# Polaris: Coupled Orbital Polar Embeddings for Hierarchical Concept Learning

arXiv: https://arxiv.org/abs/2605.00265 · HTML: https://mlsanity.com/paper/2605.00265/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 54 references that are in the map.
- First posted on arXiv 30 Apr 2026.

## Builds on (most cited first, 30 of 32)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Complex Embeddings for Simple Link Prediction](https://mlsanity.com/paper/1606.06357/) (2016)
- [RotatE: Knowledge Graph Embedding by Relational Rotation in Complex Space](https://mlsanity.com/paper/1902.10197/) (2019)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Learning Hierarchy-Aware Knowledge Graph Embeddings for Link Prediction](https://mlsanity.com/paper/1911.09419/) (2019)
- [BoxE: A Box Embedding Model for Knowledge Base Completion](https://mlsanity.com/paper/2007.06267/) (2020)
- [Hyperspherical Prototype Networks](https://mlsanity.com/paper/1901.10514/) (2019)
- [nGPT: Normalized Transformer with Representation Learning on the Hypersphere](https://mlsanity.com/paper/2410.01131/) (2024)
- [Probabilistic Embedding of Knowledge Graphs with Box Lattice Measures](https://mlsanity.com/paper/1805.06627/) (2018)
- [TaxoExpan: Self-supervised Taxonomy Expansion with Position-Enhanced Graph Neural Network](https://mlsanity.com/paper/2001.09522/) (2020)
- [Improving Local Identifiability in Probabilistic Box Embeddings](https://mlsanity.com/paper/2010.04831/) (2020)
- [STEAM: Self-Supervised Taxonomy Expansion with Mini-Paths](https://mlsanity.com/paper/2006.10217/) (2020)
- [Octet: Online Catalog Taxonomy Enrichment with Self-Supervision](https://mlsanity.com/paper/2006.10276/) (2020)
- [Taxonomy Completion via Triplet Matching Network](https://mlsanity.com/paper/2101.01896/) (2021)
- [TaxoEnrich: Self-Supervised Taxonomy Completion via Structure-Semantic Representations](https://mlsanity.com/paper/2202.04887/) (2022)
- [HyperMiner: Topic Taxonomy Mining with Hyperbolic Embedding](https://mlsanity.com/paper/2210.10625/) (2022)
- [Link Prediction with Attention Applied on Multiple Knowledge Graph Embedding Models](https://mlsanity.com/paper/2302.06229/) (2023)
- [HyperExpan: Taxonomy Expansion with Hyperbolic Representation Learning](https://mlsanity.com/paper/2109.10500/) (2021)
- [Hubs and Hyperspheres: Reducing Hubness and Improving Transductive Few-shot Learning with Hyperspherical Embeddings](https://mlsanity.com/paper/2303.09352/) (2023)
- [Normalization in Attention Dynamics](https://mlsanity.com/paper/2510.22026/) (2025)
- [A Note on the Kullback-Leibler Divergence for the von Mises-Fisher distribution](https://mlsanity.com/paper/1502.07104/) (2015)
- [Rank, Chunk and Expand: Lineage-Oriented Reasoning for Taxonomy Expansion](https://mlsanity.com/paper/2505.13282/) (2025)
- [FLAME: Self-Supervised Low-Resource Taxonomy Expansion using Large Language Models](https://mlsanity.com/paper/2402.13623/) (2024)
- [QuanTaxo: A Quantum Approach to Self-Supervised Taxonomy Expansion](https://mlsanity.com/paper/2501.14011/) (2025)
- [Producing Usable Taxonomies Cheaply and Rapidly at Pinterest Using Discovered Dynamic μ-Topics](https://mlsanity.com/paper/2301.12520/) (2023)
- [Task2Box: Box Embeddings for Modeling Asymmetric Task Relationships](https://mlsanity.com/paper/2403.17173/) (2024)
- [Global and Local Entailment Learning for Natural World Imagery](https://mlsanity.com/paper/2506.21476/) (2025)
- [Exploring Large Action Sets with Hyperspherical Embeddings using von Mises-Fisher Sampling](https://mlsanity.com/paper/2507.00518/) (2025)
- [Polar Ducks and Where to Find Them: Enhancing Entity Linking with Duck Typing and Polar Box Embeddings](https://mlsanity.com/paper/2305.12027/) (2023)

## Built on by (most cited first, 1 of 1)

- [Tree-of-Concerns: Hierarchical Multi-Agent Debate for Unstated-Limitation Extraction in Scientific Critique](https://mlsanity.com/paper/2608.20777/) (2026)

## Abstract (arXiv)

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.