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.

Logarithmic divergences from optimal transport and Rényi geometry

“…Because of the product term in (24), we cannot pass between exponential concavity and convexity simply by considering −ϕ. This is different from the classical case α = 0.…”

“…Since the matrix (g ij (ξ)) is strictly positive definite, from (24) we have that D 2 e αϕ < 0. In particular, ϕ is locally α-exponentially concave.…”

“…For example, the recent paper [4] studies the manifold of entropy-relaxed optimal transport plans and defines a divergence which interpolates between the Kullback-Leibler divergence and the Wasserstein metric (optimal transport cost). Dynamic approach are considered in [14] and the recent papers [13,24]. Also see [29] which builds on the ideas of [32] and embeds a large family of optimal transport problems in a generalized simplex.…”

Parametric Fokker-Planck Equation

“…Then G is Riemmanian metric on T Θ iff For each θ ∈ Θ, for any ξ ∈ T θ Θ (ξ = 0), we can find x ∈ M such that ∇ · (ρ θ ∂ θ T θ (T −1 θ (x)ξ) = 0. From now on, following [9,10], we call (Θ, G) Wasserstein statistical manifold.…”

Natural gradient via optimal transport

“…In particular, the displacement convexity of KL divergence relates to the Ricci curvature lower bound on sample space. We elaborate this notion in [23].…”