Klein Model for Hyperbolic Neural Networks

Yidan Mao, Jing Gu, Marcus C. Werner, Dongmian Zou
Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, PMLR 282:1181-1205, 2026.

Abstract

Hyperbolic neural networks (HNNs) have been proved effective in modeling complex data structures. However, previous works mainly focused on the Poincaré 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 a 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 Möbius operations used in the Poincaré ball model. We show numerically that Klein HNNs perform on par with HNNs using the Poincaré ball model, providing a third option for HNN that works as a building block for more complicated architectures.

Cite this Paper

BibTeX
@InProceedings{pmlr-v282-mao26a, title = {Klein Model for Hyperbolic Neural Networks}, author = {Mao, Yidan and Gu, Jing and Werner, Marcus C. and Zou, Dongmian}, booktitle = {Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations}, pages = {1181--1205}, year = {2026}, editor = {Acosta, Francisco and Azeglio, Simone and Tolooshams, Bahareh and van de Geijn, Chase and Shewmake, Christian and Sanborn, Sophia and Miolane, Nina}, volume = {282}, series = {Proceedings of Machine Learning Research}, month = {14 Dec 2024--07 Dec 2025}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v282/main/assets/mao26a/mao26a.pdf}, url = {https://proceedings.mlr.press/v282/mao26a.html}, abstract = {Hyperbolic neural networks (HNNs) have been proved effective in modeling complex data structures. However, previous works mainly focused on the Poincaré 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 a 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 Möbius operations used in the Poincaré ball model. We show numerically that Klein HNNs perform on par with HNNs using the Poincaré ball model, providing a third option for HNN that works as a building block for more complicated architectures.} }
Endnote
%0 Conference Paper %T Klein Model for Hyperbolic Neural Networks %A Yidan Mao %A Jing Gu %A Marcus C. Werner %A Dongmian Zou %B Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations %C Proceedings of Machine Learning Research %D 2026 %E Francisco Acosta %E Simone Azeglio %E Bahareh Tolooshams %E Chase van de Geijn %E Christian Shewmake %E Sophia Sanborn %E Nina Miolane %F pmlr-v282-mao26a %I PMLR %P 1181--1205 %U https://proceedings.mlr.press/v282/mao26a.html %V 282 %X Hyperbolic neural networks (HNNs) have been proved effective in modeling complex data structures. However, previous works mainly focused on the Poincaré 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 a 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 Möbius operations used in the Poincaré ball model. We show numerically that Klein HNNs perform on par with HNNs using the Poincaré ball model, providing a third option for HNN that works as a building block for more complicated architectures.
APA
Mao, Y., Gu, J., Werner, M.C. & Zou, D.. (2026). Klein Model for Hyperbolic Neural Networks. Proceedings of the 4th (2025) and 3rd (2024) NeurIPS Workshops on Symmetry and Geometry in Neural Representations, in Proceedings of Machine Learning Research 282:1181-1205 Available from https://proceedings.mlr.press/v282/mao26a.html.

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