Approximately Equivariant Graph Networks

by   Ningyuan Huang, et al.

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.


page 1

page 2

page 3

page 4


Graph Neural Networks: Architectures, Stability and Transferability

Graph Neural Networks (GNNs) are information processing architectures fo...

A General Framework For Proving The Equivariant Strong Lottery Ticket Hypothesis

The Strong Lottery Ticket Hypothesis (SLTH) stipulates the existence of ...

The passive symmetries of machine learning

Any representation of data involves arbitrary investigator choices. Beca...

On the equivalence between graph isomorphism testing and function approximation with GNNs

Graph neural networks (GNNs) have achieved lots of success on graph-stru...

Edge Entropy as an Indicator of the Effectiveness of GNNs over CNNs for Node Classification

Graph neural networks (GNNs) extend convolutional neural networks (CNNs)...

Haar Transforms for Graph Neural Networks

Graph Neural Networks (GNNs) have become a topic of intense research rec...

Generalised Implicit Neural Representations

We consider the problem of learning implicit neural representations (INR...

Please sign up or login with your details

Forgot password? Click here to reset