# Rethinking the Fully Hyperbolic Vision Transformer in Polar Coordinates

arXiv: https://arxiv.org/abs/2609.32899 · HTML: https://mlsanity.com/paper/2609.32899/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 44 of its 57 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 30 of 44)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Training data-efficient image transformers & distillation through attention](https://mlsanity.com/paper/2012.12877/) (2020)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Fine-Grained Visual Classification of Aircraft](https://mlsanity.com/paper/1306.5151/) (2013)
- [Self-Attention with Relative Position Representations](https://mlsanity.com/paper/1803.02155/) (2018)
- [Train Short, Test Long: Attention with Linear Biases Enables Input Length Extrapolation](https://mlsanity.com/paper/2108.12409/) (2021)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Skin Lesion Analysis Toward Melanoma Detection 2018: A Challenge Hosted by the International Skin Imaging Collaboration (ISIC)](https://mlsanity.com/paper/1902.03368/) (2019)
- [Kvasir-SEG: A Segmented Polyp Dataset](https://mlsanity.com/paper/1911.07069/) (2019)
- [PraNet: Parallel Reverse Attention Network for Polyp Segmentation](https://mlsanity.com/paper/2006.11392/) (2020)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [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)
- [Skin Lesion Analysis toward Melanoma Detection: A Challenge at the International Symposium on Biomedical Imaging (ISBI) 2016, hosted by the International Skin Imaging Collaboration (ISIC)](https://mlsanity.com/paper/1605.01397/) (2016)
- [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)
- [Fully Hyperbolic Neural Networks](https://mlsanity.com/paper/2105.14686/) (2021)
- [Better plain ViT baselines for ImageNet-1k](https://mlsanity.com/paper/2205.01580/) (2022)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Clipped Hyperbolic Classifiers Are Super-Hyperbolic Classifiers](https://mlsanity.com/paper/2107.11472/) (2021)
- [Fully Hyperbolic Convolutional Neural Networks for Computer Vision](https://mlsanity.com/paper/2303.15919/) (2023)
- [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)
- [Hyperbolic Busemann Learning with Ideal Prototypes](https://mlsanity.com/paper/2106.14472/) (2021)
- [Hyperbolic Fine-Tuning for Large Language Models](https://mlsanity.com/paper/2410.04010/) (2024)
- [HELM: Hyperbolic Large Language Models via Mixture-of-Curvature Experts](https://mlsanity.com/paper/2505.24722/) (2025)
- [Riemannian Residual Neural Networks](https://mlsanity.com/paper/2310.10013/) (2023)

## Abstract (arXiv)

Hyperbolic space can embed intrinsic hierarchies in data with low distortion due to the exponential growth of volume with distance from the origin. However, current Lorentz transformer blocks are formulated in ambient coordinates, where numerical errors increase at large radii due to instability in the Lorentzian inner product, leading many models to limit the radius to avoid this issue. This prevents us from utilizing the regions of hyperbolic space that motivate the geometry. To address this, we revisit the components of the transformer block in polar coordinates, where hyperbolic operations such as distance calculation and attention centroids can be computed without the numerical cancellation in their ambient-coordinate formulations. Specifically, we propose a polar fully connected layer that separately maps an embedding's direction and radius, allowing the radius of a feature to be learned rather than determined by the norm of a linear map. We additionally introduce horospherical shifts as relative positional encodings whose query-key distances grow logarithmically with the token gap. Finally, we reformulate the residual connection as average radius Lorentz boosts. Combining these components, we develop a fully hyperbolic transformer that substantially improves performance over Euclidean and hyperbolic baselines on standard vision tasks. We further evaluate our model on ImageNet and demonstrate its ability to generalize to other datasets using pre-trained weights, similar to Euclidean counterparts.