Optimal Transport (OT) provides a rigorous mathematical framework for comparing empirical distributions, such as point clouds. However, standard formulations based on the Wasserstein distances are highly sensitive to global rigid transformations in the feature space. Unlike other approaches being computationally prohibitive (e.g. Gromov-Wasserstein) the Procrustes-Wasserstein (PW) distance addresses this limitation in a more scalable way by directly incorporating alignment into the optimization problem. However, current approaches remain confined to discrete settings and lacks generalization to unseen samples. In this work, we extend the PW framework to the continuous domain by formally deriving its dual and semi-dual formulations. Leveraging results in both convex analysis and, more recently, neural optimal transport, we introduce NeuralPW, a framework that jointly learns rigid alignments and continuous transport maps. We empirically demonstrate that NeuralPW accurately recovers geometric transformations, explicitly decoupling global pose alignment from local non-rigid deformation. Finally, we validate our approach on a real-world case study in archaeozoology, demonstrating its ability to robustly match high-density point clouds where discrete primal solvers are computationally intractable.
Adamo et al. (Thu,) studied this question.