Transport information geometry I: Riemannian calculus on probability simplex

TL;DR

This paper develops the Riemannian calculus on probability simplices using the $L^2$-Wasserstein metric, establishing geometric structures like Christoffel symbols, curvature, and operators, bridging optimal transport and information geometry.

Contribution

It formulates the Riemannian calculus on probability manifolds with Wasserstein metric, introducing geometric tools and identities connecting Fisher-Rao and transport metrics, in both finite and infinite dimensions.

Findings

Derived Christoffel symbols, Levi-Civita connections, and curvature tensors.

Connected Fisher-Rao metric with optimal transport via the $ ext{Γ}_2$ operator.

Provided geometric computations and examples in finite and infinite-dimensional settings.

Abstract

We formulate the Riemannian calculus of the probability set embedded with $L^2$-Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold) supported on vertices of a finite graph. The main idea is to embed the probability manifold as a submanifold of the positive measure space with a nonlinear metric tensor. Here the nonlinearity comes from the linear weighted Laplacian operator. By this viewpoint, we establish torsion-free Christoffel symbols, Levi-Civita connections, curvature tensors and volume forms in the probability manifold by Euclidean coordinates. As a consequence, the Jacobi equation, Laplace-Beltrami and Hessian operators on the probability manifold are derived. These geometric computations are also provided in the infinite-dimensional density space (density manifold) supported on a finite-dimensional manifold. In particular, an identity is given connecting the Baker-{\'E}mery $\Gamma_2$ operator (carr{\'e} du champ it{\'e}r{\'e}) by connecting Fisher-Rao information metric and optimal transport metric. Several examples are demonstrated.