# Klein Model for Hyperbolic Neural Networks

arXiv: https://arxiv.org/abs/2410.16813 · HTML: https://mlsanity.com/paper/2410.16813/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 54 references that are in the map.
- First posted on arXiv 22 Oct 2024.

## Builds on (most cited first, 29 of 29)

- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry](https://mlsanity.com/paper/1806.03417/) (2018)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [Hyperbolic Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [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 Attention Networks](https://mlsanity.com/paper/1805.09786/) (2018)
- [Continuous Hierarchical Representations with Poincar'e Variational Auto-Encoders](https://mlsanity.com/paper/1901.06033/) (2019)
- [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)
- [Lorentzian Graph Convolutional Networks](https://mlsanity.com/paper/2104.07477/) (2021)
- [A Hyperbolic-to-Hyperbolic Graph Convolutional Network](https://mlsanity.com/paper/2104.06942/) (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)
- [Large-Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/1806.00437/) (2018)
- [Tree! I am no Tree! I am a Low Dimensional Hyperbolic Embedding](https://mlsanity.com/paper/2005.03847/) (2020)
- [HyperText: Endowing FastText with Hyperbolic Geometry](https://mlsanity.com/paper/2010.16143/) (2020)
- [Hyperbolic Geometric Latent Diffusion Model for Graph Generation](https://mlsanity.com/paper/2405.03188/) (2024)
- [Hyperbolic Relevance Matching for Neural Keyphrase Extraction](https://mlsanity.com/paper/2205.02047/) (2022)
- [HypLL: The Hyperbolic Learning Library](https://mlsanity.com/paper/2306.06154/) (2023)
- [HEAT: Hyperbolic Embedding of Attributed Networks](https://mlsanity.com/paper/1903.03036/) (2019)
- [Accelerating hyperbolic t-SNE](https://mlsanity.com/paper/2401.13708/) (2024)
- [HyperSteiner: Computing Heuristic Hyperbolic Steiner Minimal Trees](https://mlsanity.com/paper/2409.05671/) (2024)
- [Hyperbolic Delaunay Geometric Alignment](https://mlsanity.com/paper/2404.08608/) (2024)
- [Improving Hyperbolic Representations via Gromov-Wasserstein Regularization](https://mlsanity.com/paper/2407.10495/) (2024)

## Built on by (most cited first, 5 of 5)

- [Learning Along the Arrow of Time: Hyperbolic Geometry for Backward-Compatible Representation Learning](https://mlsanity.com/paper/2506.05826/) (2025)
- [Even Faster Hyperbolic Random Forests: A Beltrami-Klein Wrapper Approach](https://mlsanity.com/paper/2506.04360/) (2025)
- [Comparing Euclidean and Hyperbolic K-Means for Generalized Category Discovery](https://mlsanity.com/paper/2602.04932/) (2026)
- [Riemannian Deep Learning: Modules, Networks, and Geometries](https://mlsanity.com/paper/2607.19305/) (2026)
- [Building Transformation Layers for Riemannian Neural Networks](https://mlsanity.com/paper/2609.35436/) (2026)

## Abstract (arXiv)

Hyperbolic neural networks (HNNs) have been proved effective in modeling complex data structures. However, previous works mainly focused on the Poincar'e ball model and the hyperboloid model as coordinate representations of the hyperbolic space, often neglecting the Klein model. Despite this, the Klein model offers its distinct advantages thanks to its straight-line geodesics, which facilitates the well-known Einstein midpoint construction, previously leveraged to accompany HNNs in other models. In this work, we introduce a framework for hyperbolic neural networks based on the Klein model. We provide detailed formulation for representing useful operations using the Klein model. We further study the Klein linear layer and prove that the "tangent space construction" of the scalar multiplication and parallel transport are exactly the Einstein scalar multiplication and the Einstein addition, analogous to the M"obius operations used in the Poincar'e ball model. We show numerically that the Klein HNN performs on par with the Poincar'e ball model, providing a third option for HNN that works as a building block for more complicated architectures.