# Shadow Cones: A Generalized Framework for Partial Order Embeddings

arXiv: https://arxiv.org/abs/2305.15215 · HTML: https://mlsanity.com/paper/2305.15215/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 30 references that are in the map.
- First posted on arXiv 24 May 2023.

## Builds on (most cited first, 15 of 15)

- [Distributed Representations of Words and Phrases and their Compositionality](https://mlsanity.com/paper/1310.4546/) (2013)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Representation Tradeoffs for Hyperbolic Embeddings](https://mlsanity.com/paper/1804.03329/) (2018)
- [Low-Dimensional Hyperbolic Knowledge Graph Embeddings](https://mlsanity.com/paper/2005.00545/) (2020)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Poincar'e GloVe: Hyperbolic Word Embeddings](https://mlsanity.com/paper/1810.06546/) (2018)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Order-Embeddings of Images and Language](https://mlsanity.com/paper/1511.06361/) (2015)
- [Deep Hierarchical Semantic Segmentation](https://mlsanity.com/paper/2203.14335/) (2022)
- [Inferring Concept Hierarchies from Text Corpora via Hyperbolic Embeddings](https://mlsanity.com/paper/1902.00913/) (2019)
- [Modeling Heterogeneous Hierarchies with Relation-specific Hyperbolic Cones](https://mlsanity.com/paper/2110.14923/) (2021)
- [Random Laplacian Features for Learning with Hyperbolic Space](https://mlsanity.com/paper/2202.06854/) (2022)
- [Improved Representation Learning for Predicting Commonsense Ontologies](https://mlsanity.com/paper/1708.00549/) (2017)

## Built on by (most cited first, 3 of 3)

- [Global and Local Entailment Learning for Natural World Imagery](https://mlsanity.com/paper/2506.21476/) (2025)
- [RegD: Hierarchical Embeddings via Dissimilarity between Arbitrary Euclidean Regions](https://mlsanity.com/paper/2501.17518/) (2025)
- [Native Hierarchical and Compositional Representations with Subspace Embeddings](https://mlsanity.com/paper/2508.16687/) (2025)

## Abstract (arXiv)

Hyperbolic space has proven to be well-suited for capturing hierarchical relations in data, such as trees and directed acyclic graphs. Prior work introduced the concept of entailment cones, which uses partial orders defined by nested cones in the Poincar'e ball to model hierarchies. Here, we introduce the ``shadow cones" framework, a physics-inspired entailment cone construction. Specifically, we model partial orders as subset relations between shadows formed by a light source and opaque objects in hyperbolic space. The shadow cones framework generalizes entailment cones to a broad class of formulations and hyperbolic space models beyond the Poincar'e ball. This results in clear advantages over existing constructions: for example, shadow cones possess better optimization properties over constructions limited to the Poincar'e ball. Our experiments on datasets of various sizes and hierarchical structures show that shadow cones consistently and significantly outperform existing entailment cone constructions. These results indicate that shadow cones are an effective way to model partial orders in hyperbolic space, offering physically intuitive and novel insights about the nature of such structures.