Optimal Transport to Rényi Entropies

Abstract

Recently, an optimal transportation argument was proposed by the author to provide a simple proof of Shannon’s entropy-power inequality. Interestingly, such a proof could have been given by Shannon himself in his 1948 seminal paper. In fact, by 1948 Shannon established all the ingredients necessary for the proof and the transport argument takes the form of a simple change of variables.

In this paper, the optimal transportation argument is extended to Rényi entropies in relation to Shannon’s entropy-power inequality and to a reverse version involving a certain conditional entropy. The transportation argument turns out to coincide with Barthe’s proof of sharp direct and reverse Young’s convolutional inequalities and can be applied to derive recent Rényi entropy-power inequalities.