Asymptotically Fast Clebsch-Gordan Tensor Products with Vector Spherical Harmonics

TL;DR

This paper introduces a novel, asymptotically faster algorithm for computing Clebsch-Gordan tensor products in $E(3)$-equivariant neural networks, significantly improving efficiency from $O(L^6)$ to near $O(L^4)$, enabling more scalable 3D modeling.

Contribution

The authors develop the first complete, asymptotically efficient algorithm for Clebsch-Gordan tensor products, extending previous methods with generalized tensor spherical harmonics.

Findings

Runtime complexity reduced from $O(L^6)$ to $O(L^4\,\log^2 L)$

Generalized Gaunt formula for tensor harmonics proved

Only vector valued signals needed to recover missing interactions

Abstract

$E(3)$-equivariant neural networks have proven to be effective in a wide range of 3D modeling tasks. A fundamental operation of such networks is the tensor product, which allows interaction between different feature types. Because this operation scales poorly, there has been considerable work towards accelerating this interaction. However, recently \citet{xieprice} have pointed out that most speedups come from a reduction in expressivity rather than true algorithmic improvements on computing Clebsch-Gordan tensor products. A modification of Gaunt tensor product \citep{gaunt} can give a true asymptotic speedup but is incomplete and misses many interactions. In this work, we provide the first complete algorithm which truly provides asymptotic benefits Clebsch-Gordan tensor products. For full CGTP, our algorithm brings runtime complexity from the naive $O(L^6)$ to $O(L^4\log^2 L)$, close to the lower bound of $O(L^4)$. We first show how generalizing fast Fourier based convolution naturally leads to the previously proposed Gaunt tensor product \citep{gaunt}. To remedy antisymmetry issues, we generalize from scalar signals to irrep valued signals, giving us tensor spherical harmonics. We prove a generalized Gaunt formula for the tensor harmonics. Finally, we show that we only need up to vector valued signals to recover the missing interactions of Gaunt tensor product.