ICML 2025

An in depth look at the Procrustes-Wasserstein distance: properties and
barycenters
Davide Adamo 1 2 Marco Corneli 1 2 Manon Vuillien 1 Emmanuelle Vila 3
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. Hav-
ing that application in mind, we carefully build a
space of discrete probability measures and show
that over that space PW actually is a distance. Al-
gorithms to solve the PW problems already exist,
however we extend the PW framework by dis-
cussing 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 exist-
ing OT approaches, demonstrating superior per-
formance in scenarios requiring precise alignment
and shape preservation. We finally show the use-
fulness of the PW barycenters in an archaeologi-
cal context. Our results highlight the potential of
PW in boosting 2D and 3D point cloud analysis
for machine learning and computational geometry
applications.
1. Introduction
In force of its capability to find correspondences between
sets of objects, in the last decade computational optimal
transport (OT, Peyr´e et al., 2019) has become more and
more ubiquitous in machine learning. Notable examples
relate to learning tasks from (almost) any data type in-
cluding images (Solomon et al., 2015; Feydy et al., 2017),
1Universit´e Cˆote d’Azur, UMR 7264 CEPAM, CNRS, Nice,
France 2Universit´e Cˆote d’Azur, Inria, CNRS, Laboratoire J.A.
Dieudonn´e, Maasai team, Nice, France 3Universit´e Lumi´ere Lyon
II, UMR 5133 Arch´eorient CNRS, Lyon, France. Correspondence
to: Davide Adamo <davide.adamo@univ-cotedazur.fr>.
P...