Transport information geometry I: Riemannian calculus on probability simplex

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-{É}mery ${\Gamma }_2$ operator (carr{é} du champ it{é}r{é}) by connecting Fisher-Rao information metric and optimal transport metric. Several examples are demonstrated.

This draft is the revision of previous paper: "Geometry of probability simplex via optimal transport"

References in corpus (4)

• Population games and Discrete optimal transport
• A discrete Schrodinger equation via optimal transport on graphs
• Geodesic of minimal length in the set of probability measures on graphs
• Entropy dissipation of Fokker-Planck equations on graphs

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• Parametric Fokker-Planck equation
• Information Newton's flow: second-order optimization method in probability space
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• Constrained dynamical optimal transport and its Lagrangian formulation
• Hopf-Cole transformation via generalized Schrödinger bridge problem
• Sub-Riemannian Ricci curvature via generalized Gamma $z$ calculus
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• Hessian transport gradient flows
• Efficient Wasserstein Natural Gradients for Reinforcement Learning
• Entropy dissipation via Information Gamma calculus: Non-reversible stochastic differential equations
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