An in depth look at the Procrustes-Wasserstein distance: properties and barycenters
Authors: Davide Adamo
Marco Corneli
Manon Vuillien
Emmanuelle Vila
Venue: ICML 2025 poster (2025)
Keywords: Optimal transport
Procrustes analysis
Point clouds
Primary area: optimization

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.

An in depth look at the Procrustes- Wasseerseen distance: properties and barycenters
Davide Adamo 12
Marco Corneli 12
Manon Vuillien
Emmanuelle Vila
Abstract
Due to its invariance 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 that space PW actually is 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 new method to compute representative shapes from 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.
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1. Introduction
In force of its capability find correspondences between sets of objects, in the last decade computational optimal transport (OT, Peyré 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),
Figure 1. (Top) Two point clouds representing bird shape in dif- ferent position. OT barycenters using (a) Exact Free Wasserstein (Cuturi & Doucet, 2014) (b) Gromov- Gromov-Wasserser Wasserstein (Peyré et al.,
2016) with MDS (Borg & Groenen, 2007) (c) Gromov- Wasserstein with TSNE (Van der Maaten Hinton, 2008) (d) Procrustes- Wasserstein (our).
'Université Côte d' Azur. UMR 7264 CEPAM, CNRS, Nice, France Universsiéé Côte d' Azur. Inria, CNRS, Laboratoire J.A. Dieudonné, Maasai team, Nice, France Universstéé Lumiére Lyon II, UMR 5133 Archéorient CNRS, Lyon, France. Correspondence to: Davide Adamo davvide adamo miniv-ootedazurff2..
graphs (Vayer et al., 2019a: Vincent-Cuaz et al., 2021). shapes (Eisenberger et al., 2020) or text (Zhang et al., 2017: Grave et al., 2019). More generally and possibly more importantly OT allows one to assess the distance between
Proceedings of the ndnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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An in depth look at the Wassssssennn distance
probability distributions thus leading to applications in ma- chine learning that go far beyond the comparison of sets of objects, such as domain adaptation (Courty et al., 2016) or adversarial training (Arjovsky et al., 2017). just to cite some. However, here we keep the focus on the first frame- work we cited (i.e. data alignment and matching) since the applications we discuss in this work are of that kind.
Related works. Among the earliest PW formulations, Zhang al. (2017); Grave et al. (2019) aimed at jointly estimating orthogonal and a permutation matrix to align word embeddings across different languages. Differently from Zhang et al. (2017), which initialized the orthogo- nal matrix using an adversarial training phase, Grave et al. (2019) proposed a convex relaxation of the initialization by reformulating the problem over the convex hull. Alvarez- Melis al. (2019) extended the previous works by incor- porating global invariances directly into the optimization process. Their approach is not limited to invariances with respect to isometries but generalizes to broader invariance classes (characterized by Schatten p-norm ball) up to the recovery of Gromov- Wasserstein. This extension is particu- larly useful in scenarios where data are not simply related by rigid transformations. Additionally, employing a convexity- annealing strategy and considering relaxed PW version, they eliminate the need for an ad-hoc initialization, avoiding strong dependence on an initial guess. In contrast with pre- vious works, two-sided PW (TWP, Jin et al., 2021) adopts a two-fold transformation on both the source and target measures. Such an extension enables to handle data that lie in distinct spaces, transporting them into a common la- tent space. The optimal solution is obtained by solving component-wise convex optimization problem, combining two-sided Procrustes Analysis with a relaxed Wasserstein formulation. Aboagye et al. (2022) tackle the computational limit of PW by proposing a quantized version of the problem (qWP). The quantization step that discretizes the distribu- tions enables for the joint estimate of the alignment and transformation. qWP leverages quantization procedure inspired by Grave et al. (2019), such as k-means++, and reduces the problem to linear programming (LP). This tech- nique not only simplifies the computation but also enhances the approximation quality of OT solvers, thus leading to a more efficient solutions with a fixed computational cost. Fi- nally, Even et al. (2024) approach the problem of matching pairs of distributions using PW distances from theoretical perspective, providing convergence guarantees for the ML estimators of both the transport plan and the isometry. In more detail, they restrict their focus on discrete distributions with the same number of points in the support, further as- suming that one distribution can be obtained from the other through permutation and isometry of the support and the addition of Gaussian noise. The corresponding OT problem falls under the Monge formulation and instead of looking for doubly stochastic plans, they look for permutation matrices.
Based on the modern formulation of Kantorovich (1942), the standard optimal transport tool to compare two sets of objects is the Wasserstein distance. If we assume that each set is a point cloud, in order to fix the ideas, adopting the Wasserstein distance to quantify the similarity between the clouds specifically requires to compute the Euclidean dis- tance (or other) between the points of the first cloud and those of the second. Since, moreover, each point is equipped with a probability mass defining its importance within its cloud, we can say that the Wasserstein distance takes into account both the geometry and the distributional properties of data. However, the Wasserstein distance suffers from some limitations that makes it unfit to some applications such as point cloud matching. Indeed, it is sensitive to the way the two clouds are embedded in the space and in par- ticular to isometries. To address some of the limitations of the Wasserstein distance, Mémoli (2011) introduced the Gromov- Wassersten (GW) distance. Unlike standard OT frameworks, which assume that the compared probability measures are supported on a shared metric space, GW com- pares distributions defined on distinct spaces. In the point cloud matching examples, GW requires computing two pair- wise distance (or similarity) matrices between the points within each cloud. Points not being in the same cloud are never compared explicitly. GW is invariant to isometries and particularly suitable for the comparison of data sets with unknown correspondences or in different coodinate systems. However, GW has (at least) two main drawbacks: i) its rather prohibitive computational cost, although some solu- tions exist (Vayer et al., 2019b; Chowdhury et al., 2021) and ii) the GW barycenters matrices. If one wishes to represent them in the original features domain, dimensionalityy reduction techniques are needed. This last drawback can be severe when computing mean shapes where a high fidelity to the original is required (see Figure 1).
still pairwise distance/similarity
Mixing Procrustes and Wasserstein costs was recently done (Zhang et al., 2017; Grave et al., 2019) in order to intro- duce into the Wasserstein optimization problem invariances to global transformations such as rotations and reflections in the space. In this sense Procrustes- Wasserstein (PW) can be seen as a compromise between GW (with whom shares some invariances) and Wasserstein (since the two measures are directly compared with each other).
Contribution of our work. Despite the heterogeneous use of the PW cost in the above mentioned works, to the best of our knowledge i) it was never showed that Procrustes- Wasserstein distance actually is a distance; ii) PW barycen- ters were never defined/leamed from the data. With a focus on scenarios where the objects to compare are geometric
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An in depth look at the Frverustes-Wasseessiee distance
shapes represented as point clouds (and hence working with discrete measures) the main contribution of this paper is twofold: we define quotient space of discrete measures over which PW is a distance and we provide an estimation algorithm for the PW barycenter. We then show that one of the main advantages of PW is its capability to produce very faithful barycenters in particular conditions. the illustra- tive example Figure 1, two birds differ in both number of vertices and pose (rotation and/or reflection). As it can be seen, among the three tested OT methods, the PW barycen- ter result in more consistent geometric characteristics. We finally present a concrete application of the PW barycenters to detect morphological changes on archaeozoological data.
and Cp(X,Y) E R with
(Cp(X, = |x -y,P11.
The above minimization problem is generalization of the one described in Grave al. (2019) and can be seen as particular case of the one discussed in Alvarez-Melis et al. (2019).
By definition of C)(X,Y) it is easy to show that
Cp(X,Y) = Rx + Ry - 2XP"T""
where the i-th row of Rx E Ru is (1xx,12...... ||x and the j-th column of Ry E maxm (lly,l1,,... .... (1))) By plugging this into Eq. (2) and thanks to the bilinearity of (.,)) F get
The paper is organized as follows: we provide a background on the PW problem and the formal definitions and proof where PW is distance in section 2. We then introduce the PW barycenters and the algorithm to compute it in section 3. We investigate different intializations for specific match in point clouds and a clustering application on our barycenter in section 4. We conclude presenting a concrete real-world application in section 5.
