Weisfeiler Leman for Euclidean Equivariant Machine Learning

Understand

The $k$-Weisfeiler-Leman ($k$-WL) graph isomorphism test hierarchy is a common method for assessing the expressive power of graph neural networks (GNNs).

• Recently, GNNs whose expressive power is equivalent to the $2$-WL test were proven to be universal on weighted graphs which encode $3\mathrm{D}$ point cloud data, yet this result is limited to invariant continuous functions on point clouds.
• In this paper, we extend this result in three ways: Firstly, we show that PPGN can simulate $2$-WL uniformly on all point clouds with low complexity.
• Secondly, we show that $2$-WL tests can be extended to point clouds which include both positions and velocities, a scenario often encountered in applications.