Mathematics Colloquium
Youssef M Marzouk
MIT
Title: Sampling and generative modeling using dynamical representations of transport
Date: Friday, April 19, 2025
Place and Time: Love 101, 3:05pm
Abstract.
Drawing samples from a probability distribution is a central task in applied mathematics, statistics, and machine learning with applications ranging from Bayesian inference to computational chemistry and generative modeling. Many powerful tools for sampling employ transportation of measure, where the essential idea is to couple the target probability distribution with a simple, tractable “reference” distribution, and to use this coupling (which may be deterministic or stochastic) to generate new samples. Within this broad area, an emerging class of methods use dynamics to define a transport incrementally, e.g., via the flow map induced by trajectories of an ODE. These methods have shown great empirical success, but their consistency and convergence properties, and the ways in which they can exploit structure in the underlying distributions, are less well understood. We will discuss properties and theoretical underpinnings of these dynamical approaches to transport. In particular, we will discuss the statistical convergence of generative models based on neural ODEs. We will also present a new dynamical construction of transport: a gradient-free method which avoids complex training procedures by instead evolving an interacting particle system that approximates a Fisher-Rao gradient flow. Throughout, we will illuminate the pitfalls and opportunities of these dynamical methods, including ways to exploit the design freedom they afford.