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This paper studies separating invariants: mappings on $D$ dimensional domainswhich are invariant to an appropriate group action, and which separate orbits.The motivation for this study comes from the usefulness of separatinginvariants in proving universality of equivariant neural network architectures.  We observe that in several cases the cardinality of separating invariantsproposed in the machine learning literature is much larger than the dimension$D$. As a result, the theoretical universal constructions based on theseseparating invariants is unrealistically large. Our goal in this paper is toresolve this issue.  We show that when a continuous family of semi-algebraic separating invariantsis available, separation can be obtained by randomly selecting $2D+1 $ of theseinvariants. We apply this methodology to obtain an efficient scheme forcomputing separating invariants for several classical group actions which havebeen studied in the invariant learning literature. Examples include matrixmultiplication actions on point clouds by permutations, rotations, and variousother linear groups.  Often the requirement of invariant separation is relaxed and only genericseparation is required. In this case, we show that only $D+1$ invariants arerequired. More importantly, generic invariants are often significantly easierto compute, as we illustrate by discussing generic and full separation forweighted graphs. Finally we outline an approach for proving that separatinginvariants can be constructed also when the random parameters have finiteprecision.

Low Dimensional Invariant Embeddings for Universal Geometric Learning