The Department of Applied Mathematics weekly seminar is given by scholars and researchers working in applied mathematics, broadly interpreted.
Title: Learning in the space of probability measures
Abstract: Many datasets in modern applications - from cell gene expression and images to shapes and text documents - are naturally interpreted as probability measures, distributions, histograms, or point clouds. This perspective motivates the development of learning algorithms that operate directly in the space of probability measures. However, this space presents unique challenges: it is nonlinear and infinite-dimensional. Fortunately, it possesses a natural Riemannian-type geometry which enables meaningful learning algorithms.
This talk will introduce the fundamentals of the space of probability measures and explore approaches to unsupervised, supervised, and manifold learning within this framework. We will discuss two key ideas: (1) a linearization approach, and (2) a "curved" approach utilizing finite-dimensional Riemannian submanifolds. Additionally, we will examine time evolutions on this space, including flows involving stochastic gradient descent and trajectory inference, with applications to analyzing the evolution of gene expression in single cells.