Complete Neural Networks for Complete Euclidean Graphs

Snir Hordan, Nadav Dym, Tal Amir, Steven J. Gortler

Keywords: Graph Neural Networks, Weisfeiler Leman Test, graph isomorphism, universal approximation, equivariant neural network, Euclidean graphs

Abstract: Neural networks for point clouds, which respect their natural invariance to permutation and rigid motion, have enjoyed recent success in modeling geometric phenomena, from molecular dynamics to recommender systems. Yet, to date, no architecture with polynomial complexity is known to be complete, that is, able to distinguish between any pair of non-isomorphic point clouds. We fill this theoretical gap by showing that point clouds can be completely determined, up to permutation and rigid motion, by applying the 3-WL graph isomorphism test to the point cloud's centralized Gram matrix. Moreover, we formulate a Euclidean variant of the 2-WL test and show that it is also sufficient to achieve completeness. We then show how our complete Euclidean WL tests can be simulated by a Euclidean graph neural network of moderate size and demonstrate their separation capability on highly-symmetrical point clouds.

Submission Number: 3261