Polaris: Coupled Orbital Polar Embeddings for Hierarchical Concept Learning

2026-04-30

AI Summary

This work addresses the challenge of representation learning for real-world hierarchical knowledge, which is often hindered by structural asymmetry and semantic noise. To this end, we propose a polar-coordinate hyperspherical embedding framework that disentangles angular (semantic) and radial (hierarchical) components to learn interference-free hierarchical representations on a sphere. Our approach innovatively introduces a directional inclusion mechanism and a structure-guided candidate parent retrieval strategy, complemented by local constraints, global regularization, and an uncertainty-aware asymmetric optimization objective to effectively mitigate geometric collapse. Extensive experiments demonstrate that the model substantially outperforms 14 strong baselines across multiple hierarchical extrapolation tasks, achieving up to a 19-percentage-point improvement in Top-K retrieval accuracy and reducing mean rank by 60%.

Abstract

Real-world knowledge is often organized as hierarchies such as product taxonomies, medical ontologies, and label trees, yet learning hierarchical representations is challenging due to asymmetric structure and noisy semantics. We introduce Polaris, a polar hyperspherical embedding framework that separates semanticity from hierarchy using angular geometry and radius, enabling the learning of meaning and structure without interference. To map latent representation onto the sphere, we project it to the tangent space at the north pole, apply the exponential map, and learn unit-norm representations using spherical linear layers. Polaris then combines robust local constraints, global regularization that prevents geometric collapse, and uncertainty-aware asymmetric objectives that encourage directional containment. At inference time, Polaris uses structure-guided retrieval to efficiently narrow down candidate parents before final ranking. We evaluate Polaris on different settings of taxonomy expansion - spanning trees, multi-parent DAGs, and multimodal hierarchies, showing consistent improvements of up to ~19 points in top-K retrieval and up to ~60% reduction in mean rank over fourteen strong baselines.

Problem

• hierarchical representation learning
• asymmetric structure
• noisy semantics
• taxonomy expansion
• hierarchical concept learning

Innovation

• polar hyperspherical embedding
• hierarchical concept learning
• angular geometry
• structure-guided retrieval
• asymmetric objectives