  Hypergraphs are a common model for multiway relationships in data, and
hypergraph semi-supervised learning is the problem of assigning labels to all
nodes in a hypergraph, given labels on just a few nodes. Diffusions and label
spreading are classical techniques for semi-supervised learning in the graph
setting, and there are some standard ways to extend them to hypergraphs.
However, these methods are linear models, and do not offer an obvious way of
incorporating node features for making predictions. Here, we develop a
nonlinear diffusion process on hypergraphs that spreads both features and
labels following the hypergraph structure, which can be interpreted as a
hypergraph equilibrium network. Even though the process is nonlinear, we show
global convergence to a unique limiting point for a broad class of
nonlinearities, which is the global optimum of a interpretable, regularized
semi-supervised learning loss function. The limiting point serves as a node
embedding from which we make predictions with a linear model. Our approach is
much more accurate than several hypergraph neural networks, and also takes less
time to train.
