  We propose a new framework, called Poisson learning, for graph based
semi-supervised learning at very low label rates. Poisson learning is motivated
by the need to address the degeneracy of Laplacian semi-supervised learning in
this regime. The method replaces the assignment of label values at training
points with the placement of sources and sinks, and solves the resulting
Poisson equation on the graph. The outcomes are provably more stable and
informative than those of Laplacian learning. Poisson learning is efficient and
simple to implement, and we present numerical experiments showing the method is
superior to other recent approaches to semi-supervised learning at low label
rates on MNIST, FashionMNIST, and Cifar-10. We also propose a graph-cut
enhancement of Poisson learning, called Poisson MBO, that gives higher accuracy
and can incorporate prior knowledge of relative class sizes.
