Low Dimensional Invariant Embeddings for Universal Geometric Learning

This paper studies separating invariants: mappings on $d$-dimensional semi-algebraic subsets of $D$ dimensional Euclidean domains which are invariant to semi-algebraic group actions and separate orbits. The motivation for this study comes from the usefulness of separating invariants in proving universality of equivariant neural network architectures. We observe that in several cases the cardinality of separating invariants proposed in the machine learning literature is much larger than the ambient dimension $D$. As a result, the theoretical universal constructions based on these separating invariants is unrealistically large. Our goal in this paper is to resolve this issue. We show that when a continuous family of semi-algebraic separating invariants is available, separation can be obtained by randomly selecting $2d+1 $ of these invariants. We apply this methodology to obtain an efficient scheme for computing separating invariants for several classical group actions which have been studied in the invariant learning literature. Examples include matrix multiplication actions on point clouds by permutations, rotations, and various other linear groups.