Spin-Weighted Spherical Harmonics Enable Complete and Scalable $\mathrm{E}(3)$-Equivariant Networks

2026-07-01

AI Summary

Existing $\mathrm{E}(3)$-equivariant networks are constrained by the $O(L^6)$ computational complexity of Clebsch–Gordan tensor products, while efficient alternatives such as Gaunt tensor products sacrifice expressivity due to the absence of antisymmetric pathways. This work introduces spin-weighted spherical harmonics (SWSH) into equivariant learning for the first time and proposes the SpinGTP framework: a novel tensor product operator derived from the algebraic structure of SWSH that retains Gaunt-level computational efficiency while recovering full symmetry expressivity, including parity-odd components. Experiments demonstrate that SpinGTP achieves accuracy comparable to full Clebsch–Gordan tensor products on benchmarks such as Tetris, 3BPA, SPICE-MACE-OFF, and OC20, and further exhibits superior performance on tasks involving chiral materials and non-centrosymmetric structures.

Abstract

$\mathrm{E}(3)$-equivariant networks are promising for 3D atomistic system modeling, yet their scalability is limited by the $O(L^6)$ complexity of the Clebsch-Gordan Tensor Product (CGTP). The recently proposed Gaunt Tensor Product (GTP) reduces the complexity but is unable to capture the antisymmetric paths, resulting in incomplete expressivity. In this work, we present SpinGTP, an approach to overcome the GTP incompleteness by generalizing from scalar functions to Spin-Weighted Spherical Harmonics (SWSH). By relying on the algebraic properties of SWSH, SpinGTP recovers the missing antisymmetric interactions while maintaining the asymptotic efficiency of GTP. It also allows for a more expressive equivariant basis that naturally accounts for the parity-odd components of tensor products. We evaluate SpinGTP across diverse benchmarks, including Tetris, 3BPA, SPICE-MACE-OFF, and OC20. Our results show that SpinGTP achieves accuracies comparable to full CGTP. Notably, by explicitly capturing antisymmetric paths, SpinGTP exhibits superior performance in tasks involving chiral materials and non-centrosymmetric geometries. This work provides a complete, scalable, and mathematically rigorous path toward high-order equivariance in large-scale 3D atomistic system simulations.

Problem

• E(3)-equivariant networks
• scalability
• antisymmetric paths
• expressivity
• 3D atomistic systems

Innovation

• Spin-Weighted Spherical Harmonics
• E(3)-equivariant networks
• Gaunt Tensor Product
• antisymmetric interactions
• scalable equivariance