(Cp(X, = (Rx + Ry, 1), - FF
= =u. + (v. - 2(XPPT .T
where u E IR" is such that ", = |,0,11 and v E I such that by ||ا, 12- As such, the minimisation problem in Eq. (2) is equivalent to
2. Procrustes-aastee Wasserstein: an OT distance
Notation. We denote by Σ,, the - probability simplex. So when saying that p == (PI...., E En, we mean p; 2. 0 for all i and Σ = 1. We denote by (s))) the Frobenious dot product, hence (A, B)F == trace((" A), with A, B two compatible matrices. The set of the orthogo- nal matrices of order d is denoted by (d).
max PEO(d) rem((p)))
IXP"yT,T)e
(4)
We now consider the set Ma of all discrete measures of the same form as in Eq. (1). Namely, the generic μx E MMa is a measure supported on some X E RRa for some finite n and a fixed and for some probability vector p. With a slight abuse of notation, given a permutation a in S(x), the set of all possible permutations of elements, we denote by (XX (xT (x) ................................ x"),,))" the matrix X after the (1) permutation of its rows according to ar. Similarly, we denote by (p) (Po(I)..... PP(n)) the permuted histogram. We introduce the following equivalence relation on Ma
Consider two matrices X E RRa and Y E ga where x; (respectively x') ) is the i-th row (j-th column) of X. Sim- ilarly for Y. We attach two discrete probability measures ux and py to X and Y, respectively:
xx Σ PiOxa 1=1
E En
and
m My
nxx, ~ Pxx2 if 3P E Od), Fo E S(n)
q j=1
gyyy,
q Em
such that X, = P and p1 = (p2).
Given an orthogonal matrix P E Od), we denote by p the measure defined on the transformed support of Y, namely pyp = -y
Thus, Pxx ~ (x,, if and only if they share the same proba- bility vector, up to a permutation, and the same support up to the same permutation of the points and rigid transfor- mation (rotation, reflection combination of both).
Given Wz(ux, p the 2- 2-sasserstein distance between μa and py, we attack the following minimization problem
If we denote
min ((xx,uy)) PEO(d)
min (Cr(X,Y),T) (2) PEO(d) refl(p.))
1/2
Py)
min (C)(() PEO(d) rell(p.q)
, (5)
where [p,, q) is the set of the admissible transport plans, i..
Mpp q) = (T e Raxm TIm = p. FT =q}
then
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An in depth look at the Wasssrssiinn distance
Algorithm 1 PW problem
of X outlined in Equation (3). The minimization of PW3 (1x, joY) with respect X can be developed
Input: Locations and histograms (X, (Y) initial
correspondences To-
min PW (ux. min min (Cr(X,Y),T)
2: %% Initialization
P.T
3: UEVI + SVD(Y?T?X)
min ((u p) + 22mn (-X,TYP)))) X
4: Po + -UV,,
5: while not converged do
6: Cp + -oost(X, PP)
where constant terms in Y and q are discarded. While the first term is convex quadratic function of X (since U4 | l2),, the second term renders the optimisation of PW2 (ux, (px, MAY) with respect to X non-convex. Thus, the best we can do is to look for local minima via Newton- Raphson. Denote by (P*,I*) the optimal alignment and transport plan for PW2 (px,Hy). Calling gXX) the objec- tive function in Eq. (7)
7:
%% Update matching
8:
I' EMD(C. .. %% Earth Mover Distance
9:
%% Update P
10: UEVT + SVD((??))
11: P-UVT
12: end while
13:
Return: 1 P
x(X) (u. p) 2(X,
Theorem 2.1. (W2( is a distance on Ma/ m.
the gradient and the Hessian of g(-) with respect to X are
The proof of the above theorem is in Supplementary Mate- rial A. Moreover we have the following
V xg 2diag( p) y
Corollary 2.2. For all PX, in Ma it holds that WW2(xx,yY)> & Wy).
and
Hxg = 2diag(p).
Proof. It suffices to note that
Thus, the update of X reads
PW)( = min WY(H PEO(D)
x (k+1) x (Hxx(X vxg(X(()
< WVLL) = Wy(
Newton step
(8)
(A) (x diag(p YY"P)
diag(p "
3. Procrustes- Proorustes-Vasserrteen barycenter(s)
The update formula provides meaningful geometric inter- pretation. The matrix diag(p "))"" whose rows belong to the simplex Σm. computes weighted barycenters of points in Y, with weights defined by the optimal transport plan. This is analogous to the Wasserstein barycenter update (Cu- turi Doucet, 2014), where each point Y contributes to the updated locations in X proportionally to r" However, in the PW framework, the additional right multiplication by P* allows for simultaneous optimal alignment of the barycenter.
Now that we established that PW₂ is a distance on Ma/ m consider r empirical measures measures ,.,.....s Ax,+ in Ma. with supports (X,}==ı and probability vectors {1 We look for barycenter μa with unknown sup- port X e Raxd and weights p given by the solution to the following problem
f(p
(6)
The steps to optimize fp, X) with respect to the locations X are outlined in Algorithm Solving Problem (6) in- volves computing r independent PW distances between the barycenter (ux) and the measures μox,, Thus, the first step (lines 4-5) consists into solving all PW? (4x,,
In a general setting, we might consider positive weights dj associated with each measure μxx, with A == (A.....))p) Ar) E Σ,. For simplicity, we present the case =
and finding solutions (T PJ) following the iterative scheme introduced in (Grave et al., 2019) that we report here in Algorithm for completeness. The second step (line 7) updates the locations of the barycenter using the update formula in Eq. (8).
3.1. Differentiability of X)) with respect to X
In this section we assume that p is known. Let X E Roxd and Y E md Consider the transport cost function

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An in depth look at the Wasserssiinn distance
Algorithm 2 Procrustes- Wasserstein barycenter (PWB)
1: Input: Locations Xj E RRy, xd and histograms Pj E R" for j = ,,...,77 , initial barycenter locations Xo:
dog (2D)
UPCAW Fiedlecp Gen-Ci) Euc-CW
barycenter histogram p
2: X Xo
3: while not converged do
4: for j (I....,r) do
5: (T;; + PW2 PX, p; Xz,
6: end for
7: X X
El - 1 d
8: end while
9: Return: X*
Shape
dragon (3D)
3.2. Differentiability of X() with respect to p
BucoWW
Despite the obvious difference between the minimisation problem in Eq. (6) and its Wasserstein counterpart illustrated in Cuturi & Doucet (2014), it can be observed that
UNCAN Fedlerr Gao-SWV
X(p.X) Σ PW2 (xx,
Σ w (xx,
Shape index
where, Py is the optimal isometry aligning μx, with the barycenter. Denoting X, := Pj the analogy with the Wasserstein dual LP formulation is straightforward
Figure 2. Convergence comparison between different initialization approaches across randomly generated shapes. Each row responds to a different initialization method while each column corresponds
max (aj, p) + (3g. Pi), ag.
(9)
of the Algorithm with different shape. Red cells indicates successful convergence of the matching (in terms of rotation/reflection and couplings). while blue cells denotes failure.
where in the PW framework the couplings (aj, 35) must satisfy
O3e + Bz, S (Cp;)it = =122;- - -,,,,1
such a way to be enclosed the ID or 2D unit ball. We leave for future works extensions of the PW framework accounting for translations and scaling. The OT solvers used in the implementations are based on the POT toolbox (Flamary et al., 2021). The code is available at https: //github.... com www.//adadaoooooeeeeeeaaaa 98 PWr-bar..
where (Cp;) is the cost matrix incorporating the orthogonal alignment and j. denotes here the k-th row of X,- Eq. (9) is a linear programming (LP) problem for each j. with con- straints defined by (Cpz). The optimization of f(p, X) with respect to p can be approached analogously to Cuturi & Doucet (2014), leveraging the solutions of the dual prob- lems, e.g. a == ブドー oj. For completeness, we provide Algorithm 3 in the supplementary material, detailing the procedure for the optimization with respect to p.
