机器学习中的旋转等变性：一份全面教程
Rotational Equivariance in Machine Learning: A Comprehensive Tutorial
- Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University（海德堡大学跨学科科学计算中心（IWR））
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本教程系统介绍机器学习中旋转等变性的相关理论与方法，涵盖几何深度学习等基础，综述群卷积等融入策略，助力从业者掌握该领域并选择合适方法。
AI中文摘要：
旋转对称性是3D数据机器学习中最重要的结构原理之一，在从物理学、材料科学到3D计算机视觉的各类应用中，预测结果不应依赖于坐标系的任意选择。旋转等变性通过数学方式满足这一要求，即输入的旋转会引发模型输出发生对应的变换。本教程全面介绍旋转等变性，从坐标系无关性背后的物理与几何直觉出发，逐步构建几何深度学习、群论和表示论所需的理论基础。我们介绍欧几里得图上的消息传递、群作用与表示、球谐函数、维格纳矩阵、张量积以及克莱布施-戈登分解，并解释这些要素如何催生现代等变架构。随后，我们综述将旋转等变性融入深度学习的主要策略，包括群卷积、内部张量表示和基于规范化的方法，并讨论它们的实际优势与局限。本教程旨在通过将底层数学与实际模型设计关联起来、统一常以不同形式表述的思想、并通过清晰权衡讨论帮助从业者在不同方法间做出选择，从而降低该主题的入门门槛。
英文摘要：
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.
