Optimal Transport meets Gradient Flows: Moving Probability Distributions in Machine Learning, Statistics, and Optimization

Gaspard Monge posed the question of optimal transportation in 1781. Since then, it has grown into a mathematically rigorous theory, and a natural way to measure distances between probability distributions. It became a core tool in modern computation in the 2010s, when data became large and high-dimensional. A key insight from the 1990s, due to Otto and collaborators, is that many diffusion equations can be seen as steepest descent (a gradient flow) in the geometry defined by optimal transportation. This turns questions about randomness into questions about optimization and the calculus of variations. In this talk, I will introduce these ideas from the ground up and present my recent work on gradient flows for computation, such as the Hellinger–Kantorovich or Wasserstein–Fisher–Rao gradient flows. “I will show how this viewpoint, which echoes Lyapunov’s second method, leads to new results for computational theory and applications in optimization, Bayesian inference, sampling, and generative models.