When modeling physical properties of molecules with machine learning, it is desirable to incorporate $SO(3)$-covariance. While such models based on low body order features are not complete, we formulate and prove general completeness properties for higher order methods, and show that $6k-5$ of these features are enough for up to $k$ atoms. We also find that the Clebsch--Gordan operations commonly used in these methods can be replaced by matrix multiplications without sacrificing completeness, lowering the scaling from $O(l^6)$ to $O(l^3)$ in the degree of the features. We apply this to quantum chemistry, but the proposed methods are generally applicable for problems involving 3D point configurations.

翻译：在利用机器学习建模分子物理性质时，融入$SO(3)$-协变性是理想目标。虽然基于低体序特征的此类模型并不完备，我们针对高阶方法提出并证明了普遍的完备性性质，并证明对于包含最多$k$个原子的体系，$6k-5$个此类特征即已足够。我们还发现，这些方法中常用的Clebsch--Gordan运算可被矩阵乘法替代而不损失完备性，从而将特征阶数相关的复杂度从$O(l^6)$降低至$O(l^3)。我们将此应用于量子化学领域，但所提出的方法普遍适用于涉及三维点构型的各类问题。