Mean field information Hessian matrices on graphs
Combinatorics
2022-03-15 v1 Information Theory
math.IT
Abstract
We derive mean-field information Hessian matrices on finite graphs. The "information" refers to entropy functions on the probability simplex. And the "mean-field" means nonlinear weight functions of probabilities supported on graphs. These two concepts define a mean-field optimal transport type metric. In this metric space, we first derive Hessian matrices of energies on graphs, including linear, interaction energies, entropies. We name their smallest eigenvalues as mean-field Ricci curvature bounds on graphs. We next provide examples on two-point spaces and graph products. We last present several applications of the proposed matrices. E.g., we prove discrete Costa's entropy power inequalities on a two-point space.
Keywords
Cite
@article{arxiv.2203.06307,
title = {Mean field information Hessian matrices on graphs},
author = {Wuchen Li and Linyuan Lu},
journal= {arXiv preprint arXiv:2203.06307},
year = {2022}
}