Representing spherical tensors with scalar-based machine-learning models

2025-05-08

AI Summary

Modeling structure–property relationships in 3D point clouds while enforcing SO(3) equivariance remains challenging due to the complexity and high computational cost of explicit tensor operations. Method: We propose a scalar-dominant equivariant representation paradigm that decouples equivariant functions into a scalar neural network multiplied by predefined symmetric tensor bases—constructed from spherical harmonics and tensor representation theory—thereby avoiding explicit tensor transformations. This design strictly satisfies SO(3) equivariance while drastically reducing architectural and computational complexity. A tunable approximation mechanism further enables controllable trade-offs between expressive power and inference efficiency. Results: Experiments on molecular and materials property prediction demonstrate that our method achieves accuracy comparable to fully equivariant baselines, while delivering substantial speedups in inference. The implementation is concise, modular, and deployment-friendly—offering a practical yet theoretically grounded alternative for scalable geometric deep learning.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

Rotational symmetry plays a central role in physics, providing an elegant framework to describe how the properties of 3D objects -- from atoms to the macroscopic scale -- transform under the action of rigid rotations. Equivariant models of 3D point clouds are able to approximate structure-property relations in a way that is fully consistent with the structure of the rotation group, by combining intermediate representations that are themselves spherical tensors. The symmetry constraints however make this approach computationally demanding and cumbersome to implement, which motivates increasingly popular unconstrained architectures that learn approximate symmetries as part of the training process. In this work, we explore a third route to tackle this learning problem, where equivariant functions are expressed as the product of a scalar function of the point cloud coordinates and a small basis of tensors with the appropriate symmetry. We also propose approximations of the general expressions that, while lacking universal approximation properties, are fast, simple to implement, and accurate in practical settings.

Problem

• Representing spherical tensors with scalar-based ML models
• Addressing computational demands of equivariant 3D point cloud models
• Proposing fast, simple approximations for equivariant functions

Innovation

• Combines scalar functions with symmetric tensor basis
• Simplifies equivariant models for 3D point clouds
• Fast approximations lacking universal properties