An Efficient SE(p)-Invariant Transport Metric Driven by Polar Transport Discrepancy-based Representation
Authors: Junyi Lin
Dunyao Xue
Jun Yu
Hongteng Xu
Cheng Meng
Venue: ICLR 2026 Poster (2026)
Keywords: Distribution comparison; Optimal Transport;  Special Euclidean group; Shape matching
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We introduce SEINT, a new representation-based SE(q)-invariant metric for comparing probability distributions, with theoretical guarantees and computational efficiency. Experiments demonstrate its effectiveness on various tasks.

Abstract: We introduce SEINT, a novel Special Euclidean group-Invariant (SE(\emph{p})) metric for comparing probability distributions on $p$-dimensional measured Banach spaces. Existing SE(\emph{p})-invariant alignment methods often face high computational costs or lack metric guarantees. To overcome these limitations, we develop a polar transport discrepancy combined with distance convolution to extract SE(\emph{p})-invariant representations. These representations are then used to compute the alignment between two distributions via optimal transport. 
Theoretically, we prove that SEINT is a well-defined metric on the space of isometry classes of normed vector spaces. Beyond its inherent SE(\emph{p})-invariance, SEINT also supports cross-space distribution comparison.
Computationally, SEINT aligns two samples of size $n$ with a complexity of just $\mathcal{O}(n\log n)$ to $\mathcal{O}(n^2)$. Extensive experiments validate its advantages: As a robust metric, it outperforms or matches existing SE(\emph{p})-invariant methods in classification and cross-space tasks under isometries. As a regularizer, it greatly enhances molecular generation performance across both pre-training and fine-tuning tasks, achieving state-of-the-art (SOTA) results on key benchmarks.

Published as a conference paper at ICLR 2026
SEINT: AN EFFICIENT TRANSPORT METRIC DRIVEN BY POLAR TRANSPORT DISCREPANCY-BASED REPRESENTATION
Junyi Lin"!, Dunyao Xue" , Jun Yu', Hongteng Xu Cheng Meng
Institute of Statistics and Big Data, Renmin University of China, Beijing. China
School of Mathematics and Statistics, Beijing Institute of Technology. Beijing, China GGallngg School of Artificial Intelligence, Renmin University of China, Beijing, China *Center for Applied Statistics, Renmin University of China, Beijing, China (junyiling inn, xuedunyao1200 2022 chengmeng) @ rru. edu..eduuccom cn, cn, yu jujubbeta aejunbetabbbii @bi t. edu. on, hongtengxu ruc. edu.cn
ABSTRACT
We introduce SEINT, a novel Special Euclidean group-Invariant (SE(p) metric for comparing probability distributions on p-dimensional measured Banach spaces. Existing SE(p)-invarian alignment methods often face high computational costs or lack metric guarantees. To overcome these limitations, we develop a polar trans- port discrepancy combined with distance convolution to extract SF(p)-invariant representations. These representations are then used to compute the alignment between two distributions via optimal transport. Theoretically, we prove that SEINT is a well-defined metric on the space of isometry classes of normed vector spaces. Beyond its inherent -E(p)-invariance, SEINT also supports cross-space distribution comparison. Computationally, SEINT aligns two samples of size n with complexity of just On log n) to (n=)). Extensive experiments validate its advantages: As a robust metric, it outperforms or matches existing SE(p)-invariant methods in classification and pross-space tasks under isometries. As a regularizer, it greatly enhances molecular generation performance across both pre-training and tine-tuning tasks, achieving state-of-the-an (SOTA) results on key benchmarks. The code is available at ht // //tit//iittt tiggth.ub...
INTRODUCTION
Optimal Transport (OT) and the Wasserstein distance offer a powerful mathematical framework for comparing probability distributions (Villani et al.) 2008 Peyré et al.. 2019). These tools have driven
significant advances across a broad range of fields, including statistics (Meng et al. 2020 2019
Deb et al. 2022
Li et al., 2023c, Zhang et al. 2023), machine learning (Courty et al. 2016 6 Lac lau
ct al. 2017 Kolouri et al. 2018 Xu et al. 2022) Hu et al. 2025). computer vision (Tolstikhin et al.
