  Label spreading is a general technique for semi-supervised learning with
point cloud or network data, which can be interpreted as a diffusion of labels
on a graph. While there are many variants of label spreading, nearly all of
them are linear models, where the incoming information to a node is a weighted
sum of information from neighboring nodes. Here, we add nonlinearity to label
spreading through nonlinear functions of higher-order structure in the graph,
namely triangles in the graph. For a broad class of nonlinear functions, we
prove convergence of our nonlinear higher-order label spreading algorithm to
the global solution of a constrained semi-supervised loss function. We
demonstrate the efficiency and efficacy of our approach on a variety of point
cloud and network datasets, where the nonlinear higher-order model compares
favorably to classical label spreading, as well as hypergraph models and graph
neural networks.
