AM Seminar: A Blob Method for Diffusion and Applications to Sampling and Two Layer Neural Networks

Katy Craig 
Assistant Professor 
University of California, Santa Barbara

Join us on Zoom:

Description: Given a desired target distribution and an initial guess of that distribution, composed of finitely many samples, what is the best way to evolve the locations of the samples so that they accurately represent the desired distribution? A classical solution to this problem is to allow the samples to evolve according to Langevin dynamics, a stochastic particle method for the Fokker-Planck equation. In this talk, Assistant Professor Katy Craig will contrast this classical approach with a deterministic particle method corresponding to the porous medium equation. This method corresponds exactly to the mean-field dynamics of training a two layer neural network for a radial basis function activation function. Craig and her research team prove that, as the number of samples increases and the variance of the radial basis function goes to zero, the particle method converges to a bounded entropy solution of the porous medium equation. As a consequence, they obtain both a novel method for sampling probability distributions as well as insight into the training dynamics of two layer neural networks in the mean field regime.

Speaker bio: Katy Craig is an assistant professor at UC Santa Barbara, specializing in partial differential equations and optimal transport. She received her Ph.D. from Rutgers University in 2014, after which she spent one year at the UCLA, as an NSF Mathematical Sciences Postdoctoral Fellow and one year at UCSB as an UC President’s Postdoctoral Fellow. In January 2022, she was awarded an NSF CAREER grant to support her work on optimal transport and machine learning.