Asymptotically Fast Clebsch-Gordan Tensor Products with Vector Spherical Harmonics

TL;DR

This work provides the first complete algorithm which truly provides asymptotic benefits Clebsch-Gordan tensor products, and proves a generalized Gaunt formula for the tensor harmonics.

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 (2)

• 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.

Theorems & Definitions (24)

• Definition 2.1: Tensor product operations
• Definition 2.2: Interactability
• Definition 2.3: Triangle condition
• Theorem 3.1: Peter-Weyl Theorem
• Definition 4.1: Tensor spherical harmonics
• Remark 4.2
• Theorem 4.3
• proof : Proof sketch.
• Remark 4.4
• Theorem 5.1: Selection rules for VSTP