mlsanityData to 14 Oct 2026

The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products

arXiv 2506.13523 ↗First posted 16 Jun 2025cs.LG cs.AIcited by 15

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Abstract (arXiv)

E(3)-equivariant neural networks have demonstrated success across a wide range of 3D modelling tasks. A fundamental operation in these networks is the tensor product, which interacts two geometric features in an equivariant manner to create new features. Due to the high computational complexity of the tensor product, significant effort has been invested to optimize the runtime of this operation. For example, Luo et al. (2024) recently proposed the Gaunt tensor product (GTP) which promises a significant speedup. In this work, we provide a careful, systematic analysis of a number of tensor product operations. In particular, we emphasize that different tensor products are not performing the same operation. The reported speedups typically come at the cost of expressivity. We introduce measures of expressivity and interactability to characterize these differences. In addition, we realized the original implementation of GTP can be greatly simplified by directly using a spherical grid at no cost in asymptotic runtime. This spherical grid approach is faster on our benchmarks and in actual training of the MACE interatomic potential by 30%. Finally, we provide the first systematic microbenchmarks of the various tensor product operations. We find that the theoretical runtime guarantees can differ wildly from empirical performance, demonstrating the need for careful application-specific benchmarking. Code is available at https://github.com/atomicarchitects/PriceofFreedom.

Key facts. Cited by 15 papers in the map (data through 14 Oct 2026). Builds on 21 of its 43 references that are in the map. First posted on arXiv 16 Jun 2025.

Builds on

21 of 43 references.

References · 21

  1. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 746
  2. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 708
  3. 2022MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldscited by 474
  4. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 354
  5. 2022Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphscited by 264
  6. 2022Learning Local Equivariant Representations for Large-Scale Atomistic Dynamicscited by 256
  7. 2022FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operatorscited by 188
  8. 2022e3nn: Euclidean Neural Networkscited by 174
  9. 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 171
  10. 2023Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNscited by 117
  11. 2018N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentialscited by 42
  12. 2024Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Productscited by 36
9 more
  1. 2020Efficient Generalized Spherical CNNscited by 29
  2. 2025An Efficient Sparse Kernel Generator for O(3)-Equivariant Deep Networkscited by 22
  3. 2024Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?cited by 18
  4. 2024Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passingcited by 13
  5. 2020Hierarchical Roofline Performance Analysis for Deep Learning Applicationscited by 11
  6. 2024High-Rank Irreducible Cartesian Tensor Decomposition and Bases of Equivariant Spacescited by 6
  7. 2024Complete and Efficient Covariants for 3D Point Configurations with Application to Learning Molecular Quantum Propertiescited by 2
  8. 2020Hierarchical Roofline Analysis: How to Collect Data using Performance Tools on Intel CPUs and NVIDIA GPUscited by 1
  9. 2025Optimizing Data Distribution and Kernel Performance for Efficient Training of Chemistry Foundation Models: A Case Study with MACEcited by 1

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 4; 2026: 11.

15 papers.

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  1. 2026EquiformerV3: Scaling Efficient, Expressive, and General SE(3)-Equivariant Graph Attention Transformerscited by 11
  2. 2025Training a Foundation Model for Materials on a Budgetcited by 6
  3. 2025Universally Invariant Learning in Equivariant GNNscited by 3
  4. 2025A Cartesian-3j Framework for Machine Learning Interatomic Potentialscited by 3
  5. 2026E2Former-V2: On-the-Fly Equivariant Attention with Linear Activation Memorycited by 3
  6. 2026How the Optimizer Shapes Learned Solutions in Equivariant Neural Networkscited by 2
  7. 2025Tensor Decomposition Networks for Fast Machine Learning Interatomic Potential Computationscited by 1
  8. 2026Integral Formulas for Vector Signal Tensor Productscited by 1
  9. 2026Sobek: Streaming Equivariant Tensor Product Convolutionscited by 1
  10. 2026Diagnosing Spectral Ceilings in Equivariant Neural Force Fieldscited by 0
  11. 2026MALOQ: Massively Accelerated Learning of Operators for Quantum Transportcited by 0
  12. 2026Spin-Weighted Spherical Harmonics Enable Complete and Scalable E(3)-Equivariant Networkscited by 0
3 more
  1. 2026E3DGS: Unified Geometric-Photometric Equivariance for 3D Gaussian Splatting via Color-as-Geometry Embeddingcited by 0
  2. 2026E3J: An Efficient and Open-Source Backend for Euclidean Equivariant Operations on GPU and TPUcited by 0
  3. 2026LunaTP: Enabling Efficient and Scalable Tensor Product for Equivariant MLIPscited by 0