  Invariant and equivariant networks are useful in learning data with symmetry,
including images, sets, point clouds, and graphs. In this paper, we consider
invariant and equivariant networks for symmetries of finite groups. Invariant
and equivariant networks have been constructed by various researchers using
Reynolds operators. However, Reynolds operators are computationally expensive
when the order of the group is large because they use the sum over the whole
group, which poses an implementation difficulty. To overcome this difficulty,
we consider representing the Reynolds operator as a sum over a subset instead
of a sum over the whole group. We call such a subset a Reynolds design, and an
operator defined by a sum over a Reynolds design a reductive Reynolds operator.
For example, in the case of a graph with $n$ nodes, the computational
complexity of the reductive Reynolds operator is reduced to $O(n^2)$, while the
computational complexity of the Reynolds operator is $O(n!)$. We construct
learning models based on the reductive Reynolds operator called equivariant and
invariant Reynolds networks (ReyNets) and prove that they have universal
approximation property. Reynolds designs for equivariant ReyNets are derived
from combinatorial observations with Young diagrams, while Reynolds designs for
invariant ReyNets are derived from invariants called Reynolds dimensions
defined on the set of invariant polynomials. Numerical experiments show that
the performance of our models is comparable to state-of-the-art methods.
