Through the Looking-Glass: Efficient Parity-Complete Learning via Local O(2) Frames
Abstract
Making all symmetry-allowed tensor representations computationally accessible remains a central challenge in equivariant atomistic learning. Clebsch--Gordan tensor products (CGTPs) are computationally demanding, whereas efficient SO(2)-based alternatives lack a unified treatment of natural- and unnatural-parity representations. We first introduce a generalized Wigner-6j convolution that exactly recouples interactions involving additional node representations, replacing the first edge-level tensor-product intermediate with reusable node features. This reduces edge computation and storage but leaves the 𝒪(L5) angular scaling of the sparse CGTPs unchanged. To address this remaining bottleneck, we develop a local O(2) framework whose convolutions scale as 𝒪(L3). The framework explicitly accounts for both rotations and reflections. Its linear maps, tensor products, and gated nonlinearities, combined with frame transformations, ensure global O(3) equivariance for both spatial parities. Incorporating time-reversal labels extends this construction to O(3)×ℤ2𝒯. We apply the framework to magnetic TACE (mTACE), using distinct interactions for systems with and without spin--orbit coupling. On collinear CrN and noncollinear Fe benchmarks, mTACE achieves substantial reductions in atomic- and magnetic-force errors. We also provide EquivariantX, a library for global and local O(2)-equivariant computation.
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BibTeX
@misc{xu2026through,
title = {{Through the Looking-Glass: Efficient Parity-Complete Learning via Local $O(2)$ Frames}},
author = {Zemin Xu and Wenbo Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.16592},
eprint = {2608.16592},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.16592},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Through the Looking-Glass: Efficient Parity-Complete Learning via Local O(2) Frames
AU - Zemin Xu
AU - Wenbo Xie
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.16592
UR - https://arxiv.org/abs/2608.16592
AB - Making all symmetry-allowed tensor representations computationally accessible remains a central challenge in equivariant atomistic learning. Clebsch--Gordan tensor products (CGTPs) are computationally demanding, whereas efficient SO(2)-based alternatives lack a unified treatment of natural- and unnatural-parity representations. We first introduce a generalized Wigner-6j convolution that exactly recouples interactions involving additional node representations, replacing the first edge-level tensor-product intermediate with reusable node features. This reduces edge computation and storage but leaves the 𝒪(L^5) angular scaling of the sparse CGTPs unchanged. To address this remaining bottleneck, we develop a local O(2) framework whose convolutions scale as 𝒪(L^3). The framework explicitly accounts for both rotations and reflections. Its linear maps, tensor products, and gated nonlinearities, combined with frame transformations, ensure global O(3) equivariance for both spatial parities. Incorporating time-reversal labels extends this construction to O(3)×ℤ_2^𝒯. We apply the framework to magnetic TACE (mTACE), using distinct interactions for systems with and without spin--orbit coupling. On collinear CrN and noncollinear Fe benchmarks, mTACE achieves substantial reductions in atomic- and magnetic-force errors. We also provide EquivariantX, a library for global and local O(2)-equivariant computation.
ER -