A Appendix

G. From Eq. (4), we have: ϕ The proof is inspired by universality proofs of prior symmetrization approaches [102, 74, 41]. Let ψ: X Y be an arbitrary G equivariant function. We leave proving this as a future work. In general, we are interested in obtaining a faithful representation ρ, i.e., such that ρ(g) is distinct for each g. We now show the following: Proposition 3. The proposed distribution p We now show the following: Proposition 4. The proposed distribution p We also note that scale(Q) gives orthogonal matrix of determinant +1, as it returns Q if det(Q) = +1, otherwise (det(Q) = 1 since Q is orthogonal) scales the first column by 1 which flips determinant to +1 while not affecting orthogonality.