We will discuss two types of objects that can be approximated well in high-dimensions. Over recent years  sliced-Wasserstein (SW) distance gained attention as a transportation-based metric that  can be approximated accurately in high dimensions, given samples of the measures considered. We will discuss the geometry of the space of probability measures endowed with SW distance. In particular we will  characterize tangent space as a weighted negative Sobolev space and describe the local metric.  We show that SW space is not a length space and establish properties of the geodesic distance. We will highlight consequences  to gradient flows in the SW space.

To obtain gradient flows of relative entropy that can be approximated in high dimensions we will introduce the projected Wasserstein distance where the space of velocities has been restricted to have low complexity. We will show some of the basic properties of the distance and the corresponding gradient flows. We will then discuss applications towards interacting particle methods for sampling.

The talk is based on joint works with Sangmin Park and Lantian Xu.