Integral Formulas for Vector Spherical Tensor Products

2026-03-09

AI Summary

This work addresses the high computational complexity and lack of closed-form expressions in vector spherical harmonic tensor product calculations, particularly in antisymmetric coupling scenarios. By leveraging group representation theory, spherical harmonic analysis, and tensor algebra—combined with low-rank decomposition techniques—we derive, for the first time, an explicit closed-form solution for antisymmetric Gaunt coefficients. We formulate an efficient integral expression that replaces the conventional Clebsch–Gordan product with a single vector spherical harmonic tensor product. This approach reduces computational cost by a factor of nine while preserving SO(3) equivariance, substantially improving runtime efficiency. Moreover, it enables flexible trade-offs between representational capacity and computational overhead, offering a practical and efficient implementation pathway for equivariant neural networks.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Generative Adversarial Networks (GANs) for Vision
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We derive integral formulas that simplify the Vector Spherical Tensor Product recently introduced by Xie et al., which generalizes the Gaunt tensor product to antisymmetric couplings. In particular, we obtain explicit closed-form expressions for the antisymmetric analogues of the Gaunt coefficients. This enables us to simulate the Clebsch-Gordan tensor product using a single Vector Spherical Tensor Product, yielding a $9\times$ reduction in the required tensor product evaluations. Our results enable efficient and practical implementations of the Vector Spherical Tensor Product, paving the way for applications of this generalization of Gaunt tensor products in $\mathrm{SO}(3)$-equivariant neural networks. Moreover, we discuss how the Gaunt and the Vector Spherical 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.

Problem

• Vector Spherical Tensor Product
• Gaunt coefficients
• antisymmetric couplings
• SO(3)-equivariant neural networks
• Clebsch-Gordan tensor product

Innovation

• Vector Spherical Tensor Product
• Gaunt coefficients
• antisymmetric coupling
• SO(3)-equivariant neural networks
• low-rank decomposition