Wider Groups, Fewer Weights: Efficient Pseudo-Orthogonal Equivariance via Geometric Algebras

We introduce a new parameter-efficient method for constructing neural networks that are equivariant with respect to pseudo-orthogonal (or orthogonal) groups, based on geometric (or Clifford) algebras. The approach is based on finding and considering larger groups than the Clifford groups in geometric algebras, which perform the same action as the pseudo-orthogonal (or orthogonal) groups. By constructing networks equivariant to these larger groups, we automatically guarantee their equivariance to the corresponding pseudo-orthogonal (or orthogonal) group. This approach induces an expressiveness-efficiency trade-off: a larger equivariance group leads to stricter constraints on the equivariant layers and thus to fewer required optimizable parameters (weights). We propose and study specific groups for this framework, derive new equivariant mappings, and present several model configurations with varying levels of expressiveness and efficiency. Our models demonstrate strong performance on benchmark tasks including convex hull volume estimation, N-body dynamics prediction, high-energy jet tagging, and prediction of atom positions on the MD17 dataset.