Rotational Equivariance in Machine Learning: A Comprehensive Tutorial

2026-08-31

AI Summary

本文解决了3D数据机器学习中的旋转等变性问题，通过几何深度学习、群论和表示论的方法，介绍了实现旋转等变性的现代架构和技术。

Abstract

Rotational symmetry is one of the most important structural principles in machine learning on 3D data. In applications ranging from physics and materials science to 3D computer vision, predictions should not depend on an arbitrary choice of coordinate frame. Rotational equivariance captures this requirement mathematically by enforcing that a rotation of the input induces a corresponding transformation of the model output. This tutorial provides a comprehensive introduction to rotational equivariance, starting from the physical and geometric intuition behind coordinate independence and building up the necessary machinery from geometric deep learning, group theory, and representation theory. We introduce message passing on Euclidean graphs, group actions and representations, spherical harmonics, Wigner matrices, tensor products, and Clebsch-Gordan decomposition, and explain how these ingredients give rise to modern equivariant architectures. We then survey the principal strategies for incorporating rotational equivariance in deep learning, including group convolutions, internal tensorial representations, and canonicalization-based methods, and discuss their practical strengths and limitations. The tutorial aims to lower the barrier to the subject by connecting the underlying mathematics to practical model design, by unifying ideas that are often expressed in different formal languages, and by helping practitioners choose among competing approaches through a clear discussion of their trade-offs.

Problem

• rotational equivariance
• coordinate independence
• geometric deep learning

Innovation

• Rotational Equivariance
• Geometric Deep Learning
• Spherical Harmonics
• Group Convolution
• Tensor Products