4.1. Initialization for point cloud matching
It is well known that a primary challenge in the compu- tation of PW lies in its initialization (Grave et al., 2019; Alvarez-Mehh et al., 2019). In this section, we inspect sev- eral initialization strategies of To (Algorithm 1) for PW in the context of 2D/3D point cloud matching.
When pursuing the joint optimization of Eq. (6) with respect to X), the outlined strategy remains the one presented in Algorithm 2, except for additional equation after line 7 updating the weights according to Algorithm 3.
Let consider pivot measure μxx,, either representing 2 or 3-dimensional point cloud. Since it is assumed that each point is equipped with the same (uniform) probability mass, with a sligth abuse of notation we identify μх, with X,. We generate 50 clouds by randomly adding extra vertices, Gaussian noise and vertex permutation to X1. We also
4. Experiments
All the point clouds considered in this section are assumed to be centered at they Fuclidean barycenter and scaled in
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An in depth look at the Wasssrsstin distance
include random rotation and reflection. We thus generate X3 for i ,,...-. 50 that underline the same geomertric structure of 1 (e.g. they represent perturbed versions of the pivot). We look for a pairwise clouds registration, in terms of global alignment and couplings. We test different approaches, with the objective to compute Lo.
However, despite their invariance isometries, there are instances where the GW transport plan fails establish the correct couplings. In cases where the data underline specific geometric structure GW could reveal optimal, however its computational cost makes it use clearly prohibitive when working with larger point clouds. In contrast, the Fiedler-W W initialization consistently ensures robust convergence. In the tested scenarios, the Fiedler vectors prove to be optimal for capturing the geometry of the data. Finally, in the two cases, UPCA-W does not demonstrate effectiveness, par- ticularly in the 3D case. We leave a further investigation of this approach for future works. Additional results and visualisations are available in Supplementary Material D.
1. Euc-GW. Gromov- Wasserstein based on Euclidean pairwise distances is computed for each pair of point clouds and To is set equal to the optimal GW plan.
2. Geo-GW. Same as before but with geodesic pairwise distance in place of the Euclidean.
4.2. Clustering
3. Fiedler-W. Fiedler vector (Fiedler, 1973) is the eigen- vector associated with the algebraic connectivity (i.e.
In this section, we propose an unsupervised application of PW for performing clustering directly in the space of point clouds. Our approach draws inspiration from the k-means reformulation presented in Peyré et al. (2016) with Gromov- Wasserstein barycenters. We consider the MNIST dataset of handwritten digits with specific focus the first five digits, from 0 to 4 (Figure 3, left). For each digit class, we consider 10 images and convert them into 2-dimensional point clouds (Figure 3, center). This results in a dataset of 50 point clouds, which we aim to cluster with respect to the digit class (thus = 5). Differently from Peyré et al. (2016), we avoid applying random rotations to the dataset to highlight some benefits of PW even in scenarios where the input data are already aligned (at least in terms of reflection and rotation).
the second-smalles, eigenvalue) of the Laplacian ma- trix of a connected graph. Since point clouds can be easily transformed into graphs (Cover& & Hart, 1967; Preparata & Shamos, 2012) g. We propose to resort to a Wasserstein matching between Fiedler vectors to initialise To. More specifically, for a fixed i, we compute the Fiedler vectors of X1 and Xi,, denoted as fi and f2 respectively, and we standardize them. Furthermore, we compute both the Wasserstein dis- tance between (f1. and (fi--f) -f) (to account for the vectors orientation). The transport plan yielding the smaller distance determines l'o
4. UPCA-W.C UPCe-e.Given two point clouds, X and Y. the first step involves computing the eigenvector matrices Qx and Qy, of their covariance matrices. The multiplica- tion XQx (resp. YQr) leads to a matrix X' (respec- tively Y') that is uncorrelated, e.g. the principal axes of X' and Y' correspond, up to the directions. to the stan- dard coordinate axes of the d-dimensional Euclidean space. Moreover, fixing X, the matrix Y Vy&Qy brings Y into the same (principal component) basis as X, once more up to the direction of the axes. At this stage, a Wasserstein matching can be performed between X and Y Qy : In the case of d = 2. there are 2d possi- ble combinations of directions to check, requiring the resolution of four independent Wasserstein problems. Similarly, for d = 3. we must solve 23 Wasserstein problems. As with Fiedler-W, W. the transport plan asso- ciated to the smallest distance defines the initialization To-
To initialize the centroids, we adopt a strategy inspired by k- k-means++ as follows. We randomly select one point cloud from the 50 and label it as the first "candidate."" The first centroid is determined by applying Euclidean k-means clus- tering to the candidate cloud, where the number of clusters equals the number of points specified for the OT centroids (PW barycenters). This ensures that the points sampled from the candidate form a uniform representation. Next, we identify the point cloud among the remaining 49 that is the farthest from the first candidate, based on the PW distance. This farthest point cloud becomes the second "can- didate", and its centroid is computed using the same idea as for the first. By iterating this process: select the point cloud that is farthest from all previously selected candidates and compute its centroid, we obtain an initial configuration of five centroids. This approach ensures a well-distributed initialization with respect to the PW distance.
Convergence results of the Algorithm 1 for the four pre- sented initialization techniques are summarized in Figure 2. Red colour for the cells denotes convergence to the global minimum (matching succeeded) while blue colour denotes failure (convergence to local minima). We observe that GW initializations generally lead to good success rate.
Using the same initialization technique, we compare k- means clustering across different OT metrics. Specifically, we present comparisons between discrete Wasserstein (Earth Mover's 's Distance, EMD),, Gromov- Wasserstein with Eu- clidean distances (Euc-GW), with geodesic distances (Geo- GW) and PW with a Wasserstein initialization to establish
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An in depth look at the Wasserstenn distance
040
PM
O
2
2
2
2
2
22
Pa
8
3
3
3
3
83
3
4
4
4
4
4
4444
Figure 3. Clustering k-means of MNIST dataset. (Leftmost) Subset of considered images. (Center-left) Corresponding 2D point clouds representation. (Center-right) Clustering centroids computed with different OT barycenters. (Rightmost) Confusion matrices. Rows correspond to the digits, while columns correspond the clusters. The colour proportional the number of digits each cluster.
adapted to the socio-economic and cultural requirements of human societies. In south-west Asia, from the end of the Bronze Age onwards, archaeozoological (Vila & Helmer, 2014; Vila et al., 2021; Abrahami & Michel, 2023) and palaeogenetic data (Her et al., 2022) indicate that zootechni- cal practices were used for the management and selection of sheep morphotypes. This led to a significant increase in phenotypic diversity and a decrease in genetic diversity. While the morphological changes observed are relatively well documented by palaeogenetic data, identifying the pro- cesses linked to morphological transformations in the bones of this species remains complex: which parts of the bone are modified (anatomical characteristies)? How do they change (bone plasticity)? Why do they change adaptations linked to anthropic and environmental factors)? To track these morphological changes, traditionally archaeo- zoologists rely on visual comparisons and manual measure- ments. These methods can be time-consuming and subject to interpretational bias, especially when dealing with intra- specific variations. With the advent of 3D scanning technolo- gies, bones can now be digitized and represented as point clouds or meshes, opening new avenues for quantitative analysis and machine learning. In this context, OT offers mathematically robust framework to tackle the problem of comparing and interpreting bone shapes. The objective of this study is to highlight the morphological evolution over time, i.c. the transition from archacological to modern, by directly comparing three-dimensional representations of the astragalus (ankle bone) for one archaeological sheep dated to the Chalcolithic period and one modern sheep from the same region, the Alborz mountain in Iran. With this objective, let consider two measures Ju and Jy, with associated locations X and Y of nearly 10k vertices, repre- senting an archaeological and modern bone structure of the sheep species, respectively. By assigning weights Ax and Xy respectively, we seek for a 10k PW barycenter via Algorithm 2, that defines an interpolation between the two
Table Clustering results for MNIST dataset.
