CMX Lunch Seminar

Sampling from a probability measure given only through its density is challenging, particularly in high dimensions. A prominent approach is to discretize gradient flows on the space of probability measures. Examples include Langevin Monte Carlo and Stein Variational Gradient Descent, which correspond to gradient flows of the Kullback--Leibler (KL) divergence under the Wasserstein and Stein geometries, respectively.

In this talk, I will present recent work introducing a new geometry on the space of measures, the Radon--Wasserstein geometry. The associated KL gradient flows exhibit two key features: they admit accurate interacting-particle approximations in high dimensions, and their per-step computational cost scales linearly in both the number of particles and the dimension.

I will discuss theoretical results for these flows, including well-posedness and long-time convergence, as well as numerical experiments illustrating their behavior and performance. Time permitting, I will also describe extensions that yield affine-invariant algorithms.

This is joint work with Dejan Slepčev and Lantian Xu.