Gradient Flows in the Wasserstein Metric: From Discrete to Continuum via Regularization
Friday, November 13, 2020 - 10:40am - 11:25am
Over the past ten years, optimal transport has become a fundamental tool in statistics and machine learning: the Wasserstein metric provides a new notion of distance for classifying distributions and a rich geometry for interpolating between them. In parallel, optimal transport has led to new theoretical results on the stability and long time behavior of partial differential equations through the theory of Wasserstein gradient flows. These two lines of research recently intersected in a series of works that characterized the dynamics of training neural networks with a single hidden layer as a Wasserstein gradient flow. In this talk, I will briefly introduce the mathematical theory of Wasserstein gradient flows and describe recent results on discrete to continuum limits. In particular, I will show how passing from the discrete to continuum limit by introducing an appropriate regularization can lead to faster rates of convergence, as well as novel, deterministic particle methods for diffusive processes.