ALGOROTHM
TIME (S)
ARI
NMI
EMD
9.18
0.4069
0,5652
BUC-GW
675.19
0.5500
0.6815
GEO-GW
378.82
0.3797
0.5724
PW
130.11
0.7669
0.8361
initial correspondences.
The clustering results reported in Table 1, where we provide the computational time (in seconds), the adjusted rand index (ARI) and the normalized mutual info score (NMI) for each of the presented approaches. We also pro- vide in Figure: 3 (Center-right) and Figure (Rightmost) the estimated centroids (OT barycenters) and the confusion matrices, respectively. From the results, we observe that the PW-based clustering provides the best performances, in terms of ARI and NMI. Moreover, clustering results optimal for the digits 0, 1, and 3. Consistent with findings from Peyré et al. (2016), the digits where clustering is less effec- tive are and 4, reflecting greater variability in handwritten style. Differently form the GW-based clustering, PW-based successfully returns more representative centroids for all the five considered digits. EMD-based clustering proves to perform well for certain digits. However, it consistently fails to identify digit 3.
functional
The superior performance of PW over EMD underscores the significance of incorporating optimal rotations, even in scenarios where the input data are already aligned, at least in the sense that poses is consistent.
5. Application: tracking the morphological evolution of domestic animals
The breeding of domestic ungulates began over 9500 years ago, leading to considerable phenotypic and genetic changes,

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An in depth look at the Wassersttinn distance
rows
archaeo
n=
n=
n=
n=
modern
Figure 4. PW harycenter evolution of two 3D point clouds describing an archacological (Leftmost) and modern (Rightmost) astragalus of sheep's species. The four middle columns of the grid correspond representative interpolations each assigned with value of n-- (row)) Progressive interpolation in the euclidean space, note that the two input point clouds are not aligned and no priori knowledge on pairwise correspondence considered. The p solution of PW permits us to optimally display the frontal view (row;) and top view (row;), order to match reference manuals of morphological criteria in archaeology.
bone's structures. Set 7) [0, 1] and re- re-write Ax 1-77 and Xy = 17. By varying we can iterate the minimiza- tion problem (6) and thus model the intermediate stages of morphological changes between the two bones, enabling the study of evolutionary trajectories and species transfor- mations over "time". In order to create pipeline that is as robust and accurate possible, a priori step in this methodology is the normalization of the data. To ensure meaningful comparison and avoid any bias, we resort Solume-based normalization which consists in two key steps. First, we set to the origin the volumetric center of mass. Sec- ond, we constrain the shape to have a unit volume. This technique allows the model to compute interpolations that best capture morphological changes and are less influenced by overall distortions. We remark that the purpose of this section is not in comparing different types of normalization, however, different pre-proced pre-prccessing techniques can be tested. Figure 4 shows the evolution of the bone structure from archaeological to modern, by means of PW barycenters. In row, is reported the progressive interpolations the 3D space. We can see that the four barycenters. correspond- ing to four distinct values of 1), well representative of the input point clouds. The colouring of the bones reflect the wise similarity between each barycenter and the modern sheep (the rightmost bone) form the archacological sheep (the leftmost bone). In yellow are outlined the parts of the bone that are more similar, while in blue the parts that different the most. The colour of the archacological bone, on the other hand, reflects the distance between itself and the modern bone.
to the archaeological bone. As get closer to the modern sheep, the PW distance increases and the blue areas become more pronounced. By exploiting the solution of the PW barycenter problem, we benefit of a complete registration of the barycenters, we can thus visualize different views: the dorsal view (row:) and the proximal view (rows). These ori- entations facilitate the observation of changes in the overall proportions of the bone and more targeted changes, particu- larly to the proximal trochlea, i.e. the upper pulley-shaped articular surface. A notable observation is the widening of the lateral lip in the proximal trochlea of the modern spec- imen in comparison to the archacological specimen. The proximal view also demonstrates a narrowing of the tubercu- lus tali and development of the projecting medial ridge in the modern specimen compared to the archaeological spec- imen. For a better understanding of the bone anatomical features is provided in Figure 8 in supplementary material.
Discussion. The proposed approach allows us to trace the evolutionary trajectories of species by interpolating between bone shapes. This method provides archacozoologists with a powerful quantitative tool to infer how species adapted, evolved, or were selectively bred by humans. Furthermore, the same technique could be used to compute a "mean" representative bone shape for species for which morpho- logical criteria are not well-defined. This would reinforce and supplement studies combining machine learning and archacozoology to identify morphologically related taxa (Miele et al., 2020; Moclán et al., 2023; Vuillien et al., 2025). In conclusion, by directly working on 3D models, this approach offers a robust solution that aim avoid the
As expected, we see that the first barycenter is the closest
8

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An in depth look at the Wasssrssiinn distance
subjectivity inherent in traditional morphological analysis. The use of the PW distance and PW barycenters introduce rigorous and quantitative framework for analyzing shapes while offering to archaeologists a detailed and objective tool for interpreting species evolution, domestication pat- terns, and morphological diversity, ultimately enhancing our understanding of the past.
specifically highlighted here.
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6. Conclusions
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In this paper, we carefully defined a space of discrete proba- bility measures over which Procrustes- -asseesttsnnnnnnn is a dis- tance and provided a formal proof of such claim. This opens the door to a wider application of PW in various machine learning tasks, particularly when dealing with complex data structures. We also introduced PW barycenters extending the literature of OT barycenters. Our formulation enables the construction of representative measures that exhibit an improved visual loyalty to the geometry of the observed data. We propose applications that demonstrated the prop- erties and advantages of our approach, with comparisons with state-of-the-ate methods. Future works could explore denser formulations of the barycenter problem (via entropic regularisation) leading smoother solutions and broader applicability to large-scale datasets.
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Acknowledgements
The work has received financial support from the CNRS through the MITI interdisciplinary programs and the Ju- nior Professor Chair (Chaire de Professeur Junior, CPJ) funded by the French National Research Agency (ANR). We would like to thank Arch' Story project (Ministère de I'Enseignement Supérieur et de la Recherche and University Côte d' Azur) for funding this project. We would like to ex- press our gratitude to Dr. Hossein Davoudi (Bioarchaeology Laboratory Central Laboratory, University of Tehran, Iran) and Dr. Marjan Mashkour (Bioarchacology, Interactions laboratory-UMe UMMR 7209, CNRS, Na- tional Museum of Natural History of Paris, France) for their support and authorization to provide samples of the modern and archaeological sheep. We also wish to warmly thank Cédric Vincent-Curs Cuaz for the enlightening discussions we had with him around this work as well as for sharing with us his point of view regarding the distance. Our gratitude is finally extended to the anonymous reviewers whose contributions proved to be of substantial value in enhancing the quality of the article.
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An in depth look at the Procrustes- Waseerstes-asssssseiin distance
A. Proof of Theorem 2.1.
Proof. First we check that PW2 = 0 μx ~ ny. The left implication clear if pxx ~ Jay means that there P*) such that Y o² (X) and q = (p). Then ) is the solution of the problem in Eq. (5), with the Kantorovich formulation being equivalent the Monge's one. Vice-versa, if
PW2( = min WyPpx,MYP) PEO(d)
means that there exists à pa such that the 2-Wasserstein distance between μax and py is null, equivalently that jax and pYP p are the same measure up permutation of the points in the support together with their masses.
