UBC Math Department Colloquium: Youssef Marzouk

Transportation of measure underlies many powerful, contemporary tools for Bayesian inference, density estimation, and generative modeling. 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. 

A variety of representations and constructions of transport have been proposed in recent years. In all these constructions, however, it is advantageous to exploit the prospect of low-dimensional structure in the associated probability measures. I will discuss two notions of low-dimensional structure, and their interplay with transport-driven methods for sampling and inference. The first seeks to approximate a high-dimensional target measure as a low-dimensional update of a dominating reference measure. The second is low-rank conditional structure, where the goal is to replace conditioning variables with low-dimensional projections or summaries. In both cases, under appropriate assumptions on the reference or target measures, one can derive gradient-based upper bounds on the associated approximation error and minimize these bounds to identify good subspaces for approximation. The associated subspaces then dictate specific structural ansatzes for transport maps that represent the target of interest.

I will showcase several algorithmic instantiations of this idea, focusing on Bayesian inverse problems, data assimilation, and simulation-based inference. The main elements of the talk will be accessible to undergraduates.