# Asymptotically Fast Clebsch-Gordan Tensor Products with Vector Spherical Harmonics

arXiv: https://arxiv.org/abs/2602.21466 · HTML: https://mlsanity.com/paper/2602.21466/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Feb 2026.

## Built on by (most cited first, 4 of 4)

- [Edge Cluster Expansion with Radial Rotary Attention for Interatomic Potentials](https://mlsanity.com/paper/2607.10664/) (2026)
- [Integral Formulas for Vector Signal Tensor Products](https://mlsanity.com/paper/2603.08630/) (2026)
- [Spin-Weighted Spherical Harmonics Enable Complete and Scalable E(3)-Equivariant Networks](https://mlsanity.com/paper/2607.01408/) (2026)
- [E3J: An Efficient and Open-Source Backend for Euclidean Equivariant Operations on GPU and TPU](https://mlsanity.com/paper/2609.35099/) (2026)

## Abstract (arXiv)

E(3)-equivariant neural networks have proven to be effective in a wide range of 3D modeling tasks. A fundamental operation of such networks is the tensor product, which allows interaction between different feature types. Because this operation scales poorly, there has been considerable work towards accelerating this interaction. However, recently citetxieprice have pointed out that most speedups come from a reduction in expressivity rather than true algorithmic improvements on computing Clebsch-Gordan tensor products. A modification of Gaunt tensor product citepgaunt can give a true asymptotic speedup but is incomplete and misses many interactions. In this work, we provide the first complete algorithm which truly provides asymptotic benefits Clebsch-Gordan tensor products. For full CGTP, our algorithm brings runtime complexity from the naive O(L6) to O(L4log2 L), close to the lower bound of O(L4). We first show how generalizing fast Fourier based convolution naturally leads to the previously proposed Gaunt tensor product citepgaunt. To remedy antisymmetry issues, we generalize from scalar signals to irrep valued signals, giving us tensor spherical harmonics. We prove a generalized Gaunt formula for the tensor harmonics. Finally, we show that we only need up to vector valued signals to recover the missing interactions of Gaunt tensor product.