Kolour
2018 Arjovsky et al. [2017 Tong al. 2024). natural language processing (Kusner et al. [015
Zhao et al. 2019 Chen et al. 2023), and reinforcement learning (Metelli et al. [019]] Liu et al. 2020 He et al. 2022b). However, the Wasserstein distance has inherent limitations in addressing
structural learning problems involving geometric invariance, such as (Gianea et al.) [002 Song et al.
2023, Bose al. 2024 Yue al. 22025), point cloud registration Shen al. (2021) 1); Yu al. (2023). and multimodal alignment (Qiu et al. 2023; Alatkar & Wang 2023). For objects with translation- and rotation- invvarinnt shape properties (e.g.. molecules or 3D point clouds), an effective metric must be invariant under the Special Euclidean group. Although the Wasserstein distance provides a general framework for distribution comparison, its standard formulation lacks native SF(p)-invariance, requiring problem-specific adaptations to handle such geometric symmetries.
To the best of our knowledge, existing OT-based SE(p)-invariant methods can be classified into three main strategies, as illustrated in Fig.
Authors contributed equally. Corresponding authors.

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Point Alignment
Representation Alignment
Relation Alignment
Orthogonal Transform
Encode
G
(a) Extrinsic strategy
(b) Intrinsic strategy
(c) Representation-bast based strategy
Figure I: Overview of existing OT based SE(p) invarriat methods. (a) shows the Extrinsic strategy that jointly optimize the problem the orthogonal transformatio: (b) shows Intrinsic strategy that leverages the geometric structure of data; (c) shows the Representation strategy that align data using SF(p) -nnvaraat features.
The first strategy extrinsically enforces SE(p) invariance by jointly optimizing an orthogonal ma- trix in RP. Representative approaches include: EMD under Transformation Sets (EMD*) (Cohen & Guibasm 1999), which searches over general transformation group 9, and Softassign Pro- crustes Matching (SPM) |Rangarajan et al. 1997). which specializes this paradigm to orthogonal transformations. More recent methods in this extrinsic family include the Wasserstein Procrustes formulations (Grave et al. 2019 Alvarez- et al. 2019) and their extensions to non non-Euclidean domains (Alaya et al. 2022 Beier et al., 2025). However, these iterative schemes often lead to significant computational overhead due to repeated OT calculations. To address efficiency concerns, methods like Rotational Invariant Sliced Gromov- Wasserstein (RISGW) (Vayer et al., 2019) utilize closed-form solution by projecting data onto ID subspaces. While this reduces computational complexity, it sacrifices the metric propertics of the distance. The second strategy adopts an intrinsic approach by leveraging the geometric structure of data, c.g., Haundoff (GH) distance (Gro mov 1981) and Saamov-Wasserstere (GW) distance (Mémoli 2011). This intrinsic strategy achieves SE(p) invariance by comparing metrie spaces up isometry, offering key advantages: natural incorporation of SE(p) transformations and the ability to perform cross-space comparisons. However, this strategy usually comes with significantly higher computational complexity compared extrinsic approaches. Although some GW variants, such as Quantized GW (Chowdhury et al. 2021) and Sampled GW (Kerdoncuff al., (022), provide fast approximations the original GW distance, they may lose some of its desirable metric properties.
The third strategy employs a approach,
representation-busel which directly extracts SE(p)- invariant features from the data. This category includes classical methods such Spherical Har- monic Representations (SHR) (Kazhdan al., 2003) and statisti- cal descriptors based on signatures, histograms, or transformation- derived features (Tombrr et al. 2010 Guo al. 2013) Rusu al. 2009 Beckmann et al.) 2025
Table 1: Properties of OT-based SE(p)-invariant methods.
