Fast contracted Clebsch--Gordan tensor products for equivariant graph neural networks
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866916013326467072 |
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| author | Bochkarev, Anton Lysogorskiy, Yury Drautz, Ralf |
| author_facet | Bochkarev, Anton Lysogorskiy, Yury Drautz, Ralf |
| contents | We present an $\mathcal{O}(L^3)$ algorithm for evaluating contracted Clebsch--Gordan tensor products in $\mathrm{O}(3)$-equivariant machine learning potentials at fixed Canonical Polyadic (CP) rank. Mapping the angular integral to a structured Gauss--Legendre and Fourier tensor-product grid decouples the radial channel contractions from the angular transforms. The antisymmetric parity-odd Clebsch--Gordan channels, unreachable by the symmetric pointwise product on a scalar $S^2$ grid, are recovered through the surface-curl pairing $\hat r \cdot [\nabla_{S^2} A \times \nabla_{S^2} B]$, the spherical Poisson bracket, which supplies the $L=1$ angular momentum on the grid while preserving rotational equivariance. The construction extends to parity-aware equivariant message passing in atomic-cluster-expansion-style architectures and is verified by direct numerical quadrature. The full uncontracted Clebsch--Gordan tensor product remains subject to the $\mathcal{O}(L^4)$ output-size lower bound. A benchmark shows wall-clock scaling empirically as $L^2$ across the practical $l_{\max}$ range. For the on-site contraction this is pre-asymptotic, giving way to $L^3$ at large $l_{\max}$. For message passing it is structural and the runtime is memory-bandwidth bound on $L^2$-sized grid tensors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15073 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fast contracted Clebsch--Gordan tensor products for equivariant graph neural networks Bochkarev, Anton Lysogorskiy, Yury Drautz, Ralf Computational Physics Materials Science Chemical Physics We present an $\mathcal{O}(L^3)$ algorithm for evaluating contracted Clebsch--Gordan tensor products in $\mathrm{O}(3)$-equivariant machine learning potentials at fixed Canonical Polyadic (CP) rank. Mapping the angular integral to a structured Gauss--Legendre and Fourier tensor-product grid decouples the radial channel contractions from the angular transforms. The antisymmetric parity-odd Clebsch--Gordan channels, unreachable by the symmetric pointwise product on a scalar $S^2$ grid, are recovered through the surface-curl pairing $\hat r \cdot [\nabla_{S^2} A \times \nabla_{S^2} B]$, the spherical Poisson bracket, which supplies the $L=1$ angular momentum on the grid while preserving rotational equivariance. The construction extends to parity-aware equivariant message passing in atomic-cluster-expansion-style architectures and is verified by direct numerical quadrature. The full uncontracted Clebsch--Gordan tensor product remains subject to the $\mathcal{O}(L^4)$ output-size lower bound. A benchmark shows wall-clock scaling empirically as $L^2$ across the practical $l_{\max}$ range. For the on-site contraction this is pre-asymptotic, giving way to $L^3$ at large $l_{\max}$. For message passing it is structural and the runtime is memory-bandwidth bound on $L^2$-sized grid tensors. |
| title | Fast contracted Clebsch--Gordan tensor products for equivariant graph neural networks |
| topic | Computational Physics Materials Science Chemical Physics |
| url | https://arxiv.org/abs/2605.15073 |