Transport information geometry I: Riemannian calculus on probability simplex

Wuchen Li UCLA

Differential Geometry Information Theory Mathematical Physics Machine Learning mathscidoc:2004.19001

arXiv:1803.06360, 2018.3
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 Γ2 operator (carr{é} du champ it{é}r{é}) by connecting Fisher-Rao information metric and optimal transport metric. Several examples are demonstrated.
Optimal transport; Information geometry; Information theory
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@inproceedings{wuchen2018transport,
  title={Transport information geometry I: Riemannian calculus on probability simplex},
  author={Wuchen Li},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200427013635113904644},
  booktitle={arXiv:1803.06360},
  year={2018},
}
Wuchen Li. Transport information geometry I: Riemannian calculus on probability simplex. 2018. In arXiv:1803.06360. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200427013635113904644.
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