# Group invariant machine learning by fundamental domain projections

arXiv: https://arxiv.org/abs/2202.02164 · HTML: https://mlsanity.com/paper/2202.02164/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 43 references that are in the map.
- First posted on arXiv 4 Feb 2022.

## Builds on (most cited first, 19 of 19)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Universal approximations of invariant maps by neural networks](https://mlsanity.com/paper/1804.10306/) (2018)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [On Learning Sets of Symmetric Elements](https://mlsanity.com/paper/2002.08599/) (2020)
- [Deep Models of Interactions Across Sets](https://mlsanity.com/paper/1803.02879/) (2018)
- [Machine Learning CICY Threefolds](https://mlsanity.com/paper/1806.03121/) (2018)
- [Frame Averaging for Equivariant Shape Space Learning](https://mlsanity.com/paper/2112.01741/) (2021)
- [Learning Algebraic Structures: Preliminary Investigations](https://mlsanity.com/paper/1905.02263/) (2019)
- [Inception Neural Network for Complete Intersection Calabi-Yau 3-folds](https://mlsanity.com/paper/2007.13379/) (2020)
- [Towards Principled Design of Deep Convolutional Networks: Introducing SimpNet](https://mlsanity.com/paper/1802.06205/) (2018)

## Built on by (most cited first, 11 of 11)

- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Are demographically invariant models and representations in medical imaging fair?](https://mlsanity.com/paper/2305.01397/) (2023)
- [Stability of sorting based embeddings](https://mlsanity.com/paper/2410.05446/) (2024)
- [Interpreting Equivariant Representations](https://mlsanity.com/paper/2401.12588/) (2024)
- [Learning 3-Manifold Triangulations](https://mlsanity.com/paper/2405.09610/) (2024)
- [Calabi-Yau Four/Five/Six-folds as mathbbPnw Hypersurfaces: Machine Learning, Approximation, and Generation](https://mlsanity.com/paper/2311.17146/) (2023)
- [Approximating invariant functions with the sorting trick is theoretically justified](https://mlsanity.com/paper/2403.01671/) (2024)
- [Thinner Latent Spaces: Detecting Dimension and Imposing Invariance with Conformal Autoencoders](https://mlsanity.com/paper/2408.16138/) (2024)
- [Equivariance and Augmentation for Bayesian Neural Networks](https://mlsanity.com/paper/2606.26273/) (2026)
- [Group-invariant Coresets for Data-efficient Active Learning](https://mlsanity.com/paper/2607.01089/) (2026)
- [Triply-Scalable Equivariant Gaussian Process Modeling](https://mlsanity.com/paper/2609.21085/) (2026)

## Abstract (arXiv)

We approach the well-studied problem of supervised group invariant and equivariant machine learning from the point of view of geometric topology. We propose a novel approach using a pre-processing step, which involves projecting the input data into a geometric space which parametrises the orbits of the symmetry group. This new data can then be the input for an arbitrary machine learning model (neural network, random forest, support-vector machine etc). We give an algorithm to compute the geometric projection, which is efficient to implement, and we illustrate our approach on some example machine learning problems (including the well-studied problem of predicting Hodge numbers of CICY matrices), in each case finding an improvement in accuracy versus others in the literature. The geometric topology viewpoint also allows us to give a unified description of so-called intrinsic approaches to group equivariant machine learning, which encompasses many other approaches in the literature.