Asymptotically Fast Clebsch-Gordan Tensor Products with Vector Spherical Harmonics

YuQing Xie; Ameya Daigavane; Mit Kotak; Tess Smidt

cs.LGFeb 25, 2026

Asymptotically Fast Clebsch-Gordan Tensor Products with Vector Spherical Harmonics

Authors: YuQing Xie, Ameya Daigavane, Mit Kotak, Tess Smidt

Organizations: Department of EECS, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · Department of CSE, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

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.

Figures & tables

Figure 1: Fourier transforms from a group theoretic perspective. An action on a group induces an action on functions on the group. These functions form a vector space and hence this defines a group representation. We can decompose this representation into irreps, giving exactly the standard Fourier basis. The Peter-Weyl theorem generalizes this idea for compact Lie groups.

Figure 2: Schematic of the process in taking a vector signal tensor product. We interpret input irreps as vector SH coefficients to create vector spherical signals. We then take pointwise cross products of the two signals to create a new signal which we decompose back into vector SH coefficients.

Tensor Product	Expressivity	Runtime	Runtime / Expressivity
CGTP (Naive)	$\mathcal{O}(L^{3})$	$\mathcal{O}(L^{6})$	$\mathcal{O}(L^{3})$
CGTP (Sparse)	$\mathcal{O}(L^{3})$	$\mathcal{O}(L^{5})$	$\mathcal{O}(L^{2})$
GTP (Fourier)	$\mathcal{O}(L)$	$\mathcal{O}(L^{3})$	$\mathcal{O}(L^{2})$
GTP (Grid)	$\mathcal{O}(L)$	$\mathcal{O}(L^{3})$	$\mathcal{O}(L^{2})$
GTP (Healy)	$\mathcal{O}(L)$	$\mathcal{O}(L^{2}\log^{2}L)$	$\mathcal{O}(L\log^{2}L)$
VSTP (Grid)	$\mathcal{O}(L)$	$\mathcal{O}(L^{3})$	$\mathcal{O}(L^{2})$

Table 1: Asymptotic runtimes and expressivity of various tensor product implementations. Note that the fast spherical transform algorithm by Healy et al. (2003) gives the best runtime/expressivity tradeoff ratios.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

$SO(n)$	Group of rotations in $n$ -dimensional space
$O(n)$	Group of rotations and inversion in $n$ -dimensional space
$S^{2}$	The 2-sphere, surface defined by $x^{2}+y^{2}+z^{2}=1$
$Y^{m}_{\ell}$	Spherical harmonic function of degree $\ell$ and order $m$
$\mathbf{Y}_{\ell}$	The collection of spherical harmonic functions of degree $\ell$ for all orders $m$
$\mathbf{Y}^{\ell,s}_{j,m}$	Tensor spherical harmonic function
$\mathbf{Y}^{\ell,s}_{j}$	Collection of tensor spherical harmonic functions for all $m$

Appendix

Table 2: Notation used throughout this paper

$G$	Denotes a group
$\rho(g)$	Representation of a group
$A$	First input space of a constructed bilinearity
$B$	Second input space of a constructed bilinearity
$C$	Output space of a constructed bilinearity
$X$	First input space of a tensor product operation (fixed equivariant bilinearity)
$Y$	Second input space of a tensor product operation (fixed equivariant bilinearity)

Appendix

Table 3: Commonly used meanings of symbols

	SISO	SIMO	MIMO
Encode	$\mathcal{O}((s_{1}+s_{2})L^{2}\log L)$	$\mathcal{O}((s_{1}+s_{2})L^{2}\log L)$	$\mathcal{O}((s_{1}^{2}+s_{2}^{2})L^{2}+(s_{1}+s_{2})L^{2}\log^{2}L)$
Interact	$\mathcal{O}(s_{1}s_{2}s_{3}L^{2}/\tilde{s})$	$\mathcal{O}(s_{1}s_{2}s_{3}L^{2}/\tilde{s})$	$\mathcal{O}(s_{1}s_{2}s_{3}L^{2}/\tilde{s})$
Decode	$\mathcal{O}(s_{3}L^{2}\log L)$	$\mathcal{O}(s_{3}^{2}L^{2}+s_{3}L^{2}\log^{2}L)$	$\mathcal{O}(s_{3}^{2}L^{2}+s_{3}L^{2}\log^{2}L)$
Total	$\mathcal{O}(s_{1}s_{2}s_{3}L^{2}/\tilde{s}+\tilde{s}L^{2}\log L)$	$\mathcal{O}((s_{1}+s_{2})L^{2}\log L+s_{3}^{2}L^{2}+s_{3}L^{2}\log^{2}L)$	$\mathcal{O}(\tilde{s}^{2}L^{2}+\tilde{s}L^{2}\log^{2}L)$

Appendix

Table 4: Asymptotic runtimes of ISTPs assuming a naive grid implementation.

