Approximately Equivariant Graph Networks
This work addresses a fundamental limitation in GNN design for practitioners dealing with asymmetric graphs, offering a principled approach to improve generalization, though it is incremental in refining symmetry concepts.
The paper tackles the problem of graph neural networks (GNNs) lacking natural symmetries on real-world asymmetric graphs by formalizing approximate symmetries via graph coarsening, showing theoretically and empirically that choosing a suitably larger group than graph automorphism but smaller than the permutation group achieves the best generalization performance in tasks like image inpainting, traffic flow prediction, and human pose estimation.
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 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 signals (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, but smaller than the permutation group.