A Appendix A.1 Proofs A.1.1 Proof of Theorem 1 (Section 2.1) Theorem 1. If p

Let ψ: X Y be an arbitrary G equivariant function. We leave proving this as a future work. We now show the following: Proposition 3. The proposed distribution p We now show the following: Proposition 6. From Eq. (29), we have: ϕ Proposition 7. The proposed symmetrization From Eq. (29), we have: ϕ This is after handling the translation component of the Euclidean group E ( d) / SE (d) as in Eq. (29). We now show the following: Proposition 8. Therefore, probabilistic symmetrization can become frame averaging.