Integral Formulas for Vector Spherical Tensor Products

Abstract

We derive integral formulas that simplify the Vector Spherical Tensor Product recently introduced by Xie et al., which generalizes the Gaunt tensor product to antisymmetric couplings. In particular, we obtain explicit closed-form expressions for the antisymmetric analogues of the Gaunt coefficients. This enables us to simulate the Clebsch-Gordan tensor product using a single Vector Spherical Tensor Product, yielding a $9\times$ reduction in the required tensor product evaluations. Our results enable efficient and practical implementations of the Vector Spherical 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 Spherical 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.

Figures (2)

• Figure 1: Decomposition quality for $(\tilde{V}_{l_1 l_2}^{l_3})^{-1}$ vs $L_{\max}$. (a)$\sigma_{\!\log}$ (log scale): rank 2 remains below 0.04, while rank 1 exhibits order-of-magnitude errors. (b) 100% of rank-2 entries lie within a factor of $2$. (c)$R^2$ for the rank-2 fit remains above $0.9$.
• Figure 2: $\sigma_{\!\log}$ vs $L_{\max}$ for $(\tilde{V}^{l_3}_{l_1 l_2})^{-1}$ (left) and $(\widetilde{G}^{l_3}_{l_1,l_2})^{-1}$ (right). The dashed line indicates $2\times$ scatter. We see that $\tilde{V}^{-1}$ requires rank 2, whereas $\widetilde{G}^{-1}$ a single rank approximation is sufficient for normalization in a neural network.

Theorems & Definitions (4)

• Theorem 1
• proof
• Theorem 2
• proof