The concavity of generalized entropy powers

TL;DR

This paper introduces a new family of entropy powers based on Sharma-Mittal entropies and proves their concavity along diffusion processes, extending previous results on Re9nyi entropy powers and connecting to entropy power inequalities.

Contribution

It establishes the concavity of Sharma-Mittal entropy powers along Wasserstein gradient flows, generalizing prior work on Re9nyi entropy powers.

Findings

Proves concavity of Sharma-Mittal entropy powers during diffusion processes.

Extends previous results on Re9nyi entropy power concavity.

Links new entropy powers to established entropy inequalities.

Abstract

In this note we introduce a new family of entropy powers which are related to generalized entropies, called Sharma-Mittal entropies, and we prove their concavity along diffusion processes generated by $L^2$-Wasserstein gradient flows of corresponding entropy functionals. This result extends the result of Savar\'e and Toscani on the concavity of R\'enyi entropy powers (IEEE Trans. Inf. Theory, 2014) and reveals a connection to R\'enyi entropy power inequalities by Bobkov and Marsiglietti (IEEE Trans. Inf. Theory, 2017).