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Complete and Efficient Covariants for 3D Point Configurations with Application to Learning Molecular Quantum Properties

Hartmut Maennel, Oliver T. Unke, Klaus-Robert Müller

TL;DR

This work 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 in the degree of the features.

Abstract

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.

Complete and Efficient Covariants for 3D Point Configurations with Application to Learning Molecular Quantum Properties

TL;DR

This work formulate and prove general completeness properties for higher order methods, and show that of these features are enough for up to atoms in the degree of the features.

Abstract

When modeling physical properties of molecules with machine learning, it is desirable to incorporate -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 of these features are enough for up to 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 to 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.
Paper Structure (50 sections, 20 theorems, 156 equations, 1 figure, 4 tables, 1 algorithm)

This paper contains 50 sections, 20 theorems, 156 equations, 1 figure, 4 tables, 1 algorithm.

Table of Contents

  1. Overview diagram statements and proofs
  2. Variants of G, X, and function spaces
  3. Description of variants
  4. Separation properties of radial functions
  5. Capabilities to distinguish configurations
  6. Weaker conditions on radial functions
  7. General theorems
  8. Pseudo code for algorithm
  9. Incompleteness of 3 body functions
  10. Some background from representation theory
  11. The irreducible representations of SO(3)
  12. Equivariant maps from the sphere to an irreducible representation
  13. Schur's lemma
  14. Isotypic decomposition
  15. Dual, tensor product, and Hom
  16. ...and 35 more sections

Key Result

Theorem 5

Any $SO(3)$--covariant feature with values in a ${\mathcal{H}}^{(l)}$ can be written as a linear combination of the $SO(3)$--covariants eq:matrix_invariants with $a_m=l$. For $O(3)$--covariants it is enough to use those features given by eq:matrix_invariants with the appropriate parity of $l_1+...+l

Figures (1)

  • Figure 1: A: Given an atom (red) in a local chemical environment (translucent red sphere), the aim is to find a descriptor, i.e. a fixed-size feature vector $\mathbf{x}$, such that all different environments are also mapped to different descriptors (uniqueness). The features can be either invariant or covariant w.r.t. rotations, meaning that when the environment is rotated, the new features $\mathbf{x'}$ are either identical (invariant) or "rotate in the same way" (covariant). B: Examples for local chemical environments (taken from Ref. Pozdnyakov2020), which cannot be distinguished by features constructed from $m$-body information. From the perspective of the central black atom, the environments 1a and 1b appear identical when considering only 2-body information (e.g. distances), but are readily distinguished by 3-body information (e.g. angles). For environments 2a and 2b, 4-body information (e.g. dihedral angles) is necessary to distinguish them, whereas environments 3a and 3b require even higher-order information. C: Computational cost for evaluating learned invariants. While the straightforward GPU/TPU-friendly implementation of invariants using Clebsch--Gordan operations scales with $O(l^6)$, the proposed implementation replacing them by matrix multiplications scales with $O(l^3)$, enabling the use of higher degrees. D: Examples of invariants that distinguish the pairs of B above. According to Theorem \ref{['theorem:MatMultComplete']} there must be a distinguishing invariant which is given by matrix multiplication as in \ref{['eq:matrix_invariants']}, here we used $l=2$. Products of $k$ matrices give $(k+1)$--body invariants, so for structure pairs 1, 2, and 3 (see panel B) we need 2, 3, and 4 matrix factors, respectively. While in general these invariants are $SO(3)$--invariants, the sum of the matrix indices chosen here is even, so the invariants are actually $O(3)$--invariants and these numbers show that the pairs are also in distinct $O(3)$--equivalent classes.

Theorems & Definitions (38)

  • Theorem 5: Algebraic Completeness for features from matrix multiplication
  • Lemma 1
  • proof
  • Lemma 2: Separation of radial functions
  • proof
  • Proposition 3
  • proof
  • Theorem 1: Topological Completeness
  • Theorem 2: Finiteness
  • Theorem 3: Algebraic completeness when using Spherical Harmonics and Clebsch--Gordon operations
  • ...and 28 more