Wasserstein geometry of Gaussian measures

By Asuka Takatsu. 2011

294 citations recorded by the source corpus.

This paper concerns the Riemannian/Alexandrov geometry of Gaussian measures, from the view point of the L 2 -Wasserstein geometry. The space of Gaussian measures is of finite dimension, which allows to write down the ex plicit Riemannian metric which in turn induces the L 2 -Wasserstein distance. Moreover, its completion as a metric space provides a complete picture of the singular behavior of the L 2 Wasserstein geometry. In particular, the singular set is st ratified according to the dimension of the support of the Gaussian measures, providing an explicit nontrivial example of Alexandrov space with extremal sets.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/bf970af9e0d3d8082385beaeae0ac13536fc0299

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Fréchet regression for random objects with Euclidean predictors (2016, background): “We note that Example 1 refers to objects in the Wasserstein space, a complex smooth manifold that is characterized by the Wasserstein geodesics (Takatsu 2011) and thus provides an example of random objects on a manifold, for which we obtain consistent estimation of global and local Fréchet regressio”
Statistical inference for Bures–Wasserstein barycenters (2019, background): “It is interesting to note, that the dBW distance appears not only one of the central distances, used in quantum mechanics, but also an object of extensive investigation in transportation theory Takatsu et al. (2011). Let N(0, Q) and N(0, S) be two centred Gaussian distributions.”
On the Bures–Wasserstein distance between positive definite matrices (2017, background): “The material in this section is based on [5, 20, 33, 34].”
Wasserstein Riemannian geometry of Gaussian densities (2018, methodology): “4, where we use the presentation given by [36], cf.”
Statistical Aspects of Wasserstein Distances (2018, background): “The restriction of W 2 ( R d ) to Gaussian measures leads to a stratiﬁed space , whose geometry was studied carefully by Takatsu (2011), including expressions for the curvature.”