  Invariant models, one important class of geometric deep learning models, are
capable of generating meaningful geometric representations by leveraging
informative geometric features in point clouds. 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 a wide range of
invariant models. We first rigorously bound the expressiveness of the most
classic invariant model, message-passing neural networks incorporating distance
(DisGNN), restricting its unidentifiable cases to be only highly symmetric
point clouds. We then show that GeoNGNN, the geometric counterpart of one of
the simplest subgraph graph neural networks (subgraph GNNs), can effectively
break these corner cases' symmetry and thus achieve E(3)-completeness. By
leveraging GeoNGNN as a theoretical tool, we further prove that: 1) most
subgraph GNNs developed in traditional graph learning can be seamlessly
extended to geometric scenarios with E(3)-completeness; 2) DimeNet, GemNet and
SphereNet, three well-established invariant models, are also all capable of
achieving E(3)-completeness. Our theoretical results fill the gap in the
theoretical power of invariant models, contributing to a rigorous and
comprehensive understanding of their capabilities. We also empirically
evaluated GeoNGNN, the simplest model within the large E(3)-complete family we
established, which achieves competitive results to models relying on high-order
invariant/equivariant representations on molecule-relevant tasks.
