Weisfeiler Leman for Euclidean Equivariant Machine Learning

• URL: http://arxiv.org/abs/2402.02484v3
• Date: Wed, 26 Jun 2024 12:11:16 GMT
• Title: Weisfeiler Leman for Euclidean Equivariant Machine Learning
• Authors: Snir Hordan, Tal Amir, Nadav Dym,
• Abstract summary: 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.
• Score: 3.0222726571099665
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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. Finally, we provide a general framework for proving equivariant universality and leverage it to prove that a simple modification of this invariant PPGN architecture can be used to obtain a universal equivariant architecture that can approximate all continuous equivariant functions uniformly. Building on our results, we develop our WeLNet architecture, which sets new state-of-the-art results on the N-Body dynamics task and the GEOM-QM9 molecular conformation generation task.

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