# Exact Symmetry as Algebra: Group-Structured Tensors that Enforce Physical Selection Rules

arXiv: https://arxiv.org/abs/2605.20440 · HTML: https://mlsanity.com/paper/2605.20440/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 95 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 22 of 22)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SVD-LLM: Truncation-aware Singular Value Decomposition for Large Language Model Compression](https://mlsanity.com/paper/2403.07378/) (2024)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [ASVD: Activation-aware Singular Value Decomposition for Compressing Large Language Models](https://mlsanity.com/paper/2312.05821/) (2023)
- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [Orb: A Fast, Scalable Neural Network Potential](https://mlsanity.com/paper/2410.22570/) (2024)
- [The Lie Derivative for Measuring Learned Equivariance](https://mlsanity.com/paper/2210.02984/) (2022)
- [VN-Transformer: Rotation-Equivariant Attention for Vector Neurons](https://mlsanity.com/paper/2206.04176/) (2022)
- [Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution](https://mlsanity.com/paper/2310.02299/) (2023)
- [Bispectral Neural Networks](https://mlsanity.com/paper/2209.03416/) (2022)
- [A Space Group Symmetry Informed Network for O(3) Equivariant Crystal Tensor Prediction](https://mlsanity.com/paper/2406.12888/) (2024)
- [Symmetry Discovery for Different Data Types](https://mlsanity.com/paper/2410.09841/) (2024)
- [A Unified Framework for Discovering Discrete Symmetries](https://mlsanity.com/paper/2309.02898/) (2023)
- [The Selective G-Bispectrum and its Inversion: Applications to G-Invariant Networks](https://mlsanity.com/paper/2407.07655/) (2024)
- [Revisiting the Canonicalization for Fast and Accurate Crystal Tensor Property Prediction](https://mlsanity.com/paper/2410.02372/) (2024)
- [Representation Learning for Equivariant Inference with Guarantees](https://mlsanity.com/paper/2505.19809/) (2025)
- [Any-Subgroup Equivariant Networks via Symmetry Breaking](https://mlsanity.com/paper/2603.19486/) (2026)
- [Discovering Abstract Symbolic Relations by Learning Unitary Group Representations](https://mlsanity.com/paper/2402.17002/) (2024)
- [Tensor-Tensor Products, Group Representations, and Semidefinite Programming](https://mlsanity.com/paper/2507.12729/) (2025)
- [A Single Architecture for Representing Invariance Under Any Space Group](https://mlsanity.com/paper/2512.13989/) (2025)
- [Efficient Equivariant High-Order Crystal Tensor Prediction via Cartesian Local-Environment Many-Body Coupling](https://mlsanity.com/paper/2602.04323/) (2026)

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

- [The Token Is a Group Element: On Lie-Algebra Attention over Matrix Lie Groups](https://mlsanity.com/paper/2606.20547/) (2026)

## Abstract (arXiv)

Symmetry determines which physical responses of molecules and crystals are allowed and underlies many models used throughout the physical sciences. Yet several widely used machine-learned models learn symmetry only approximately from data, raising the question of whether approximate symmetry is sufficient for scientific prediction. Using the Orb-v3 force field, we show that it is often not: the relaxed space group of a crystal depends on the orientation of the input for 47% of a 150-crystal subsample, and relaxation lowers the space group for half of all 399 crystals studied. To characterize, quantify, and correct such failures, we introduce a tensor algebra, starG, defined by convolution over a finite symmetry group and applicable to tensors with noncommuting entries. We prove that the symmetry group determines the algebra's block-diagonal transform and use this structure to analyze model errors on symmetric targets. The resulting projections decompose the average squared error into orthogonal components arising from selection-rule violations, inconsistencies across symmetry-related inputs, and residual error. The removable components reveal how much error arises from broken symmetry, provide reference-free lower bounds on prediction error, and can be eliminated without sacrificing accuracy. For operators, projection followed by blockwise truncation yields the optimal symmetric low-rank approximation. Applying these projections to force predictions reduces both rates of space-group change to at most 2%. Exact selection rules, reference-free error bounds, and symmetry-preserving corrections are therefore available for any trained model without retraining.