Mathematics Colloquium
Dejan Slepcev
CMU
Title: Gradient flows and interacting particle dynamics for sampling in high dimensions
Date: Friday, December 5th
Place and Time: Love 101, 3:05-3:55 pm
Abstract. Providing samples of a measure given by its density is a challenging problem, especially in high dimensions. We will first describe standard approaches based on Markov chains whose invariant measure is the desired target measure, and highlight the difficulties. At the mean-field level the distributions of particles in these processes are governed by evolution equations. In particular, the Fokker–Planck equation, the gradient flow of Kullback-Leibler (KL) divergence in Wasserstein geometry, corresponds to Langevin Monte Carlo sampling. We will consider several gradient flows of KL-divergence for sampling based on different geometries on the spaces of probability measures, including the gradient flows with respect to Stein geometry and a new, Radon-Wasserstein, geometry. In addition to discussing foundational questions regarding the flows and their asymptotic properties, we will describe the discretizations used and focus on interacting particle systems. Particular attention will be given to methods that are applicable in high dimensions. The talk is based on joint works with Lantian Xu and Elias Hess-Childs