Complete and Efficient Covariants for Three-Dimensional Point Configurations with Application to Learning Molecular...

When physical properties of molecules are being modeled 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(l⁶) to O(l³) in the degree of the features. We apply this to quantum chemistry, but the proposed methods are generally applicable for problems involving three-dimensional point configurations.