Type
Method
Complexity
Metric Isometry
EMD
Onn)) log
Extrinsic
SPM
O(n" log O(n log
x
RISGW
GH
NP-hard On²))-O(m1)
Intrinsic
GW
SHR
Representation RIT
X
X
SEINT (uurs) (og))))O0
Piening & Beinert 2025). ) as well modern learning-based approaches such as neural networks (Yew &
Complexity here evaluated under the assumption p. Metrie indicates whether the method a well-defined metric. Isometry indicates whether the method's 's applicability extends to comparing general isometry classes.
Lee 2020; Salch et al. 2022; Wang ct al. 2022) and Rotation-Invaal Invariant
While these methods demonstrate strong empirical performance, they may induce a pseudometric rather than true metric, since the representation process can potentially discard some geometric information.
Transformers (RIT) (Yu al. 2023).
The comparative analysis in Table reveals that existing strategies whether extrinsic, intrinsic, or representation-basee based fail to simultaneously optimize for computational efficiency, rigorous metric
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properties, and general applicability across isometry classes. To address these limitations, we propose the SElp)-Invariant Transport (SEINT) metric, which aims to achieve this crucial balance.
Our contributions can be summarized as follows:
1) Novel Representation Route:
We introduce two innovative unsupervised representation tech- Polar Transport Discrepancy (DePTD), which generate ID SE(p)-invarian features without training. These two techniques form the foundation for the SEINT distance through optimal transport.
niques, called Polar Transport Discrepancy (PTD) and
2) Strong Metric Foundation: We prove that SEINT is a proper metric on isometry classes of normed spaces, which inherently guarantees SEGp)-invariance for data embedded in the normed space. In addition. SEINT also enables cross-space comparison of distributions.
3) Efficient Numerical Implementation: The SEINT metric achieves O(n²) time complexity in general cases. For scenarios where the ground distance matrix is decomposable, we demonstrate that this complexity can be further reduced to On log())). enabling large-scale applications.
Experimental results show that SEINT performs well on point cloud classification and cross-space tasks, and it also works effectively as regularization term. For molecule generation, SEINT brings substantial improvements to baseline methods and achieves state-of-the-art (SOTA) performance on key metrics. These results validate SEINT as practical SF(p)-invariant metric for structured data and effective tool for regularizing complex generative models.
2
PROPOSED METHOD
2.1 MOTIVATION AND BRIEF PRINCIPLE
Developing effective SE(p)-img +E(p))invariant metrics is crucial for solving structural learning problems. We address this challenge through optimal transport-based framework that utilizes novel SE(p)- invariant features constructed from measures in the measured Banach space.
Throughout this paper, denote measured Banach space via the triplet II-I|- ,xx). Here, (IIlx) is Banach space (a complete, separable normed vector space), ((X) is its Borel a-algebra, and jax be a probability measure on the measurable space (X, B x Consider two measured Banach spaces (X, II-Ilx ,x)) and (Y, II-llly : NY) with their induced metries dx and dy. Without loss of generality, we assume the distributions under consideration are centered. Furthermore, we assume that any relevant isomorphism maps the base point 0 to the base point O. These assumptions effectively standardize the spaces with respect to translation, meaning we disregard differences arising purely from translational shifts.
Drawing inspiration from optimal transport in polar coordinates, we introduce two novel unsupervised representation techniques: the Polar Transport Discrepancy (PTD) and its enhanced variant, the Distance-convolute Polar Transport Discrepancy (DePTD). These training- free techniques effectively extract SB(p) -invaraat features from measures in Banach spaces. We now provide formal definitions of PTD and DcPTD, followed by the proposed SEINT approach.
2.2 POLAR TRANSPORT DISCREPANCY AND DISCREPANCY
POLAR TRANSPORT
Given two measured Banach spaces (X, Il px) and (Y, I1-lly pY) with induced distance dx and dy, a probability measure æ on their product space X x Y is coupling of μx and by if its marginals satisfy (projx) = μx and (projy) ) fy. where projx and projy are the canonical projections. The set of all such couplings is denoted by I[ H.). For measurable map f: X Y. the push-forward measure s lxx on (Y,B(Y) is defined by fa (A) == px (x( for any set A E B(Y). If f is surjection satisfying dy (f())), (f(x), = (x(x, x') for all σ E X, it is called an isometry between X and Y. If such an isometry exists, the two spaces are said to be isometrid Consider reference probability measure Pz E P(R) on the ID Banach space
Since norm induces metric via dx (xr, xx' := || x' llx it follows that any measured Banach space (X, Il-llx x (xx) is also metric measure space (X, dxxxx) with this induced metric.
