Hyperbolic Latent Position Models for Hierarchical Bipartite Data

2026-08-13

AI Summary

This study addresses the challenge of jointly modeling bipartite data heterogeneity and hierarchical interactions by proposing HELPI. The method embeds dyadic objects into hyperbolic space to disentangle geometric interactions from additive biases, utilizing root-invariant projections to identify hierarchical structures and augmented variational inference for efficient posterior computation. Theoretically, we establish the likelihood equivalence between hyperbolic distances and root Gromov product predictors, alongside posterior contraction properties. Empirical results demonstrate that HELPI accurately recovers hierarchical structures and yields interpretable geometric representations on educational and voting datasets. The model outperforms existing baselines in predictive performance and successfully uncovers latent partisan and policy structures without supervision, validating its effectiveness in capturing complex relational patterns within heterogeneous bipartite data.

Abstract

Binary bipartite data often combine row and column heterogeneity with hierarchical interaction, as when students answer exercises organized by prerequisites or legislators vote on nested policy areas. Fixed-dimensional Euclidean volume grows only polynomially with radius, whereas branching hierarchies expand exponentially. Hyperbolic space matches this growth, and its rooted Gromov product measures shared ancestry. We propose the \emph{HypErbolic Latent Position model for bipartite Interaction data} (HELPI), which places both object types in hyperbolic space while separating geometric interaction from additive propensities. With unrestricted main effects, a hyperbolic-distance predictor is likelihood-equivalent to a rooted Gromov-product predictor. Radial depth is absorbed by additive terms, leaving a root-invariant projected hierarchy signal as the identified geometric target. We establish identification and posterior contraction results and develop augmented variational procedures for binary outcomes. Simulations support recovery of hierarchical interaction and clarify weak-branch regimes. In Junyi Academy data, curriculum anchors yield an interpretable geometry with temporal prediction close to item-response benchmarks, while distinguishing attempted exercises from unconditional mastery. In U.S. House roll calls, flexible HELPI variants improve on additive and two-parameter logistic benchmarks and recover party and policy structure without using party labels during fitting.

Problem

• Hierarchical Bipartite Data
• Hyperbolic Space
• Latent Position Model
• Binary Data

Innovation

• Hyperbolic Latent Position Model
• Hierarchical Bipartite Data
• Rooted Gromov Product
• Augmented Variational Inference
• Geometric Interaction Separation