Formal Riemannian Structure of the Wasserstein metric: Difference between revisions
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Given a closed and convex space <math> X \subseteq R^d </math>, two probability measures on the same space, <math> \mu, \nu </math>, the Wasserstein metric is defined as | |||
:<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 | |||
==Basic Structure of Riemannian Manifolds== | ==Basic Structure of Riemannian Manifolds== | ||
Revision as of 06:06, 6 June 2020
Given a closed and convex space , two probability measures on the same space, , the Wasserstein metric is defined as
where
Basic Structure of Riemannian Manifolds
Tangent Space Induced by Wasserstein Metric
Riemannian Metric Induced by Wasserstein Metric
References
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