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Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products

ICLR 2024 · Architectures and efficiency › CNN design · approach: Group symmetry equivariant networks · 4 citations (OpenAlex)

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Shows tensor products of irreps equal multiplication of spherical functions via Gaunt coefficients, then computes them in a 2D Fourier basis with FFT, reducing full tensor product cost from O(L^6) to O(L^3).

Problem

Full tensor products of irreducible representations used in E(3)-equivariant networks scale as O(L^6) in the max degree L, limiting practical degree to 2-3 and slowing inference on large systems.

Approach

Connects Clebsch-Gordan coefficients to Gaunt coefficients (integrals of triple products of spherical harmonics) so that irreps tensor products become multiplication of spherical functions; changes basis from spherical harmonics to a 2D Fourier basis so the products become 2D convolutions computed via the convolution theorem and FFT.

Key results

• On OC20 S2EF validation, EquiformerV2 + Gaunt-Selfmix (L=6) reaches 276 meV energy MAE, 20.1 meV/Å force MAE and 1.95% EFwT, versus 285 / 20.5 / 1.67 for EquiformerV2 at L=6.
• On 3BPA, MACE-Gaunt achieves 43.7x speed-up over the e3nn implementation with 5.8% memory cost, while keeping accuracy comparable to MACE.
• Efficiency experiments show multiple orders of magnitude speed-up over e3nn for full tensor products when L>7.

Measured effects reported in the paper

Surprising findings

• Gaunt-based parameterization, which excludes pseudovector outputs, performs competitively with Clebsch-Gordan tensor products on the SEGNN N-body task.

Limitations

• Gaunt coefficients impose more symmetry constraints and exclude pseudovector outputs, so the method is not equivalent to Clebsch-Gordan products in general.
• Efficiency results are reported on sampled random features and specific architectures rather than broad end-to-end deployments.

Experimental setup

Datasets: OC20 2M 3BPA N-body simulation

Benchmarks: OC20 S2EF validation 3BPA

Metrics: energy MAE force MAE force cosine EFwT energy MAE (meV) force MAE (meV/Å) speed-up memory cost

Baselines: e3nn eSCN MACE EquiformerV2 SCN GemNet-OC NequIP Allegro BOTNet sGDML

Builds on: e3nn eSCN MACE SEGNN EquiformerV2

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