Complete Neural Networks for Complete Euclidean Graphs

TL;DR

This paper introduces a theoretically complete neural network model for point clouds that can distinguish any non-isomorphic configurations by leveraging Euclidean graph isomorphism tests, filling a key gap in geometric deep learning.

Contribution

It develops a novel complete Euclidean graph neural network framework based on 3-WL and 2-WL tests, capable of distinguishing all non-isomorphic point clouds.

Findings

The 3-WL test applied to the Gram matrix achieves completeness.

The Euclidean 2-WL variant is also sufficient for completeness.

The proposed networks can distinguish highly symmetrical point clouds.

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 model 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 an 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 an Euclidean graph neural network of moderate size and demonstrate their separation capability on highly symmetrical point clouds.