2022/03/12 by Wuchen Li, Linyuan Lu, Li, Wuchen +2 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #cs.IT #math.CO #math.IT
paper · pdf · doi:10.48550/arxiv.2203.06307
arxiv created 2022/03/12 · arxiv updated 2022/03/15
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.