  This paper studies separating invariants: mappings on $D$ dimensional domains
which are invariant to an appropriate group action, and which 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 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.
  Often the requirement of invariant separation is relaxed and only generic
separation is required. In this case, we show that only $D+1$ invariants are
required. More importantly, generic invariants are often significantly easier
to compute, as we illustrate by discussing generic and full separation for
weighted graphs. Finally we outline an approach for proving that separating
invariants can be constructed also when the random parameters have finite
precision.
