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>. These plans are used to define the [[Kantorovich Problem]]. The Wasserstein metric is indeed a metric in the sense that it satisfies the desired properties of a distance function between probability measures on <math> \mathcal{P}_2(X)</math>. Moreover, the Wasserstein metric can be used to define a Riemannian metric, which gives a formal Riemannian structure on
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]]. The Wasserstein metric is indeed a metric in the sense that it satisfies the desired properties of a distance function between probability measures on <math> \mathcal{P}_2(X)</math>. Moreover, the Wasserstein metric can be used to define a Riemannian metric on <math> \mathcal{P}_2(X) </math>. Such a metric allows one to define angles and lengths of vectors at each point in our ambient space. This structure can then be used to apply tools and phenomena found in Riemannian geometry, such as [[Geodesics and generalized geodesics|geodesics]].


==Basic Structure of Riemannian Manifolds==
==Basic Structure of Riemannian Manifolds==

Revision as of 06:32, 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. The Wasserstein metric is indeed a metric in the sense that it satisfies the desired properties of a distance function between probability measures on . Moreover, the Wasserstein metric can be used to define a Riemannian metric on . Such a metric allows one to define angles and lengths of vectors at each point in our ambient space. This structure can then be used to apply tools and phenomena found in Riemannian geometry, such as geodesics.

Basic Structure of Riemannian Manifolds

Tangent Space Induced by the Wasserstein Metric

Riemannian Metric Induced by the Wasserstein Metric

References

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