On the Completeness of Invariant Geometric Deep Learning Models
Zian Li Xiyuan Wang Shijia Kang Muhan Zhang
Invariant models, one important class of geometric deep learning models, are
capable of generating meaningful geometric representations by leveraging
informative geometric features. These models are characterized by their
simplicity, good experimental results and computational efficiency. However,
their theoretical expressive power still remains unclear, restricting a deeper
understanding of the potential of such models. In this work, we concentrate on
characterizing the theoretical expressiveness of invariant models. We first
rigorously bound the expressiveness of the most classical invariant model,
Vanilla DisGNN (message passing neural networks incorporating distance),
restricting its unidentifiable cases to be only those highly symmetric
geometric graphs. To break these corner cases' symmetry, we introduce a simple
yet E(3)-complete invariant design by nesting Vanilla DisGNN, named GeoNGNN.
Leveraging GeoNGNN as a theoretical tool, we for the first time prove the
E(3)-completeness of three well-established geometric models: DimeNet, GemNet
and SphereNet. Our results fill the gap in the theoretical power of invariant
models, contributing to a rigorous and comprehensive understanding of their
capabilities. Experimentally, GeoNGNN exhibits good inductive bias in capturing
local environments, and achieves competitive results w.r.t. complicated models
relying on high-order invariant/equivariant representations while exhibiting
significantly faster computational speed.