Asymptotically Fast Clebsch-Gordan Tensor Products with Vector Spherical Harmonics

$E(3)$-equivariant neural networks have proven to be extremely 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 [Xie et al. 2025](https://openreview.net/forum?id=EvIwwGYTLc) 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 ([Luo et al.](https://openreview.net/forum?id=mhyQXJ6JsK)) 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 ([Luo et al.](https://openreview.net/forum?id=mhyQXJ6JsK)). 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.

Symmetry is extremely useful for simplifying our understanding of nature. Therefore there has been significant effort towards building neural networks which use those symmetries. A fundamental operation in rotationally symmetric networks is the tensor product. However this operation is slow and scales poorly. Our work introduces the first algorithm for this operation which is provably faster at large enough scales. We draw inspiration from FFT convolution which uses a classic trick of speeding up computation. By changing the basis we work in (using a Fourier transform), a complex interaction (convolution) becomes simple one (multiplication). Naively applying the same idea for our operation, we explain why a previously proposed algorithm is fast, but it is limited to only part of the tensor product operation. We realize that the naive attempt tries to use too simple an interaction in the new basis (pointwise multiplication) and making it slightly more complex (pointwise cross product) can resolve the limitations while only making things slightly slower. In the process, we also derived new formulas that may be useful for science outside of geometric deep learning.