# Poincar'e Embeddings for Learning Hierarchical Representations

arXiv: https://arxiv.org/abs/1705.08039 · HTML: https://mlsanity.com/paper/1705.08039/

## Key facts

- Cited by 684 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 May 2017.

## Built on by (most cited first, 30 of 684)

- [The Geometry of Categorical and Hierarchical Concepts in Large Language Models](https://mlsanity.com/paper/2406.01506/) (2024)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [Hyperbolic Image-Text Representations](https://mlsanity.com/paper/2304.09172/) (2023)
- [Hyperbolic Vision Transformers: Combining Improvements in Metric Learning](https://mlsanity.com/paper/2203.10833/) (2022)
- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [Hyperbolic Image Segmentation](https://mlsanity.com/paper/2203.05898/) (2022)
- [Deep Hierarchical Semantic Segmentation](https://mlsanity.com/paper/2203.14335/) (2022)
- [ConE: Cone Embeddings for Multi-Hop Reasoning over Knowledge Graphs](https://mlsanity.com/paper/2110.13715/) (2021)
- [Compositional Entailment Learning for Hyperbolic Vision-Language Models](https://mlsanity.com/paper/2410.06912/) (2024)
- [Hyperbolic Deep Learning in Computer Vision: A Survey](https://mlsanity.com/paper/2305.06611/) (2023)
- [Lorentzian Graph Convolutional Networks](https://mlsanity.com/paper/2104.07477/) (2021)
- [Discrete-time Temporal Network Embedding via Implicit Hierarchical Learning in Hyperbolic Space](https://mlsanity.com/paper/2107.03767/) (2021)
- [The Limits of Word Level Differential Privacy](https://mlsanity.com/paper/2205.02130/) (2022)
- [Hyperbolic Contrastive Learning for Visual Representations beyond Objects](https://mlsanity.com/paper/2212.00653/) (2022)
- [Plug & Play Attacks: Towards Robust and Flexible Model Inversion Attacks](https://mlsanity.com/paper/2201.12179/) (2022)
- [Rickrolling the Artist: Injecting Backdoors into Text Encoders for Text-to-Image Synthesis](https://mlsanity.com/paper/2211.02408/) (2022)
- [Clipped Hyperbolic Classifiers Are Super-Hyperbolic Classifiers](https://mlsanity.com/paper/2107.11472/) (2021)
- [A Hyperbolic-to-Hyperbolic Graph Convolutional Network](https://mlsanity.com/paper/2104.06942/) (2021)
- [Hyperbolic Variational Graph Neural Network for Modeling Dynamic Graphs](https://mlsanity.com/paper/2104.02228/) (2021)
- [Probing BERT in Hyperbolic Spaces](https://mlsanity.com/paper/2104.03869/) (2021)
- [Learning the Predictability of the Future](https://mlsanity.com/paper/2101.01600/) (2021)
- [HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections](https://mlsanity.com/paper/2106.03306/) (2021)
- [The Numerical Stability of Hyperbolic Representation Learning](https://mlsanity.com/paper/2211.00181/) (2022)
- [Hypformer: Exploring Efficient Transformer Fully in Hyperbolic Space](https://mlsanity.com/paper/2407.01290/) (2024)
- [von Mises-Fisher Loss: An Exploration of Embedding Geometries for Supervised Learning](https://mlsanity.com/paper/2103.15718/) (2021)
- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [Hyperbolic Busemann Learning with Ideal Prototypes](https://mlsanity.com/paper/2106.14472/) (2021)
- [Knowledge Graph Embedding: A Survey from the Perspective of Representation Spaces](https://mlsanity.com/paper/2211.03536/) (2022)
- [Hyperbolic Graph Neural Networks: A Review of Methods and Applications](https://mlsanity.com/paper/2202.13852/) (2022)
- [Unsupervised Discovery of the Long-Tail in Instance Segmentation Using Hierarchical Self-Supervision](https://mlsanity.com/paper/2104.01257/) (2021)

## Abstract (arXiv)

Representation learning has become an invaluable approach for learning from symbolic data such as text and graphs. However, while complex symbolic datasets often exhibit a latent hierarchical structure, state-of-the-art methods typically learn embeddings in Euclidean vector spaces, which do not account for this property. For this purpose, we introduce a new approach for learning hierarchical representations of symbolic data by embedding them into hyperbolic space -- or more precisely into an n-dimensional Poincar'e ball. Due to the underlying hyperbolic geometry, this allows us to learn parsimonious representations of symbolic data by simultaneously capturing hierarchy and similarity. We introduce an efficient algorithm to learn the embeddings based on Riemannian optimization and show experimentally that Poincar'e embeddings outperform Euclidean embeddings significantly on data with latent hierarchies, both in terms of representation capacity and in terms of generalization ability.