Many-to-many Relationship Modeling via Bi-order Entailment Cones

Hyperbolic entailment cones are effective for modeling hierarchical and multimodal relationships and have been applied to diverse fields. However, existing multimodal formulations typically assign one modality to be universally more general than the other. This one-sided inductive bias is suitable for one-to-many relationships, but becomes suboptimal in many-to-many settings such as recommendation, protein–ligand binding, and protein function annotation, where neither modality is inherently more general. To alleviate this limitation, we propose Bi-Order Entailment (BiOE), a product-manifold framework that decomposes many-to-many relationships into directional latent orders modeled by specialized hyperbolic entailment factors. To handle noisy and incomplete data, BiOE further introduces confidence-weighted pseudo-order supervision from neighborhood and semantic-cluster coverage, together with a lightweight residual compatibility factor for non-hierarchical interactions. We theoretically and empirically show that single-entailment models cannot faithfully capture both compatibility and induced order structure, and that they introduce direction-dependent retrieval biases. Across recommender systems, protein–ligand binding, and protein function prediction, BiOE significantly improves over hyperbolic baselines and single-entailment variants. Our experiments also introduce new hyperbolic downstream pipelines, including the first hyperbolic Collaborative Filtering–LLM alignment method and one of the first hyperbolic gene ontology classification models