Graph neural networks (GNNs) are commonly described as being permutation equivariant with respect to node relabeling in the graph. This symmetry of GNNs is often compared to the translation equivariance symmetry of Euclidean convolution neural networks (CNNs). However, these two symmetries are fundamentally different: The translation equivariance of CNNs corresponds to symmetries of the fixed domain acting on the image signal (sometimes known as active symmetries), whereas in GNNs any permutation acts on both the graph signals and the graph domain (sometimes described as passive symmetries). In this work, we focus on the active symmetries of GNNs, by considering a learning setting where signals are supported on a fixed graph. In this case, the natural symmetries of GNNs are the automorphisms of the graph. Since real-world graphs tend to be asymmetric, we relax the notion of symmetries by formalizing approximate symmetries via graph coarsening. We present a bias-variance formula that quantifies the tradeoff between the loss in expressivity and the gain in the regularity of the learned estimator, depending on the chosen symmetry group. To illustrate our approach, we conduct extensive experiments on image inpainting, traffic flow prediction, and human pose estimation with different choices of symmetries. We show theoretically and empirically that the best generalization performance can be achieved by choosing a suitably larger group than the graph automorphism group, but smaller than the full permutation group.

翻译：图神经网络（GNNs）通常被描述为对图节点重标号具有排列等变性。这种GNN的对称性常被类比于标准卷积神经网络（CNNs）的平移等变性对称性。然而，这两种对称性存在本质差异：CNNs的平移等变性对应固定域作用于图像信号的对称性（有时称为主动对称性），而GNNs中任何排列同时作用于图信号和图域（有时描述为被动对称性）。本研究聚焦于GNNs的主动对称性，考虑信号支撑于固定图上的学习场景。在此情形下，GNNs的自然对称性是图的自同构。由于现实图数据往往具有非对称性，我们通过图粗化形式化近似对称性，从而放宽对称性概念。我们提出偏差-方差公式，量化了表达能力损失与学习器正则性增益之间的权衡，该权衡取决于所选对称群。为验证方法有效性，我们在图像修复、交通流预测和人体姿态估计任务上，采用不同对称性选择进行了大量实验。理论与实证结果共同表明：选择比图自同构群更大但小于全排列群的对称群，可实现最优泛化性能。