Transport information geometry: Riemannian calculus on probability simplex

Abstract

We formulate the Riemannian calculus of the probability set embedded with L2-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 weighted graph 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, Hessian operators and diffusion processes 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, we present an identity connecting among Baker–Émery Γ 2 operator (carré du champ itéré), Fisher–Rao metric and optimal transport metric. Several examples are demonstrated.