Proof of Theorem 1

where W and Z represent the random 3D point clouds with k points subsampled from T and T ∗ uniformly at random without replacement, respectively. We use Φ to denote the joint space of W and Z, where each element is a 3D point cloud with k points subsampled from T or T ∗. We denote by E the set of intersection points between T and T ∗, i.e., E = T ∩ T ∗. Before proving our theorem, we first describe a variant of the Neyman-Pearson Lemma [21] that will be used in our proof. The variant is from [9].