# Fully Hyperbolic Convolutional Neural Networks for Computer Vision

arXiv: https://arxiv.org/abs/2303.15919 · HTML: https://mlsanity.com/paper/2303.15919/

## Key facts

- Cited by 49 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 54 references that are in the map.
- First posted on arXiv 28 Mar 2023.

## Builds on (most cited first, 28 of 28)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Improved Regularization of Convolutional Neural Networks with Cutout](https://mlsanity.com/paper/1708.04552/) (2017)
- [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 Image Embeddings](https://mlsanity.com/paper/1904.02239/) (2019)
- [Riemannian Adaptive Optimization Methods](https://mlsanity.com/paper/1810.00760/) (2018)
- [A guide to convolution arithmetic for deep learning](https://mlsanity.com/paper/1603.07285/) (2016)
- [Hyperbolic Deep Neural Networks: A Survey](https://mlsanity.com/paper/2101.04562/) (2021)
- [From Variational to Deterministic Autoencoders](https://mlsanity.com/paper/1903.12436/) (2019)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [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)
- [Clipped Hyperbolic Classifiers Are Super-Hyperbolic Classifiers](https://mlsanity.com/paper/2107.11472/) (2021)
- [Differentiating through the Fr'echet Mean](https://mlsanity.com/paper/2003.00335/) (2020)
- [The Numerical Stability of Hyperbolic Representation Learning](https://mlsanity.com/paper/2211.00181/) (2022)
- [Large-Margin Classification in Hyperbolic Space](https://mlsanity.com/paper/1806.00437/) (2018)
- [Capturing implicit hierarchical structure in 3D biomedical images with self-supervised hyperbolic representations](https://mlsanity.com/paper/2012.01644/) (2020)
- [Skin Lesion Recognition with Class-Hierarchy Regularized Hyperbolic Embeddings](https://mlsanity.com/paper/2209.05842/) (2022)
- [Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design](https://mlsanity.com/paper/2112.03402/) (2021)

## Built on by (most cited first, 30 of 49)

- [HELM: Hyperbolic Large Language Models via Mixture-of-Curvature Experts](https://mlsanity.com/paper/2505.24722/) (2025)
- [Hyperbolic Category Discovery](https://mlsanity.com/paper/2504.06120/) (2025)
- [Position: Beyond Euclidean -- Foundation Models Should Embrace Non-Euclidean Geometries](https://mlsanity.com/paper/2504.08896/) (2025)
- [Lorentzian Residual Neural Networks](https://mlsanity.com/paper/2412.14695/) (2024)
- [Hyperbolic Dataset Distillation](https://mlsanity.com/paper/2505.24623/) (2025)
- [HyperCore: The Core Framework for Building Hyperbolic Foundation Models with Comprehensive Modules](https://mlsanity.com/paper/2504.08912/) (2025)
- [Geo-Sign: Hyperbolic Contrastive Regularisation for Geometrically Aware Sign Language Translation](https://mlsanity.com/paper/2506.00129/) (2025)
- [Hyperbolic Learning with Multimodal Large Language Models](https://mlsanity.com/paper/2408.05097/) (2024)
- [Neural Networks on Symmetric Spaces of Noncompact Type](https://mlsanity.com/paper/2601.01097/) (2026)
- [Is Hyperbolic Space All You Need for Medical Anomaly Detection?](https://mlsanity.com/paper/2505.21228/) (2025)
- [Hyperbolic Busemann Neural Networks](https://mlsanity.com/paper/2602.18858/) (2026)
- [Robust Hyperbolic Learning with Curvature-Aware Optimization](https://mlsanity.com/paper/2405.13979/) (2024)
- [Learning Along the Arrow of Time: Hyperbolic Geometry for Backward-Compatible Representation Learning](https://mlsanity.com/paper/2506.05826/) (2025)
- [Hyperbolic Metric Learning for Visual Outlier Detection](https://mlsanity.com/paper/2403.15260/) (2024)
- [Fully Hyperbolic Rotation for Knowledge Graph Embedding](https://mlsanity.com/paper/2411.03622/) (2024)
- [Adversarial Attacks on Hyperbolic Networks](https://mlsanity.com/paper/2412.01495/) (2024)
- [Hyperbolic Genome Embeddings](https://mlsanity.com/paper/2507.21648/) (2025)
- [Understanding and Improving Hyperbolic Deep Reinforcement Learning](https://mlsanity.com/paper/2512.14202/) (2025)
- [Fast and Geometrically Grounded Lorentz Neural Networks](https://mlsanity.com/paper/2601.21529/) (2026)
- [Intrinsic Lorentz Neural Network](https://mlsanity.com/paper/2602.23981/) (2026)
- [HYDEN: Hyperbolic Density Representations for Medical Images and Reports](https://mlsanity.com/paper/2408.09715/) (2024)
- [On the Universal Statistical Consistency of Expansive Hyperbolic Deep Convolutional Neural Networks](https://mlsanity.com/paper/2411.10128/) (2024)
- [Hgformer: Hyperbolic Graph Transformer for Recommendation](https://mlsanity.com/paper/2502.15693/) (2024)
- [Hyperbolic Dual Feature Augmentation for Open-Environment](https://mlsanity.com/paper/2506.08906/) (2025)
- [Geometry-aware Distance Measure for Diverse Hierarchical Structures in Hyperbolic Spaces](https://mlsanity.com/paper/2506.18533/) (2025)
- [Learning Protein-Ligand Binding in Hyperbolic Space](https://mlsanity.com/paper/2508.15480/) (2025)
- [HIDISC: A Hyperbolic Framework for Domain Generalization with Generalized Category Discovery](https://mlsanity.com/paper/2510.17188/) (2025)
- [HEEGNet: Hyperbolic Embeddings for EEG](https://mlsanity.com/paper/2601.03322/) (2026)
- [EEG-Based Multimodal Learning via Hyperbolic Mixture-of-Curvature Experts](https://mlsanity.com/paper/2604.12579/) (2026)
- [Siegel Neural Networks](https://mlsanity.com/paper/2511.09577/) (2025)

## Abstract (arXiv)

Real-world visual data exhibit intrinsic hierarchical structures that can be represented effectively in hyperbolic spaces. Hyperbolic neural networks (HNNs) are a promising approach for learning feature representations in such spaces. However, current HNNs in computer vision rely on Euclidean backbones and only project features to the hyperbolic space in the task heads, limiting their ability to fully leverage the benefits of hyperbolic geometry. To address this, we present HCNN, a fully hyperbolic convolutional neural network (CNN) designed for computer vision tasks. Based on the Lorentz model, we generalize fundamental components of CNNs and propose novel formulations of the convolutional layer, batch normalization, and multinomial logistic regression. {Experiments on standard vision tasks demonstrate the promising performance of our HCNN framework in both hybrid and fully hyperbolic settings.} Overall, we believe our contributions provide a foundation for developing more powerful HNNs that can better represent complex structures found in image data. Our code is publicly available at https://github.com/kschwethelm/HyperbolicCV.