Second we prove that PW2(ux: py) = PW2(py, px) for all px, py in Ma. Let us assume that (P*,I*) is solution of Problem (4), with pa E (d) and 1* E Π(,, q). Then d( is defined by the maximization over Q E ((d) and
E [l p) of
(YQ"X"" x======"00""xx===========)==>=>= ((XQQT))
= (XQV" ,0))""0)))))
((X()))Y""r")
by optimality of (P*, ,T*) and where we used that the trace of a matrix equals the trace its transposed and the trace is invariant under cyelic permutations of its arguments. The above equation shows that if (P", r*) is the stationary point leading to PW2(ux, ny) then, ((P*)*(T**) is the solution leading to PW2(uY, px) and vice-versa. Thanks to Eq. (3), it is now immediate to verify that PWa(ux, My) PW2(HY,As) px).
Third, we show that the triangular inequality is satisfied: PW2(Hx,HY) S PW2(ux, + PWa(Hz, for all Mx,, py, Az in M(d). For all pz E MM it holds that
W W2(px Hyp) < W2(ux, (zr) + Wa(pyp, = W2(xx:P22 + W2(My,
where the first inequality holds since the Wasserstein distance is symmetric and satisfies the triangular inequality (as any distance) and the second equality comes from the fact that if we equally rotate or reflect the supports of two measures the Euclidean distances between any pair of points in the supports will be unchanged. From the above equation, paired with Eq. (2) we deduce that, for any pz E &Ma
PWe(ux. & PW2(Xx, u2) + W2(Hy,
Now, if we replace Z with Z = 2P where pa is solution of
min PEO(d)
Wg(Ry,HZP).
we obtain
(y) S PW2(ux, - W2(uy, pz-) = PW2(ux + PWa(Uy, (uz);
where the last equality holds by the definition of PW. Finally, since Z and Z only differ by night-multpllication with an orthogonal matrix, PW2(ux,H2 p2-) = [W2(xx,uz).

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An in depth look at the Procrustes-Wasserere distance
B. Optimization of 2W2 with respect to p
In this section we provide for clarity the algorithm for the optimization the weights p. already proposed by Cuturi Doucet (2014) within the context of Wasserstein barycenters. In our framework we assume p En and denote the Schur's product.
Algorithm 3 Optimization of
1: Input: Cost matrices with orthogonal alignments Op, R"x"y and histograms Pj E R for j I....,r
2: Set ý - 1/n
3: while not converged do
4: B = (++ 1)/2
5:
p x (1- 1-3-1)0+3-3
6:
a =j= using all dual optima a3 of Eq. (9)
7:
p p 1oāa
8:
- 4-10/""1 1n
9:
0 p1 1--18- 1 3
10:
6-|+1
end while
12: Return: p
C. Regularized PW barycenter(s)
As common the OT community, we extend in this section the PW barycenter problem by adding an entropic regularization. Consider the following relaxed version of the PW problem
min (Cx(X,Y))T)x+++HH0)
(10)
PEO(d) rell(p.a)
where non-negative parameter controlling the strength of the regularization and H(T) = E,,, Fej log(Tig) the entropy.
Looking for measure μx with unknown support X E Raxd and weights p. given measures JX,, Mx,, translates into the following minimization problem
f(p. : Σ X3 P, 5=1
(11)
The resolution of this problem can be done similarly as for the classical case. Given the probability vector p and assuming (L,, P?) are the solutions the regularized PW problem (ux, ,x,), the Newton update is still given by Eq. (8). Notably, at the optimum, both the gradient and the Hessian of Eq. (11) are independent of the regularization term H-). The optimization scheme for the computation of the barycenter equivalent Algorithm 2, where in this case solve each PW sub-problem independently using the Sinkhorn algorithm Cuturi (2013). Under this framework, the optimization of Eq. (11) can be seen as particular case of the one discussed in Alvarez-Melis et al. (2019).
To illustrate the effectiveness of the proposed barycenter, we present a 2D toy example where we consider two measures μx and μy, with associated locations X and Y, representing different instances of the "same" object. Specifically, X represents a point cloud depicting a bird in a particular pose. We define as a modified version of X with the following transformations: addition of Gaussian perturbations and random vertex permutation; addition of extra vertices; application of random rotation. Within this setting, our goal is generate interpolated shapes that progressively move from X to Y. Set thus E [0, 1] and re-write Ax = 1 and λy 1). By varying we can iterate the minimization problem and model the intermediate stages. We compute PW barycenters using both the classical problem formulation (Problem 6) and the regularized version (Problem 11).
In Figure we can follow the progressive interpolations between the two point clouds. The four central columns correspond PW barycenters different interpolation steps, while the two rows compare the classical PW barycenter (top) with the
12

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An in depth look at the Procrustes- Wasssrsttinn distance
PW
Reg- PW
X
n=
n= 3
Figure Progressive interpolations two point clouds describing in different positions. four central columns the grid correspond to PW barycenters, each reflecting specific interpolation step (defined by 1).. The two rows represent barycenters calculated using the classical formulation (PW) and considering an entropic relaxation of the problem (Reg-PW).
relaxed version (bottom). The colors of the barycenters are given by transporting the colors of X via the optimal plan T This highlights how the cloud evolves while maintaining the structural consistency of the original shape.
13

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An in depth look at the ia distance
D. Additional results
2-dimensional larget (X1)
(b) 2-dimensonal perturbed sources (X5).
(c) Matching results with Fuc-GW.
(d) Matching results with Geo-GW
(c) Matching results with Fiedler-W.
(f) Matching results with UPCA-W.
Figure 6. Comparisons of PW matching results with different initialization approaches (2. dog). Matchings reflect the convergence results reported in Figure 2
14

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An in depth look at the Procrustes- Wasserstein iroerustes-Wasserstein
distance
(a) 3-dimensional target (X1)
(b) 3-dimensonal perturbed sources (X²
(c) Matching results with Euc-GW.
(d) Matching results with Geo-GW.
Matching results Fiedler
Matching results with UPCA-W.
