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）常被描述为对图中节点重标号具有置换等变性。这种GNN的对称性常被类比为卷积神经网络（CNNs）的平移等变对称性。然而，这两种对称性存在根本差异：CNNs的平移等变性对应于固定域作用于图像信号的对称性（有时称为主动对称性），而在GNNs中，任何置换同时作用于图信号和图域（有时称为被动对称性）。本文聚焦于GNNs的主动对称性，考虑信号定义在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。由于真实世界图结构往往是非对称的，我们通过图粗化对近似对称性进行形式化定义，从而放宽对称性概念。我们提出一个偏差-方差公式，定量刻画了在选定对称群下，表达能力损失与学习估计器正则性增益之间的权衡。为展示该方法，我们针对图像修复、交通流预测和人体姿态估计任务，采用不同对称群选择进行了大量实验。理论和实验均表明，选择比图自同构群更大但小于全置换群的群，可获得最佳泛化性能。