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.
View on arXiv@article{bellaard2025_2504.03299,
title={ Universal Collection of Euclidean Invariants between Pairs of Position-Orientations },
author={ Gijs Bellaard and Bart M. N. Smets and Remco Duits },
journal={arXiv preprint arXiv:2504.03299},
year={ 2025 }
}