Ricci curvature for parametric statistics via optimal transport

Abstract

We elaborate the notion of a Ricci curvature lower bound for parametrized statistical models. Following the seminal ideas of Lott-Strum-Villani, we define this notion based on the geodesic convexity of the Kullback-Leibler divergence in a Wasserstein statistical manifold, that is, a manifold of probability distributions endowed with a Wasserstein metric tensor structure. Within these definitions, the Ricci curvature is related to both, information geometry and Wasserstein geometry. These definitions allow us to formulate bounds on the convergence rate of Wasserstein gradient flows and information functional inequalities in parameter space. We discuss examples of Ricci curvature lower bounds and convergence rates in exponential family models.

References in corpus (4)

• Entropic Ricci curvature bounds for discrete interacting systems
• Transport information geometry I: Riemannian calculus on probability simplex
• Population games and Discrete optimal transport
• A discrete Schrodinger equation via optimal transport on graphs

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