# A Hypertoroidal Covering for Perfect Color Equivariance

arXiv: https://arxiv.org/abs/2603.04256 · HTML: https://mlsanity.com/paper/2603.04256/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 47 references that are in the map.
- First posted on arXiv 4 Mar 2026.

## Builds on (most cited first, 26 of 26)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The Many Faces of Robustness: A Critical Analysis of Out-of-Distribution Generalization](https://mlsanity.com/paper/2006.16241/) (2020)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [AugMix: A Simple Data Processing Method to Improve Robustness and Uncertainty](https://mlsanity.com/paper/1912.02781/) (2019)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Generalization and Representational Limits of Graph Neural Networks](https://mlsanity.com/paper/2002.06157/) (2020)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Polar Transformer Networks](https://mlsanity.com/paper/1709.01889/) (2017)
- [Zero-Shot Day-Night Domain Adaptation with a Physics Prior](https://mlsanity.com/paper/2108.05137/) (2021)
- [Equivariant Multi-View Networks](https://mlsanity.com/paper/1904.00993/) (2019)
- [Surface Networks via General Covers](https://mlsanity.com/paper/1812.10705/) (2018)
- [CIRCLe: Color Invariant Representation Learning for Unbiased Classification of Skin Lesions](https://mlsanity.com/paper/2208.13528/) (2022)
- [Planckian Jitter: countering the color-crippling effects of color jitter on self-supervised training](https://mlsanity.com/paper/2202.07993/) (2022)
- [You Only Look Around: Learning Illumination Invariant Feature for Low-light Object Detection](https://mlsanity.com/paper/2410.18398/) (2024)
- [Color Equivariant Convolutional Networks](https://mlsanity.com/paper/2310.19368/) (2023)
- [Learning Color Equivariant Representations](https://mlsanity.com/paper/2406.09588/) (2024)
- [Empowering Networks With Scale and Rotation Equivariance Using A Similarity Convolution](https://mlsanity.com/paper/2303.00326/) (2023)
- [Tomato Maturity Recognition with Convolutional Transformers](https://mlsanity.com/paper/2307.01530/) (2023)
- [Image Intrinsic Scale Assessment: Bridging the Gap Between Quality and Resolution](https://mlsanity.com/paper/2502.06476/) (2025)

## Built on by (most cited first, 1 of 1)

- [Transformation Laws in Neural Representations: Structure, Realisability, and Construction](https://mlsanity.com/paper/2609.18190/) (2026)

## Abstract (arXiv)

When the color distribution of input images changes at inference, the performance of conventional neural network architectures drops considerably. A few researchers have begun to incorporate prior knowledge of color geometry in neural network design. These color equivariant architectures have modeled hue variation with 2D rotations, and saturation and luminance transformations as 1D translations. While this approach improves neural network robustness to color variations in a number of contexts, we find that approximating saturation and luminance (interval valued quantities) as 1D translations introduces appreciable artifacts. In this paper, we introduce a color equivariant architecture that is truly equivariant. Instead of approximating the interval with the real line, we lift values on the interval to values on the circle (a double-cover) and build equivariant representations there. Our approach resolves the approximation artifacts of previous methods, improves interpretability and generalizability, and achieves better predictive performance than conventional and equivariant baselines on tasks such as fine-grained classification and medical imaging tasks. Going beyond the context of color, we show that our proposed lifting can also extend to geometric transformations such as scale.