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Mathematics > Differential Geometry

Title: Geometry of probability simplex via optimal transport

Authors: Wuchen Li
Abstract: We study the Riemannian structures of the probability simplex on a weighted graph introduced by $L^2$-Wasserstein metric. The main idea is to embed the probability simplex as a submanifold of the positive orthant. From this embedding, we establish the geometry formulas of the probability simplex in Euclidean coordinates. The geometry computations on discrete simplex guide us to introduce the ones in the Fr{\'e}chet manifold of densities supported on a finite dimensional base manifold. Following the steps of Nelson, Bakery-{\'E}mery, Lott-Villani-Strum and the geometry of density manifold, we demonstrate an identity that connects the Bakery-{\'E}mery $\Gamma_2$ operator (carr{\'e} du champ it{\'e}r{\'e}) and Yano's formula on the base manifold. Several examples of differential equations in probability simplex are demonstrated.
Subjects: Differential Geometry (math.DG)
Cite as: arXiv:1803.06360 [math.DG]
  (or arXiv:1803.06360v1 [math.DG] for this version)

Submission history

From: Wuchen Li [view email]
[v1] Fri, 16 Mar 2018 18:15:04 GMT (27kb)