How to sample brand-new hyperlinks from an observed hypergraph without operating a generative model directly on the discrete space of node subsets. A conditional likelihood model attaches to every node a latent embedding and a scalar degree parameter, and to every hyperlink its own embedding, so that hyperlinks are conditionally independent given the embeddings; constrained maximum likelihood then recovers all three sets of parameters up to stated identifiability conditions. A score-based diffusion model is trained on the recovered hyperlink embeddings alone, and new hyperlinks are produced by sampling embeddings from it and pushing them back through the likelihood model. Students learn why this two-stage design reduces generating an n-dimensional binary vector to generating a K-dimensional continuous one, how the total-variation and KL bounds on generative error transfer from the hyperlink space to the embedding space, and how hypergraph dimensionality, node degree heterogeneity, and hyperlink sparsity govern the error when the embeddings must be estimated rather than assumed known.
Hypergraph data, which capture multi-way interactions among entities, are increasingly prevalent in the big data era. Generating new hyperlinks from an observed, usually high-dimensional hypergraph i…