An in depth look at the Procrustes-Wasserstein distance: properties and barycenters
Journal:
arXiv
Published Date:
Jul 1, 2025
Abstract
Due to its invariance to rigid transformations such as rotations and
reflections, Procrustes-Wasserstein (PW) was introduced in the literature as an
optimal transport (OT) distance, alternative to Wasserstein and more suited to
tasks such as the alignment and comparison of point clouds. Having that
application in mind, we carefully build a space of discrete probability
measures and show that over that space PW actually is a distance. Algorithms to
solve the PW problems already exist, however we extend the PW framework by
discussing and testing several initialization strategies. We then introduce the
notion of PW barycenter and detail an algorithm to estimate it from the data.
The result is a new method to compute representative shapes from a collection
of point clouds. We benchmark our method against existing OT approaches,
demonstrating superior performance in scenarios requiring precise alignment and
shape preservation. We finally show the usefulness of the PW barycenters in an
archaeological context. Our results highlight the potential of PW in boosting
2D and 3D point cloud analysis for machine learning and computational geometry
applications.
