Geometry and analytic properties of the sliced Wasserstein space
Speaker: Sangmin Park
Abstract: The sliced Wasserstein metric (SW) compares probability measures on Rd by taking averages of the Wasserstein distances between projections of the measures to lines. The distance has found a range of applications in statistics and machine learning, as it is easier to approximate and compute than the Wasserstein distance in high dimensions. This talk will focus on analytic properties of the sliced Wasserstein distance related to its statistical properties. Namely, when the measures considered are ``nice'' then the SW metric is comparable to the (homogeneous) negative Sobolev norm H˙−(d+1)/2. In particular, this is connected to the parametric finite sample approximation rate noted by Manole, Balakrishnan, and Wasserman '22. This allows us to view SW as a `curved' version of Maximum Mean Discrepancy. This talk is based on a joint work with Dejan Slepcev. https://arxiv.org/abs/2311.05134