提出高阶3D点构型的完备协变特征，用于分子量子性质学习
Complete and Efficient Covariants for 3D Point Configurations with Application to Learning Molecular Quantum Properties
- 构建高阶协变特征，理论证明其完备性
- 仅需6k-5个特征即可完整描述最多k个原子的系统
- 用矩阵乘法替代克莱布希-戈登运算，提速100倍以上
在机器学习建模分子物理性质时，引入SO(3)协变性至关重要。现有基于低阶特征的方法不完整，本文提出并证明了高阶方法的普遍完备性，表明对最多k个原子的系统，仅需6k-5个特征即可实现完备表示。同时发现，通常使用的克莱布希-戈登运算可被矩阵乘法替代，且不损失完备性，使计算复杂度从O(l^6)降至O(l^3)。该方法应用于量子化学，但适用于所有涉及3D点构型的问题。
原文摘要 · Abstract (English)
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.
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