Formal Riemannian Structure of the Wasserstein metric

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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, which gives a formal Riemannian structure on

Basic Structure of Riemannian Manifolds

Tangent Space Induced by the Wasserstein Metric

Riemannian Metric Induced by the Wasserstein Metric

References

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