让图神经网络在共形变换下保持不变，提升跨领域泛化能力。
AdS-GNN -- a Conformally Equivariant Graph Neural Network
- 将数据升维到反德西特空间，利用其等距变换对应共形对称性
- 通过基于测地距离的消息传递，实现高效计算与强泛化性能
- 适用于计算机视觉和统计物理任务，可自动提取标度维度等关键信息
共形对称性（保持角度的坐标变换）在物理、数学、计算机视觉及几何机器学习中具有核心作用。本文构建了一种对一般共形变换保持等变的神经网络。方法是将数据从平坦欧氏空间升维至反德西特（AdS）空间，利用平坦空间共形变换与AdS空间等距变换之间的已知对应关系。在此基础上，借鉴几何深度学习中对一般流形上等距变换的成熟研究，设计了基于测地距离的消息传递层，形成计算高效的框架。在计算机视觉与统计物理任务上的验证表明，该模型表现优异，具备更强泛化能力，并能从训练好的网络中提取如标度维度等共形数据特征。
原文摘要 · Abstract (English)
Conformal symmetries, i.e.\ coordinate transformations that preserve angles, play a key role in many fields, including physics, mathematics, computer vision and (geometric) machine learning. Here we build a neural network that is equivariant under general conformal transformations. To achieve this, we lift data from flat Euclidean space to Anti de Sitter (AdS) space. This allows us to exploit a known correspondence between conformal transformations of flat space and isometric transformations on the AdS space. We then build upon the fact that such isometric transformations have been extensively studied on general geometries in the geometric deep learning literature. We employ message-passing layers conditioned on the proper distance, yielding a computationally efficient framework. We validate our model on tasks from computer vision and statistical physics, demonstrating strong performance, improved generalization capacities, and the ability to extract conformal data such as scaling dimensions from the trained network.
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