Wasserstein Riemannian geometry on statistical manifold

Wasserstein Riemannian Geometry of Positive-definite Matrices∗

The Wasserstein distance on multivariate non-degenerate Gaussian densities is a Riemannian distance. After reviewing the properties of the distance and the metric geodesic, we derive an explicit form of the Riemannian metrics on positive-definite matrices and compute its tensor form with respect to the trace scalar product. The tensor is a matrix, which is the solution of a Lyapunov equation. W...

On a class of paracontact Riemannian manifold

We classify the paracontact Riemannian manifolds that their Riemannian curvature satisfies in the certain condition and we show that this classification is hold for the special cases semi-symmetric and locally symmetric spaces. Finally we study paracontact Riemannian manifolds satisfying R(X, ξ).S = 0, where S is the Ricci tensor.

Dimensional contraction in Wasserstein distance for diffusion semigroups on a Riemannian manifold

We prove a refined contraction inequality for diffusion semigroups with respect to the Wasserstein distance on a Riemannian manifold taking account of the dimension. The result generalizes in a Riemannian context, the dimensional contraction established in [BGG13] for the Euclidean heat equation. The theorem is proved by using a dimensional coercive estimate for the Hodge-de Rham semigroup on 1...

F-Geometry and Amari's α-Geometry on a Statistical Manifold

Riemannian Geometry and Statistical Machine Learning

Statistical machine learning algorithms deal with the problem of selecting an appropriate statistical model from a model space Θ based on a training set {xi}i=1 ⊂ X or {(xi, yi)}i=1 ⊂ X × Y. In doing so they either implicitly or explicitly make assumptions on the geometries of the model space Θ and the data space X . Such assumptions are crucial to the success of the algorithms as different geo...