Suppose that $\{ P_i \}$ is a dataset of point clouds(PCDs), i.e., set of collections of 3-d real vectors. I have a query PCD $Q$ and I want to find all $P_i$ such that $P_i$ contains a subset (which is also a PCD) which has a small difference with $Q$. To do so we typically define metrics (or loss function, distance etc.) in the space of PCDs, which could be regarded as $\mathcal{M}(R^3)$, the space of discrete measures on $R^3$.
I have roughly read about some theories like Hausdorff distances, Fisher-Rao metric and optimal transport. But I don't know which type of metrics have the property that if $\exists T, T(Q)\subset P$, where $T$ is a rigid transformation, then $d(Q, P) = 0$ and could be computed fast by computers.