An in depth look at the Procrustes-Wasserstein distance: properties and barycenters

Davide Adamo, Marco Corneli, Manon Vuillien, Emmanuelle Vila
Proceedings of the 42nd International Conference on Machine Learning, PMLR 267:444-459, 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 advancing 2D and 3D point cloud analysis for machine learning and computational geometry applications.

Cite this Paper

BibTeX
@InProceedings{pmlr-v267-adamo25a, title = {An in depth look at the Procrustes-{W}asserstein distance: properties and barycenters}, author = {Adamo, Davide and Corneli, Marco and Vuillien, Manon and Vila, Emmanuelle}, booktitle = {Proceedings of the 42nd International Conference on Machine Learning}, pages = {444--459}, year = {2025}, editor = {Singh, Aarti and Fazel, Maryam and Hsu, Daniel and Lacoste-Julien, Simon and Berkenkamp, Felix and Maharaj, Tegan and Wagstaff, Kiri and Zhu, Jerry}, volume = {267}, series = {Proceedings of Machine Learning Research}, month = {13--19 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v267/main/assets/adamo25a/adamo25a.pdf}, url = {https://proceedings.mlr.press/v267/adamo25a.html}, 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 advancing 2D and 3D point cloud analysis for machine learning and computational geometry applications.} }
Endnote
%0 Conference Paper %T An in depth look at the Procrustes-Wasserstein distance: properties and barycenters %A Davide Adamo %A Marco Corneli %A Manon Vuillien %A Emmanuelle Vila %B Proceedings of the 42nd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2025 %E Aarti Singh %E Maryam Fazel %E Daniel Hsu %E Simon Lacoste-Julien %E Felix Berkenkamp %E Tegan Maharaj %E Kiri Wagstaff %E Jerry Zhu %F pmlr-v267-adamo25a %I PMLR %P 444--459 %U https://proceedings.mlr.press/v267/adamo25a.html %V 267 %X 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 advancing 2D and 3D point cloud analysis for machine learning and computational geometry applications.
APA
Adamo, D., Corneli, M., Vuillien, M. & Vila, E.. (2025). An in depth look at the Procrustes-Wasserstein distance: properties and barycenters. Proceedings of the 42nd International Conference on Machine Learning, in Proceedings of Machine Learning Research 267:444-459 Available from https://proceedings.mlr.press/v267/adamo25a.html.

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