Optimal mass transportation map from a Riemannian manifold to itself transforms one probability measure to the other in the most economical way, the transportation cost is the so-called Wasserstein distance between the two measures. Wasserstein distance is a Riemannian metric in the space of all probability measures on the Riemannian manifold.
Finding the optimal mass transportation map is equivalent to solve the Monge-Ampere equation, which has intrinsic relations with Minkowski and Alexandrov problems in convex geometry. In this talk, we introduce a variational approach to solve the optimal mass transportation problem, which gives a constructive proof for the classical Alexdrov theorem and leads to a practical algorithm.
We also cover some direct applications of optimal mass transportation , such as surface and volume measure-preserving parameterization, and shape classification based on Wasserstein distance and so on.