Some inequalities on Riemannian manifolds linking Entropy,Fisher information, Stein discrepancy and Wasserstein distance

TL;DR

This paper establishes new inequalities on Riemannian manifolds that connect entropy, Fisher information, Stein discrepancy, and Wasserstein distance, extending classical inequalities like log-Sobolev and transportation-cost inequalities.

Contribution

It extends the HSI inequality from Euclidean space to Riemannian manifolds, linking key probabilistic and geometric quantities in this setting.

Findings

Derived inequalities linking entropy, Fisher information, Stein discrepancy, and Wasserstein distance on manifolds.

Extended the HSI inequality from Euclidean space to Riemannian manifolds.

Strengthened classical log-Sobolev and transportation-cost inequalities.

Abstract

For a complete connected Riemannian manifold $M$ let $V\in C^2(M)$ be such that $\mu(d x)={\rm e}^{-V(x)} \mbox{vol}(d x)$ is a probability measure on $M$. Taking $\mu$ as reference measure, we derive inequalities for probability measures on $M$ linking relative entropy, Fisher information, Stein discrepancy and Wasserstein distance. These inequalities strengthen in particular the famous log-Sobolev and transportation-cost inequality and extend the so-called Entropy/Stein-discrepancy/Information (HSI) inequality established by Ledoux, Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space to the setting of Riemannian manifolds.