Figure 7. Comparisons of PW matching results with different initialization approaches (3D dragon). Matchings reflect the convergence results reported in Figure
15

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-nerrustes-Wasserseenn An in depth look at the Procrustes- Wasserstein distance
Tuberculum toli
Proximal trochlea
Lateral
Medial lip
Medial ridge
Figure 8. Proximal view of the modern astragalus and its main anatomical features presented in the Scction
16


Reviews:
[{"review_id": "96V5c12Zu1", "reviewer": "Reviewer_nX3Q", "summary": "The authors propose that in the quotient space of the discrete measures over the rigid transformation equivalence, the Procrustes Wasserstein is a metric. To calculate the Procrustes Wasserstein distance, they introduce several initialization methods, i.e. Euc-GW, Geo-GW, Fiedler-W and UPCA-W. The barycenter problem and algorithms are also provided.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for reviewing the paper and checking the proofs.  We are quite disappointed by the gap between the assigned score (reject) and a review which sounds quite positive. We would appreciate further explanations on the concerns raised about lack of novelty: PW barycenters are new. However, we provide in the following detailed responses to the specific questions mentioned.\n> 1. Lack of comparisons with other methods\n\nThe purpose of the archaeological section (S5) is not to propose a new morphological interpolation technique that outperforms the state of the art, but rather to demonstrate that in the OT univers, PW offers significant advantages compared with existing alternatives. In our archaeological application, the point clouds $X$ and $Y$ are approximately made of 10k points. At this scale, GW barycenters cannot be computed (well-known computational limitations). Nonetheless, we have conducted additional experiments by computing Wasserstein barycenters using the free support barycenters (FBS) method from the POT library, allowing for a comparison with the PW results shown in Figure 4. We will include these additional experiments in the appendix of the camera ready version of the paper. The computed Wasserstein barycenters can clearly be seen to be of lower quality, similarly to what is observed in Figure 1. Furthermore, we report below the corresponding objective functional values: Eq. (6) from our paper for PW barycenters, and Eq. (5) from [1] for the Wasserstein barycenters. The values for, respectively, the four interpolation steps are as follows:\n- Wasserstein (FBS): [0.00943, 0.01499, 0.01496, 0.00917]\n- PW: [0.00092, 0.00155, 0.00155, 0.00091]\n\nOur method achieves the lowest barycenter functional value for each interpolation step.\n\n> 2. In Figure 1, can the authors comment on the Gromov-Wasserstein results? Why would there be some outlier points?\n\nThe GW barycenter is computed from pairwise distance matrices of size $N\\times N$ and $M\\times M$, where $N$ and $M$ correspond to the number of points in $X$ and $Y$, respectively. The barycenter itself is a $K\\times K$ matrix (with $K$ fixed a priori) and must then be projected back into the original Euclidean space $\\mathbb{R}^D$. In case of Figure 1, $D$ is set to 2. Dimensionality reduction techniques such as Multi-Dimensional Scaling (MDS, [3]) and t-Distributed Stochastic Neighbor Embedding (TSNE, [4]) can be used for this projection. However, this step can lead to suboptimal representations of the GW barycenter, even in simple cases such as the introductory example presented in Figure 1. We have made the caption of Figure 1 clearer in the final version of the paper (for more details please refer to the answer of Q1 for Reviewer HyoR).\n\n\nReferences:\n\n[1] Cuturi, Marco, and Arnaud Doucet. \"Fast computation of Wasserstein barycenters.\" International conference on machine learning. PMLR, 2014.\n\n[2] Peyré, Gabriel, Marco Cuturi, and Justin Solomon. \"Gromov-wasserstein averaging of kernel and distance matrices.\" International conference on machine learning. PMLR, 2016.\n\n[3] Borg, Ingwer, and Patrick JF Groenen. Modern multidimensional scaling: Theory and applications. Springer Science & Business Media, 2007.\n\n[4] Maaten, Laurens van der, and Geoffrey Hinton. \"Visualizing data using t-SNE.\" Journal of machine learning research 9.Nov (2008): 2579-2605.\n\n[5] Cuturi, Marco. \"Sinkhorn distances: Lightspeed computation of optimal transport.\" Advances in neural information processing systems 26 (2013).", "claims_and_evidence": "The claims are supported by evidence.", "methods_and_evaluation_criteria": "Although the authors show the application of barycenter in tracking the morphological evolution of domestic animals, no comparisons are made with other existing methods, can you show the results by the baselines for morphological interpolations?", "theoretical_claims": "The proof for Theorem 2.1 and Corollary 2.2 are correct.", "experimental_designs_or_analyses": "Details of the experimental setup are provided. It would be better if the authors could provide the comparisons with baselines for morphological evolution applications, especially with the Gromov-Wasserstein methods.", "relation_to_broader_scientific_literature": "The proposed method could be applied to point cloud analysis where rigid transformations are prevalent but should not affect the comparisons.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "My main concern is the novelty of this work. The quotient space and metric properties are very incremental, and the methods for calculating the initial plan are based on previous methods.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "In Figure 1, can the authors comment on the gromov-Wasserestein results? Why would there be some outlier points?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "axsSDbzptm", "reviewer": "Reviewer_6RWS", "summary": "This paper extends the Wasserstein barycenter to the Procrustes-Wasserstein (PW) barycenter, offering a novel approach to computing representative shapes from a collection of point clouds. Additionally, it provides a proof that the Procrustes-Wasserstein distance satisfies the properties of a valid distance metric.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your kind remarks as well as for reviewing the paper and checking the correctness of the main proof.\n\n> 1. What is the convergence behavior of the proposed Procrustes-Wasserstein (PW) barycenter?\n\nInspecting the convergence properties of the PW barycenters is a crucial homework and a very important topic. We are currently working on it, but we honestly believe that having introduced the PW barycenters might be enough for one paper. Moreover, we respectfully point out that the convergence behavior of OT barycenters is not reported in related papers such as (for instance) [1] and [2], both accepted at this conference. \n\n> 2. How does the performance change when replacing EMD with the Sinkhorn algorithm?\n\nThe relevance of the proposed method for computing the barycenter also lies in the quality of the obtained solution, specifically in terms of local geometric details. However, as common in OT, an entropic penalisation can be introduced in the objective function to stabilize the solution and speed up computations. To this end, we conducted new experiments adopting the Sinkhorn algorithm in place of EMD for the pairwise OT computations (line 8 of Alg. 1) and denoted the corresponding barycenter method as Procrustes-Sinkhorn barycenter (PSb). In the camera ready version of the paper, we might dedicate an additional section to provide the formal definition of the regularized problem and the derivation of the update formulas. We exactly observed what is known from the literature: the entropic penalty can be used to stabilize the solution and speed up computations but it comes at a price. A higher $\\epsilon$ (the strength of the penalty term) is required to have a faster convergence (see Table here after) but leads to less accurate, deformed barycenters. However, for ML pipelines where there is no need for such a high level of barycentric detail, PSb certainly represents a valuable option. \n\n|   |      PWb with EMD (Alg. 1)      |  PSb ($\\epsilon=10$) |  PSb  ($\\epsilon=1$) | PSb ($\\epsilon=0.01$) |\n|----------|:-------------:|:------------:|:------------:|:------------:|\n| Mean time & std. (s) |  1.91 ± 0.14 | 1.18 ± 0.26 | 5.69 ± 1.73 | 13.74 ± 1.12\n\nResults over 10 barycenter computations (10 runs) considering two input measures of nearly 500 points (as in Figure 1). \n\n> 3. How does the computational complexity of PW barycenters compare to Gromov-Wasserstein (GW) barycenters in theory?\n\nThanks for this question. In practical applications, the PW algorithm (Alg. 1) is dominated by the OT computation. Since the Procrustes update has an explicit formula and the dimension D is small (in our case 2 or 3), the cost of the Procrustes step is negligible. Thus, the main computational bottleneck remains the EMD computations, which scales as O(n3 log(n)) (or O(n^2) if one adopts Sinkhorn divergences) according to [3].  The algorithmic complexity of computing PW is thus O(Kn3(log(n)), where K is the number of iterations between the Wasserstein and the Procrustes optimisation (also the number of Ping-Pong steps in [Even et al. 24], with reference to answer to Review GNtQ). By direct analogy with Alg. 2 in [1], the computational complexity of the PW barycenter algorithm follows the same structure as that of Wasserstein barycenters, with the key difference that each Wasserstein update is replaced by a PW computation. The theoretical complexity of a PW barycenter in Alg. 2 is O(TNKn3log(n)) (respectively O(TNKn^2) with Sinkhorn divergences) where N is the number of input measures, n is the maximal number of points in the input clouds and T is the number of barycenter’s updates (line 7, Alg. 2) which are essentially equivalent to the ones of a Wassersetin barycenter and so negligible with respect to the PW updates. Noting that in all our experiments we set the maximum value of K to 100 and assuming this value as a constant << N, which is reasonable when working with real point clouds (thousands of points) the computational complexity of  a PW barycenter is the same as the one of  a Wassesrtain barycenter, obviously up to a constant.  