# Integral Formulas for Vector Signal Tensor Products

arXiv: https://arxiv.org/abs/2603.08630 · HTML: https://mlsanity.com/paper/2603.08630/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 16 references that are in the map.
- First posted on arXiv 9 Mar 2026.

## Builds on (most cited first, 7 of 7)

- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products](https://mlsanity.com/paper/2401.10216/) (2024)
- [The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products](https://mlsanity.com/paper/2506.13523/) (2025)
- [E2Former: An Efficient and Equivariant Transformer with Linear-Scaling Tensor Products](https://mlsanity.com/paper/2501.19216/) (2025)
- [Differentiable and accelerated spherical harmonic and Wigner transforms](https://mlsanity.com/paper/2311.14670/) (2023)
- [Asymptotically Fast Clebsch-Gordan Tensor Products with Vector Spherical Harmonics](https://mlsanity.com/paper/2602.21466/) (2026)

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

- [Edge Cluster Expansion with Radial Rotary Attention for Interatomic Potentials](https://mlsanity.com/paper/2607.10664/) (2026)

## Abstract (arXiv)

We derive integral formulas that simplify the Vector Signal Tensor Product recently introduced by Xie et al., which generalizes the Gaunt tensor product to anti-symmetric couplings. In particular, we obtain explicit closed-form expressions for the anti-symmetric analogues of the Gaunt coefficients. This enables us to simulate the Clebsch-Gordan tensor product using a single Vector Signal Tensor Product, yielding up to a 9× reduction in the required tensor product evaluations. Our results enable efficient and practical implementations of the Vector Signal Tensor Product, paving the way for applications of this generalization of Gaunt Tensor Products in SO(3)-equivariant neural networks. Moreover, we discuss how the Gaunt and the Vector Signal Tensor Products allow to control the expressivity-runtime tradeoff associated with the usual Clebsch-Gordan Tensor Products. Finally, we investigate low rank decompositions of the normalizations of the considered tensor products in view of their use in equivariant neural networks.