Geometry of sliced optimal transport and projected-transport gradient flows

We will discuss two types of objects that can be approximated in high-dimensions. Recent results have established that sliced-Wasserstein (SW) distance can be approximated accurately in high dimensions based on samples of the measures considered. We will discuss the geometry of the SW distance. In particular we will characterize tangent space to the SW space as a certain weighted negative Sobolev space and obtain the local metric. We show that SW space is not a length space and establish properties of the geodesic distance, relevant to gradient flows in the space.

To obtain gradient flows that can be approximated in high dimensions we 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. Application towards interacting particle methods for sampling will also be discussed.

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