That complexity is smaller than those of GW barycenters  (the simple evaluation of the GW$_2$ cost function, without considering the network flow algorithms to solve it, is at least cubic). \n\n\nReferences:\n\n[1] Cuturi, Marco, and Arnaud Doucet. \"Fast computation of Wasserstein barycenters.\" International conference on machine learning. PMLR, 2014.\n\n[2] Peyré, Gabriel, Marco Cuturi, and Justin Solomon. \"Gromov-wasserstein averaging of kernel and distance matrices.\" International conference on machine learning. PMLR, 2016.\n\n[3] Cuturi, Marco. \"Sinkhorn distances: Lightspeed computation of optimal transport.\" Advances in neural information processing systems 26 (2013).", "claims_and_evidence": "The claims made in the submission are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "I have checked the correctness that the Procrustes-Wasserstein distance satisfies the properties of a metric.", "experimental_designs_or_analyses": "Yes, it is soundness.", "relation_to_broader_scientific_literature": "Procrustes-Wasserstein (PW) barycenter is simple and interesting.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "The paper does not provide any convergence proof for the Procrustes-Wasserstein barycenter, which raises concerns about the theoretical guarantees of the proposed method. Additionally, it lacks an analysis of the computational complexity, which is crucial for understanding the feasibility and scalability of the approach. A detailed complexity analysis would help clarify the efficiency of the algorithm, especially in large-scale point cloud applications.", "other_comments_or_suggestions": "None", "questions_for_authors": "1. What is the convergence behavior of the proposed Procrustes-Wasserstein (PW) barycenter?\n2. How does the performance change when replacing EMD with the Sinkhorn algorithm?\n3. How does the computational complexity of PW barycenters compare to Gromov-Wasserstein (GW) barycenters in theory?\n4. How about the performance in partial overlapping shape matching", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {"ethics_expertise_needed": ["Other expertise"]}}, {"review_id": "ch3exRMkhm", "reviewer": "Reviewer_GNtQ", "summary": "The paper proposes a method for aligning and matching point clouds based on Optimal Transport using the Procrustes-Wasserstein distance (PW), allowing the alignment to be calculated by also taking pose transformations into account. \n\nThe paper shows that PW is indeed a distance in the space of discrete probability measures. It also introduces an algorithm to calculate the barycentres of the measures with respect to PW. The contribution is validated by showing how to initialise optimal transport, testing four main approaches on a synthetic dataset. The method is then tested for clustering on a subset of MNIST and compared with other Optimal Transport metrics.   A final application shows how the method was crucial for tracing the morphological evolution of sheep based on astragalus alignments from the Chalcolithic period up to the modern age.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "First of all, thank you very much for the careful reading of the paper, as well as for the analysis and comments, they are truly appreciated. We have addressed all the grammatical errors identified in the text, and the corrections will be included in the final version of the paper. We also appreciate the suggestions regarding [Zemel et al., 2019] and [Even et al., 2024], which are now references in the main paper. However, while both studies align with the general problems discussed in our paper, they differ from the specific contributions presented in our work. Below, we provide a detailed discussion on the connections between our work and these references, along with responses to the specific points raised in the review. \n\n> 1. Relation to [Zemel et al., 2019]:\n\nIt is true that the barycenter problem is, by definition, equivalent to computing the Fréchet mean. However, the work of [Zemel et al., 2019] aims to highlight the analogy between the Fréchet mean computation, i.e. barycenters, computed in the Wasserstein space and Procrustes analysis as methods for the “registration” of distributions. Unlike our work, [Zemel et al., 2019] does not minimize a Procrustes-Wasserstein cost, but rather a classical Wasserstein one. Specifically, is it worth to note that Algorithm 1 in [Zemel et al., 2019] consists of N pairwise OT computations, followed by a barycenter update performed directly in the Wasserstein space. We apologize if this distinction was not sufficiently clear in our manuscript. Our work directly addresses the PW problem, introducing the first formalization of PW barycenters along with a practical and efficient algorithm for their computation (Alg. 2). This approach is fundamentally different from [Zemel et al., 2019].\n\n> 2. Relation to [Even et al., 2024]:\n\nThe problem addressed in [Even et al., 2024] tackles the matching of pairs of distributions from a theoretical perspective, providing thorough results about the convergence of their ML estimates. However, it represents a special case of the more general Procrustes-Wasserstein problem which is approached in the current work. Indeed [Even et al., 2024] specifically considers distributions (pairs of point clouds) that have the same number of points; in their model, the signal only differs from the dependent point cloud by a permutation of the points plus an isometry, then Gaussian noise is added to each point. The OT problem they address corresponds thus to the Monge formulation of OT. In contrast, the problem approached in the current work generalizes to cases where the distributions may have different numbers of points (as can be also observed in the experiments we propose) thereby corresponding to the more general Kantorovich formulation of OT. We look for doubly stochastic solutions (i.e. transport plans), they look for permutation matrices.  \n\n> 3. Initialization of $\\Gamma_0$\n\nWe agree with you, the initialization of the Procrustes-Wasserstein matching in real-world scenarios remains an open challenge and addressing it is not the primary goal of our work. \nHowever, it is important to note that, differently from what you state, in the matching problem we tackled, noise is present in the Laplacian (although at a level such that the global geometric structure of the shape remains visible). As detailed in our section, the generated clouds $X^i_2$ (meant to be matched to the pivot $X$) also include Gaussian noise applied to each point of the pivot cloud and thus represent a perturbation of the pivot thus adding noise to the corresponding Laplacian matrices. We emphasize that the perturbations applied to the clouds $X^i_2$ are designed to simulate real data scenarios. However, the Wasserstein distance between Fiedler vectors can still find a good and sufficient matching, which can then be used as the initial $\\Gamma_0$​ for solving Alg. 1. \n\n> 4. “You never mention algorithms for computing OT.”\n\nWe apologize for any lack of clarity in the text. Please refer to responses 2 and 3 to reviewer 6RWS for details.\n\nReferences :\n\n[1] Cuturi, Marco, and Arnaud Doucet. \"Fast computation of Wasserstein barycenters.\" International conference on machine learning. PMLR, 2014.\n\n[2] Peyré, Gabriel, Marco Cuturi, and Justin Solomon. \"Gromov-wasserstein averaging of kernel and distance matrices.\" International conference on machine learning. PMLR, 2016.\n\n[3] Borg, Ingwer, and Patrick JF Groenen. Modern multidimensional scaling: Theory and applications. Springer Science & Business Media, 2007.\n\n[4] Maaten, Laurens van der, and Geoffrey Hinton. \"Visualizing data using t-SNE.\" Journal of machine learning research 9.Nov (2008): 2579-2605.\n\n[5] Cuturi, Marco. \"Sinkhorn distances: Lightspeed computation of optimal transport.\" Advances in neural information processing systems 26 (2013).", "claims_and_evidence": "The claims of the article are:\n\n1) Using PW, it is possible to encompass geometric transformations. Other articles have already proved this claim (we discuss this in another section).\n\n 2) The Procrustes-Wasserstein distance is a distance on the space of discrete probability measures, and this claim is proven in the supplementary.\nThis claim is used to define the barycentres.\n\n3) The calculation of the barycentres is relevant for obtaining clusters that take geometric transformations into account.\nThe algorithm is as precise as the derivations that lead to its optimisation.\n\nI'm unsure if it is the first algorithm (as claimed in the introduction). The Frechet mean can actually be considered a barycentre of a discrete measure, and its computation was introduced by Zemel et *Al.* [1], referring to the OT (the authors do not cite the article).\n\nThe last claim concerns the initialisation of $\\Gamma_{0}$ or admissible transport plans.\n\nFrom the two pictures in Figure 2 and the comments in the text, it appears that only the Fiedler vector ensures convergence. However, it should be noted that the Fiedler vector only works if the Laplacian matrix is noise-free, as in the case studied, which is defined by a synthetic set. \n\nThus, the problem of the initialisation of $\\Gamma\\_{0}$ remains open for real cases.\n\n[1] Zemel, Y. and Panaretos, V.M., 2019. *Fréchet means and Procrustes analysis in Wasserstein space.