Displacement convexity of entropy and related inequalities on graphs
Probability
2012-07-24 v1 Functional Analysis
Abstract
We introduce the notion of an interpolating path on the set of probability measures on finite graphs. Using this notion, we first prove a displacement convexity property of entropy along such a path and derive Prekopa-Leindler type inequalities, a Talagrand transport-entropy inequality, certain HWI type as well as log-Sobolev type inequalities in discrete settings. To illustrate through examples, we apply our results to the complete graph and to the hypercube for which our results are optimal -- by passing to the limit, we recover the classical log-Sobolev inequality for the standard Gaussian measure with the optimal constant.
Keywords
Cite
@article{arxiv.1207.5116,
title = {Displacement convexity of entropy and related inequalities on graphs},
author = {Nathaël Gozlan and Cyril Roberto and Paul-Marie Samson and Prasad Tetali},
journal= {arXiv preprint arXiv:1207.5116},
year = {2012}
}