Formal Riemannian Structure of the Wasserstein metric: Difference between revisions
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:<math> W_2(\mu, \nu) := \min_{\gamma \in \Gamma(\mu, \nu)} \left( \int |x_1 - x_2|^2 \, d\gamma(x_1, x_2) \right)^{1/2} </math> | :<math> W_2(\mu, \nu) := \min_{\gamma \in \Gamma(\mu, \nu)} \left( \int |x_1 - x_2|^2 \, d\gamma(x_1, x_2) \right)^{1/2} </math> | ||
where <math> \Gamma(\mu, \nu) </math> is a transport plan from <math> \mu </math> to <math> \nu </math>. | where <math> \Gamma(\mu, \nu) </math> is a transport plan from <math> \mu </math> to <math> \nu </math>. These plans are used to define the [[Kantorovich Problem]]. | ||
==Basic Structure of Riemannian Manifolds== | ==Basic Structure of Riemannian Manifolds== | ||
Revision as of 06:18, 6 June 2020
Given a closed and convex space , two probability measures on the same space, , the Wasserstein metric is defined as
where is a transport plan from to . These plans are used to define the Kantorovich Problem.
Basic Structure of Riemannian Manifolds
Tangent Space Induced by Wasserstein Metric
Riemannian Metric Induced by Wasserstein Metric
References
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