Representing spherical tensors with scalar-based machine-learning models.

Area of Science:

• Physics
• Computational Chemistry
• Materials Science

Background:

• Rotational symmetry is crucial for understanding 3D object properties across scales.
• Equivariant models ensure consistency with rotation group structures using spherical tensors.
• Current methods face computational challenges and implementation complexity.

Purpose of the Study:

• To develop a more computationally tractable method for learning rotational symmetry in 3D point clouds.
• To explore a novel approach by expressing equivariant functions as products of scalar functions and tensor bases.
• To investigate the separation of learning into learnable scalars and fixed geometric terms.

Main Methods:

• Representing equivariant functions as a product of a scalar function and a basis of symmetric tensors.
• Decomposing the learning of equivariant properties into learnable scalars and fixed geometric terms derived from interatomic vectors.
• Developing approximations for practical, efficient, and accurate implementations.

Main Results:

• Demonstrated that learning equivariant properties can be separated into learnable scalars and fixed geometric components.
• Proposed approximations that are computationally fast, simple to implement, and accurate.
• Showcased a viable alternative to computationally demanding fully equivariant or unconstrained models.

Conclusions:

• The proposed method offers an efficient and practical approach to incorporating rotational symmetry in 3D point cloud analysis.
• This technique balances the need for symmetry adherence with computational feasibility.
• The findings pave the way for more accessible and scalable equivariant deep learning models in scientific applications.