Polaris: Coupled Orbital Polar Embeddings for Hierarchical Concept Learning

TL;DR

Polaris introduces a hyperspherical embedding framework that effectively captures hierarchical structures in knowledge representations by separating semantic meaning from hierarchy using angular geometry.

Contribution

The paper proposes Polaris, a novel polar hyperspherical embedding method that disentangles semanticity from hierarchy, improving hierarchical concept learning and retrieval performance.

Findings

Achieves up to 19 points improvement in top-K retrieval accuracy.

Reduces mean rank by up to 60% compared to baseline methods.

Effective across various hierarchy types including trees, DAGs, and multimodal hierarchies.

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.