Pith Number

pith:54HDAFAW

pith:2026:54HDAFAWNSEV5AVEC2ADM4UGJC

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Fast contracted Clebsch--Gordan tensor products for equivariant graph neural networks

Anton Bochkarev, Ralf Drautz, Yury Lysogorskiy

An O(L^3) algorithm evaluates contracted Clebsch-Gordan tensor products for O(3)-equivariant machine learning potentials using a structured grid and surface-curl pairing.

arxiv:2605.15073 v1 · 2026-05-14 · physics.comp-ph · cond-mat.mtrl-sci · physics.chem-ph

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Claims

C1strongest claim

We present an O(L^3) algorithm for evaluating contracted Clebsch--Gordan tensor products in O(3)-equivariant machine learning potentials at fixed Canonical Polyadic (CP) rank.

C2weakest assumption

The structured Gauss-Legendre and Fourier grid combined with the surface-curl pairing accurately recovers all required Clebsch-Gordan channels, including antisymmetric parity-odd ones, without introducing numerical errors or breaking equivariance for the target L range.

C3one line summary

An O(L^3) algorithm computes contracted Clebsch-Gordan tensor products for equivariant ML potentials using a structured angular grid and spherical Poisson bracket to handle parity-odd terms at fixed CP rank.

References

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[1] Parity-odd identity recovery For each parity-odd triple (l1, l2, l) we evaluate Eq.(15) on every (m 1, m2, m) entry withm=m 1 +m 2 and ex- tractκ −1(l1, l2, l) = integral/G lm l1m1 l2m2 from the non-

[2] Across all parity-even triples up to (4,4,8), the maximum integral was∼10 −10, set by the quadrature

[3] Antisymmetry under1↔2exchange The surface-curl integrand ˆr·[∇ S2 Yl1m1 × ∇S2 Yl2m2] flips sign exactly under exchange of the two factors. Di- rect evaluation across all parity-odd triples up to (4,4,

[4] End-to-end consistency on random synthetic fields To verify that the surface-curl construction reproduces the contracted CG output not only on plain spheri- cal harmonics but on arbitrary input fields

[5] Φ(sym/anti) lm = X n1,n2 M (sym/anti) n1n2 × X m1m2 Glm l1m1 l2m2 An1l1m1 An2l2m2 14 withA nlm random Gaussian,l 1 =l 2,M sym symmetric, M anti antisymmetric

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First computed	2026-05-17T21:40:26.020099Z
Last reissued	2026-05-17T21:57:19.328196Z
Builder	pith-number-builder-2026-05-17-v1
Signature	unsigned_v0
Schema	pith-number/v1.0

Canonical hash

ef0e3014166c895e82a41680367286489085e6560f68ba3dac95e55188660f86

Aliases

arxiv: 2605.15073 · arxiv_version: 2605.15073v1 · pith_short_12: 54HDAFAWNSEV · pith_short_16: 54HDAFAWNSEV5AVE · pith_short_8: 54HDAFAW

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Canonical record JSON

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    "abstract_canon_sha256": "7a242647d6f4b7d650096015e4c5c6a83787a9c779bde28124a74f1377b9d1f3",
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    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "physics.comp-ph",
    "submitted_at": "2026-05-14T16:59:00Z",
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