization of Talagrand's celebrated transportation costinequality. Following Marton's approach, we show that the new transportationcost inequality can be used to recover old and new concentration of measureresults. Finally, we provide an application of this new inequality to networkinformation theory. We show that it can be used to recover almost immediately arecent solution to a long-standing open problem posed by Cover regarding thecapacity of the relay channel.

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