Polaris: Coupled Orbital Polar Embeddings for Hierarchical Concept Learning

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 $\sim$19 points in top-$K$ retrieval and up to $\sim$ 60\% reduction in mean rank over fourteen strong baselines.

Many real-world systems organize knowledge as hierarchies: products sit inside categories and subcategories on shopping sites, medical terms are arranged from broad diseases down to specific conditions, and dictionary words branch from general concepts into more specific incomplete because new products, diseases, and ideas appear all the time. Automatically deciding where a new concept belongs in such a hierarchy is a difficult problem: a method must judge both what a concept means and how general or specific it is, without confusing the two. We introduce \texttt{Polaris}, a method that represents each concept on the surface of a sphere and separately assigns it a "depth" from the center. Concepts placed near each other on the sphere are similar in meaning, while the depth captures how broad or specific the concept is---broad concepts (such as "beverage") sit on inner rings, and specific ones (such as "green tea") sit on outer rings. Keeping these two pieces of information on different axes lets the method learn meaning and structure at the same time without one signal interfering with the other. To place a new concept, \texttt{Polaris} first looks at the right depth ring and then ranks candidates by how well their meaning matches the new concept. We tested \texttt{Polaris} on text-based hierarchies (scientific, environmental, and medical taxonomies) and on an image-based hierarchy (a fine-grained dataset of bird species). It consistently finds the correct parent for a new concept more often and ranks it higher in the candidate list than fourteen earlier methods, with the correct parent often retrieved several times more reliably. The same approach works on words and images alike, suggesting that separating "what something means" from "where it belongs" is a useful general principle for organizing knowledge.