# Universal Collection of Euclidean Invariants between Pairs of Position-Orientations

arXiv: https://arxiv.org/abs/2504.03299 · HTML: https://mlsanity.com/paper/2504.03299/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 15 references that are in the map.
- First posted on arXiv 4 Apr 2025.

## Builds on (most cited first, 10 of 10)

- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Spherical Message Passing for 3D Graph Networks](https://mlsanity.com/paper/2102.05013/) (2021)
- [PPF-FoldNet: Unsupervised Learning of Rotation Invariant 3D Local Descriptors](https://mlsanity.com/paper/1808.10322/) (2018)
- [Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Space](https://mlsanity.com/paper/2310.02970/) (2023)
- [Mesh Neural Networks for SE(3)-Equivariant Hemodynamics Estimation on the Artery Wall](https://mlsanity.com/paper/2212.05023/) (2022)
- [LaB-GATr: geometric algebra transformers for large biomedical surface and volume meshes](https://mlsanity.com/paper/2403.07536/) (2024)

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

- [Generalized Reduction to the Isotropy for Flexible Equivariant Neural Fields](https://mlsanity.com/paper/2603.08758/) (2026)

## Abstract (arXiv)

Euclidean E(3) equivariant neural networks that employ scalar fields on position-orientation space M(3) have been effectively applied to tasks such as predicting molecular dynamics and properties. To perform equivariant convolutional-like operations in these architectures one needs Euclidean invariant kernels on M(3) x M(3). In practice, a handcrafted collection of invariants is selected, and this collection is then fed into multilayer perceptrons to parametrize the kernels. We rigorously describe an optimal collection of 4 smooth scalar invariants on the whole of M(3) x M(3). With optimal we mean that the collection is independent and universal, meaning that all invariants are pertinent, and any invariant kernel is a function of them. We evaluate two collections of invariants, one universal and one not, using the PONITA neural network architecture. Our experiments show that using a collection of invariants that is universal positively impacts the accuracy of PONITA significantly.