# Constructing 3D Rotational Invariance and Equivariance with Symmetric Tensor Networks

arXiv: https://arxiv.org/abs/2508.12596 · HTML: https://mlsanity.com/paper/2508.12596/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 40 references that are in the map.
- First posted on arXiv 18 Aug 2025.

## Builds on (most cited first, 20 of 20)

- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [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)
- [Tensor Networks Meet Neural Networks: A Survey and Future Perspectives](https://mlsanity.com/paper/2302.09019/) (2023)
- [Einconv: Exploring Unexplored Tensor Network Decompositions for Convolutional Neural Networks](https://mlsanity.com/paper/1908.04471/) (2019)
- [Deep Learning and Quantum Entanglement: Fundamental Connections with Implications to Network Design](https://mlsanity.com/paper/1704.01552/) (2017)
- [Equivariant Graph Attention Networks for Molecular Property Prediction](https://mlsanity.com/paper/2202.09891/) (2022)
- [Brauer's Group Equivariant Neural Networks](https://mlsanity.com/paper/2212.08630/) (2022)
- [Machine learning and invariant theory](https://mlsanity.com/paper/2209.14991/) (2022)
- [Tensor cumulants for statistical inference on invariant distributions](https://mlsanity.com/paper/2404.18735/) (2024)
- [Cost-efficient Gaussian Tensor Network Embeddings for Tensor-structured Inputs](https://mlsanity.com/paper/2205.13163/) (2022)
- [Unifying O(3) Equivariant Neural Networks Design with Tensor-Network Formalism](https://mlsanity.com/paper/2211.07482/) (2022)
- [Tensor learning with orthogonal, Lorentz, and symplectic symmetries](https://mlsanity.com/paper/2406.01552/) (2024)

## Abstract (arXiv)

Symmetry-aware architectures are central to geometric deep learning. We present a systematic approach for constructing continuous rotationally invariant and equivariant functions using symmetric tensor networks. The proposed framework supports inputs and outputs given as a tuple of Cartesian tensors of different rank as well as spherical tensors of different type. We introduce tensor network generators for invariant maps and obtain equivariant maps via differentiation. Specifically, we derive general continuous equivariant maps from vector inputs to Cartesian or spherical tensor output. Finally, we clarify how common equivariant primitives in geometric graph neural networks arise within our construction.