Incompleteness of graph convolutional neural networks for points clouds in three dimensions

Graph convolutional neural networks (GCNN) are very popular methods in machine learning, and have been applied very successfully to the prediction of the properties of molecules and materials: atoms form the vertices of a graph, and bonds are the edges connecting them. This first-order approach is known to be incomplete, i.e. there are graphs that are distinct, but appear identical when seen through the lens of the GCNN. More complicated schemes have thus been designed to increase the resolving power. Applications to molecules (and more generally, point clouds), however, add a geometric dimension to the problem. Edges can be decorated with the distance between atoms, and the resulting “distance graph convolution NNs” have empirically demonstrated excellent resolving power, and are widely used in chemical ML. Here we show that, even for the restricted case of graphs induced by 3D atom clouds, dGCNNs are not complete. We construct pairs of distinct point clouds that are, for any cutoff radius, indistinguishable by first-order graph convolutions that rely on node labels and distance-labelled edges. This class of degenerate structures include chemically-plausible configurations, setting an ultimate limit to the expressive power of some of the well-established GCNN architectures for atomistic machine learning. Models that explicitly use angular information in the description of atomic environments can resolve these degeneracies.