*", "methods_and_evaluation_criteria": "Regarding the experiments, the article follows the literature somewhat (see, for example, Cuturi and Doucet) with no tables (here, only Table 1 to compare with other OT metrics) and only pictures without showing measures of similarities and failure values. Even for the main application, there is only one image.\n\nHowever, this is partly justified by the fact that the pictures serve as proof of concept, and the work is not experimental but more theory-oriented.", "theoretical_claims": "I checked.", "experimental_designs_or_analyses": "As specified above, I am critical about the initialisation of $\\Gamma_{0}$. \n\nAs shown in the paper, only with the Fiedler vector is convergence obtained at the minimum distance detecting $\\Gamma_{0}$.  However, the test was carried out on synthetic data and, thus, without noise.  In the presence of noise, the Fiedler Vector does not separate the components correctly.\n\nFurthermore,  the presupposition on the space of measures is very strong. As noted in Zemel [1] an optimal transport map may fail to exist, and instead, one may need to solve the relaxed Monge problem.", "relation_to_broader_scientific_literature": "The article is well placed, it is definitely a contribution. \n\nHowever, the paper *Aligning Embeddings and Geometric Random Graphs: Informational Results and Computational Approaches for the Procrustes-Wasserstein Problem' by Even et Al., appeared at Neurips 2024, which deals with the problem in a very general way. \n\nAlthough the authors limit their method to discrete measures, they should have mentioned this and, therefore, related their findings to the more general ones of Even et *Al.* [2]. \nIndeed, this should be done in the revised version. \n \nAnother limitation is perhaps not looking at the Frechet mean (e.g. Zemel's [1]).\n\n[2] Even, M., Ganassali, L., Maier, J. and Massoulié, L., 2024. *Aligning Embeddings and Geometric Random Graphs: Informational Results and Computational Approaches for the Procrustes-Wasserstein Problem*", "essential_references_not_discussed": "Essential references not discussed have been highlighted in the previous paragraph.", "other_strengths_and_weaknesses": "The article makes a simple and useful contribution but has to be compared with the works mentioned above.\n\nIt has many limitations:\n1. It is not well written; there are many spelling mistakes, which I point out in the next paragraph. \n\n2. It is limited in its experiments. Perhaps it should have reduced the space for describing the application and compared the results better with the literature, see, for example, [3].\n\n3. The initialisation of $\\Gamma0_{0}$ should have been analysed more deeply and with real examples.\n\n[3] Jin, K., Liu, C., and Xia, C. *Two-sided Wasserstein Procrustes analysis.* In IJCAI, pp. 3515–3521, 2021. (this paper was cited)", "other_comments_or_suggestions": "Some spelling errors (line:column)\n\n\n“062:1  the repeated twice\"\n\n“96:2   discrtete”\n\n“112:1  matchin”\n\n“113:   applicaton”\n\n“123:1  the the”\n\n“210:2 interative”\n\n“289:1  space.Morover”\n\n“325:2 calstering”\n\n“325:2 provide”\n\n“348:1 cluetsring”", "questions_for_authors": "Please look into both Even's and Zemel's papers.\n\nYou never mention algorithms for computing OT. \n\nFor example, Cuturi & Doucet  refer, in their Algorithm 3 Cuturi fast algorithm [Cuturi 2013]", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9UGmJOJUPs", "reviewer": "Reviewer_HyoR", "summary": "This paper defines a quotient space of discrtete measures over which PW is a distance and provides an estimation algorithm for the PW barycenter. This paper then shows that one of the main advantages of PW is its capability to produce very faithful barycenters in particular conditions. Experiments have verified its effectiveness in point cloud alignment and shape analysis, and its application in archaeological data shows the practical significance of the proposed method.\n\n\n## update after rebuttal\nI appreciate the response provided in the rebuttal, which addresses some of my concerns, and I maintain the score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "First of all, thanks for the kind remarks and for reviewing the paper and checking the supplementary. We provide in the following the answer to the specific questions.\n\n> 1-2. Figure 1 lacks more explanation. The caption shows: OT barycenters using (a) Free Wasserstein (b) Gromov-Wasserstein with MDS (c) Gromov-Wasserstein with TSNE. However, (a-c) is not introduced in the paper. In addition, the caption should introduce what task this is, such as a matching task or another. MDS and TSNE only have abbreviations, no references or full names are provided.\n\nWe apologize for the lack of clarity regarding Figure 1. Below, we provide additional details about the experiment and include references to the MDS and TSNE methods, which were indeed missing from the text. Here is the new caption for Figure 1, which will be updated in the final version of the paper:\n\n“(Top) Two point clouds representing a bird shape in different positions. OT barycenters using (a) Exact Free Wasserstein [1] (b) Gromov-Wasserstein [2] with MDS [3] (c) Gromov-Wasserstein with TSNE [4] (d) Procrustes-Wasserstein.”\n\nIn this motivational example, we compare different OT techniques for computing barycenters. The two point clouds at the top represent the input measures (depicting a bird shape, without and with noise, and in different positions). The techniques used in panels (a-d) are, respectively: exact Wasserstein barycenters [1], Gromov-Wasserstein barycenters [2] with two different visualization techniques, Multidimensional Scaling (MDS, [3]) and t-distributed Stochastic Neighbor Embedding (TSNE, [4]), and finally, the proposed PW method. The results highlight how the PW barycenter provides the clearest and most precise representation, even for local geometric structures. Additionally, the barycenters are colored by transporting the color of the first bird (top left) onto the computed barycenter.\n \n> 3. The archaeological application experiment only shows the interpolation results. Are there some quantitative indicators to evaluate the results?\n\nThank you for this central question. Quantitative methods are available for the quantification of morphological deformations based on simplified morphological patterns in archaeology (e.g. geometric morphometric methods). However, no quantitative method exists that can be used with the entire 3D bone. Ensuring that the observed morphological deformations on the barycentre correspond to a biological reality based on archaeological data is a significant challenge. While acknowledging these challenges, we firmly believe that this preliminary approach signifies a significant opportunity for interdisciplinary research.\n\nReferences:\n\n[1] Cuturi, Marco, and Arnaud Doucet. \"Fast computation of Wasserstein barycenters.\" International conference on machine learning. PMLR, 2014.\n\n[2] Peyré, Gabriel, Marco Cuturi, and Justin Solomon. \"Gromov-wasserstein averaging of kernel and distance matrices.\" International conference on machine learning. PMLR, 2016.\n\n[3] Borg, Ingwer, and Patrick JF Groenen. Modern multidimensional scaling: Theory and applications. Springer Science & Business Media, 2007.\n\n[4] Maaten, Laurens van der, and Geoffrey Hinton. \"Visualizing data using t-SNE.\" Journal of machine learning research 9.Nov (2008): 2579-2605.\n\n[5] Cuturi, Marco. \"Sinkhorn distances: Lightspeed computation of optimal transport.\" Advances in neural information processing systems 26 (2013).", "claims_and_evidence": "yes", "methods_and_evaluation_criteria": "yes", "theoretical_claims": "Yes, I don’t find any issues with the proofs for theoretical claims.", "experimental_designs_or_analyses": "The archaeological application experiment only shows the interpolation results. Are there some quantitative indicators to evaluate the results?", "relation_to_broader_scientific_literature": "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. This paper builds a space of discrete probability measures and shows that over that space PW actually is a distance.", "essential_references_not_discussed": "No", "other_strengths_and_weaknesses": "Strengths\n1. This paper proves the distance property of PW distance on discrete measure quotient space.\n2. This paper introduces the notion of PW barycenter.\n\nWeaknesses\n1. Figure 1 lacks more explanation. The caption shows: OT barycenters using (a) Free Wasserstein (b) Gromov-Wasserstein with MDS (c) Gromov-Wasserstein with TSNE. However, (a-c) is not introduced in the paper. In addition, the caption should introduce what task this is, such as a matching task or another.\n2. MDS and TSNE only have abbreviations, no references or full names are provided.\n3. The archaeological application experiment only shows the interpolation results. Are there some quantitative indicators to evaluate the results?", "other_comments_or_suggestions": "As above", "questions_for_authors": "As above", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper introduces the Procrustes-Wasserstein (PW) distance, a metric on the quotient space of discrete measures under rigid transformations. It extends Optimal Transport to account for pose alignment, proving PW is a valid distance and proposing algorithms for computing PW barycenters. The method is validated on synthetic data, MNIST clustering, and archaeological shape analysis, showing improved alignment and representative shape recovery. While most reviews are positive, one reviewer questioned the novelty and limited comparisons. In rebuttal, the authors emphasized that PW barycenters are new, clarified why many suggested baselines are not applicable due to lack of isometry invariance, and provided additional experimental results and visual comparisons.

While more extensive benchmarks could further strengthen the empirical side, the paper makes a meaningful theoretical and methodological contribution to the OT literature. I lean toward acceptance.
Decision: Accept (poster)