HWI inequalities in discrete spaces via couplings

Abstract

HWI inequalities are interpolation inequalities relating entropy, Fisher information and optimal transport distances. We adapt an argument of Y. Wu for proving the Gaussian HWI inequality via a coupling argument to the discrete setting, establishing new interpolation inequalities for the discrete hypercube and the discrete torus. In particular, we obtain an improvement of the modified logarithmic Sobolev inequality for the discrete hypercube of Bobkov and Tetali.