Transport information geometry: Riemannian calculus on probability simplex
| Autor: | Wuchen Li |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Statistics and Probability
Christoffel symbols Applied Mathematics Curvature Submanifold Manifold Computer Science Applications Computational Theory and Mathematics Differential geometry Wasserstein metric Metric (mathematics) Calculus Mathematics::Differential Geometry Geometry and Topology Information geometry Mathematics |
| Zdroj: | Information Geometry. 5:161-207 |
| ISSN: | 2511-249X 2511-2481 |
| Popis: | 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 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–Emery $$\Gamma _2$$ operator (carre du champ itere), Fisher–Rao metric and optimal transport metric. Several examples are demonstrated. |
| Databáze: | OpenAIRE |
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