	SISO	SIMO	MIMO
Encode	$\mathcal{O}((s_{1}+s_{2})L^{2}\log L)$	$\mathcal{O}((s_{1}+s_{2})L^{2}\log L)$	$\mathcal{O}((s_{1}^{2}+s_{2}^{2})L^{2}+(s_{1}+s_{2})L^{2}\log^{2}L)$
Interact	$\mathcal{O}(s_{1}s_{2}s_{3}L^{2}/\tilde{s})$	$\mathcal{O}(s_{1}s_{2}s_{3}L^{2}/\tilde{s})$	$\mathcal{O}(s_{1}s_{2}s_{3}L^{2}/\tilde{s})$
Decode	$\mathcal{O}(s_{3}L^{2}\log L)$	$\mathcal{O}(s_{3}^{2}L^{2}+s_{3}L^{2}\log^{2}L)$	$\mathcal{O}(s_{3}^{2}L^{2}+s_{3}L^{2}\log^{2}L)$
Total	$\mathcal{O}(s_{1}s_{2}s_{3}L^{2}/\tilde{s}+\tilde{s}L^{2}\log L)$	$\mathcal{O}((s_{1}+s_{2})L^{2}\log L+s_{3}^{2}L^{2}+s_{3}L^{2}\log^{2}L)$	$\mathcal{O}(\tilde{s}^{2}L^{2}+\tilde{s}L^{2}\log^{2}L)$

Appendix

Table 5: Asymptotic runtimes of ISTPs assuming asymptotically fast S2FFT grid implementation.

Tensor Product	SISO	SIMO	MIMO
Clebsch-Gordan (Naive)	$\mathcal{O}(L^{3})$	$\mathcal{O}(L^{4})$	$\color[rgb]{0.8008,0.1992,0}\mathcal{O}(L^{6})$
Clebsch-Gordan (Sparse)	$\color[rgb]{0.0391,0.4063,0.2773}\mathcal{O}(L^{2})$	$\mathcal{O}(L^{3})$	$\color[rgb]{0.8008,0.1992,0}\mathcal{O}(L^{5})$
Gaunt (Original)	$\mathcal{O}(L^{2}\log L)$	$\mathcal{O}(L^{3})$	$\mathcal{O}(L^{3})$
Gaunt (Naive Grid)	$\mathcal{O}(L^{2}\log L)$	$\mathcal{O}(L^{3})$	$\mathcal{O}(L^{3})$
Gaunt (S2FFT Grid)	$\mathcal{O}(L^{2}\log L)$	$\color[rgb]{0.0391,0.4063,0.2773}\mathcal{O}(L^{2}\log^{2}L)$	$\color[rgb]{0.0391,0.4063,0.2773}\mathcal{O}(L^{2}\log^{2}L)$
Vector Signal (Naive Grid)	$\mathcal{O}(L^{2}\log L)$	$\mathcal{O}(L^{3})$	$\mathcal{O}(L^{3})$

Appendix

Table 6: Asymptotic runtimes of various tensor products for different output settings. The best performing tensor products for each output settings are highlighted in green . In the MIMO setting, the Clebsch-Gordan tensor products are highlighted in red to indicate that they can output irreps with multiplicity

>1

, unlike the Gaunt tensor products.

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Integral Formulas for Vector Signal Tensor Products

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InstaDeep, 1 Triton Square, London, NW1 3BF, United Kingdom

Equivariant Neural Networks

Open paper AskPDF arXiv

We derive integral formulas that simplify the Vector Signal Tensor Product recently introduced by Xie et al., which generalizes the Gaunt tensor product to anti-symmetric couplings. In particular, we obtain explicit closed-form expressions for the anti-symmetric analogues of the Gaunt coefficients. This enables us to simulate the Clebsch-Gordan tensor product using a single Vector Signal Tensor Product, yielding up to a

9\times

reduction in the required tensor product evaluations. Our results enable efficient and practical implementations of the Vector Signal Tensor Product, paving the way for applications of this generalization of Gaunt Tensor Products in

\mathrm{SO}(3)

-equivariant neural networks. Moreover, we discuss how the Gaunt and the Vector Signal Tensor Products allow to control the expressivity-runtime tradeoff associated with the usual Clebsch-Gordan Tensor Products. Finally, we investigate low rank decompositions of the normalizations of the considered tensor products in view of their use in equivariant neural networks.

Abstract

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