# Tensor learning with orthogonal, Lorentz, and symplectic symmetries

arXiv: https://arxiv.org/abs/2406.01552 · HTML: https://mlsanity.com/paper/2406.01552/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 74 references that are in the map.
- First posted on arXiv 3 Jun 2024.

## Builds on (most cited first, 30 of 44)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [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)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (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)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Deep Signature Transforms](https://mlsanity.com/paper/1905.08494/) (2019)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [A statistical model for tensor PCA](https://mlsanity.com/paper/1411.1076/) (2014)
- [Certified Robustness of Graph Neural Networks against Adversarial Structural Perturbation](https://mlsanity.com/paper/2008.10715/) (2020)
- [Graph Metanetworks for Processing Diverse Neural Architectures](https://mlsanity.com/paper/2312.04501/) (2023)
- [Signature moments to characterize laws of stochastic processes](https://mlsanity.com/paper/1810.10971/) (2018)
- [TensorNet: Cartesian Tensor Representations for Efficient Learning of Molecular Potentials](https://mlsanity.com/paper/2306.06482/) (2023)
- [Introduction to Tensor Decompositions and their Applications in Machine Learning](https://mlsanity.com/paper/1711.10781/) (2017)
- [Approximation-Generalization Trade-offs under (Approximate) Group Equivariance](https://mlsanity.com/paper/2305.17592/) (2023)
- [N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials](https://mlsanity.com/paper/1803.01588/) (2018)
- [Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors](https://mlsanity.com/paper/1512.02337/) (2015)
- [Provably Strict Generalisation Benefit for Invariance in Kernel Methods](https://mlsanity.com/paper/2106.02346/) (2021)
- [The Exact Sample Complexity Gain from Invariances for Kernel Regression](https://mlsanity.com/paper/2303.14269/) (2023)
- [A note on the sample complexity of the Er-SpUD algorithm by Spielman, Wang and Wright for exact recovery of sparsely used dictionaries](https://mlsanity.com/paper/1601.02049/) (2016)
- [Noether Networks: Meta-Learning Useful Conserved Quantities](https://mlsanity.com/paper/2112.03321/) (2021)

## Built on by (most cited first, 3 of 3)

- [Representing spherical tensors with scalar-based machine-learning models](https://mlsanity.com/paper/2505.05404/) (2025)
- [Towards Coordinate- and Dimension-Agnostic Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2505.16549/) (2025)
- [Constructing 3D Rotational Invariance and Equivariance with Symmetric Tensor Networks](https://mlsanity.com/paper/2508.12596/) (2025)

## Abstract (arXiv)

Tensors are a fundamental data structure for many scientific contexts, such as time series analysis, materials science, and physics, among many others. Improving our ability to produce and handle tensors is essential to efficiently address problems in these domains. In this paper, we show how to exploit the underlying symmetries of functions that map tensors to tensors. More concretely, we develop universally expressive equivariant machine learning architectures on tensors that exploit that, in many cases, these tensor functions are equivariant with respect to the diagonal action of the orthogonal, Lorentz, and/or symplectic groups. We showcase our results on three problems coming from material science, theoretical computer science, and time series analysis. For time series, we combine our method with the increasingly popular path signatures approach, which is also invariant with respect to reparameterizations. Our numerical experiments show that our equivariant models perform better than corresponding non-equivariant baselines.