Note that, if the spaces (X, Illlx (xx) and (Y, I1-Ily /uy) are isometric via such isometry, their push-
forward measure of the norm must coincide, i.e., (II- / = (II-lly )apy.
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| |I). where P(R) denotes the space of probability measures R. We define cost function c: C: x + R2o by c(z, 2) = =|||=||x - 2/. This is equivalent to computing the optimal transport between the distribution of ||X]|x x and uzz. and hence the existence of optimal coupling is guaranteed. We denote the set of all such optimal couplings by:
inf 1xx,2 := { 11 pz) Exx | - 2| a (ET((x
Ex x =
For any "x,2 2. we can disintegrate it as d x (z, z) dx 2 (z)dux (2), where π 2 is the conditional probability measure on R given Z E X. We define the Polar transport discrepancy as follows.
Definition 1 (Polar Transport Discrepancy (PTD)) Given
measured Banach space
(X, |+ jx), a reference distribution μzz E P(R), and an optimal coupling x xxx z E 1xx,z from
equation the Polar Transport Discrepancy Gxx X + R2o is defined as:
S=x (x) = |||=||x - 2 di =/
(2)
Note that PTD exhibits key property that enhances its flexibility. Specifically, the value of PTD depends on the selection of both the reference distribution μlz and the particular optimal coupling Tx.2 z (in cases of non-unine non-unique optimal couplings). This dependency enables the generation of multiple distinct feature representations, making PTD particularly versatile for capturing different aspects of the measure x-
However, one key question remains: Do PTDD-derieed features suffice for building a true isometry- invariant metric between two measures, rather than just pscudometric? Existing literature suggests this may not hold, as intrinsic distance information proves indispensable for constructing proper coometry-invariin metrics, demonstrated in foundational works on Gromov-type distances (Mé) moli] 2011 Sturr (023). To address this limitation, we develop an enhanced approach that synergis- tically combines PTD with intrinsic distance information through kernel-like transformation. This leads to our novel isometry-invariant feature extraction function, formally defined as follows.
Definition 2 (Distance-convoluted Polar Transport Discrepancy (DePTD)) Given a measured Banach space (X, II. (xx with induced metric dx, a reference measure Pz E P(R). an op- timal coupling xx E TT'x from equation 7 and the corresponding polar transport discrepancy Saz,2' the Distance-convoluted Polar Transport Discrepancy 0xx,2 :X- + Ryo is defined as:
0x 2x (z) == J dx (x((2,2) (x') (x').
(3)
Remark DcPTD exhibits two key properties that ensure its soundness as a representation. (I) It satisfies rigorous isometric consistency: for any isometry f +YY,, we have Φnx,2 (a) 2 (x))), thus guaranteeing the invariance of extracted representations under rigid isometric transformations. (2) It yields a dimension- inneeendenn representation, as the output Φx (I) is always non-negative scalar value on the real line space (R. I-), regardless of the ambient dimension of the input space.
These two unique properties of DcPTD enable a novel approach for comparing distributions across different spaces. Given two measured Banach spaces (X, |1-1Ix (xx) and (Y, II-ll , with induced diimension-independee characteristics
metrics dx and dy. we leverage DcPTD's ' isometry-invarian through the mappings (Ørx 2 Way.)) to encode structural information of two input distributions μx and py into a common representation domain. The encoding step transforms the cross-space comparison problem into tractable optimal transport computation in the same space, i.e., by computing
inf (E=11 a (x) &rū ()) =EI((xxXX))
(4)
This Optimal Transport computation capitalizes on DePTD's ability to maintain geometric invariance while providing a unified framework for distribution comparison across potentially heterogeneous spaces. One key question that remains is how to select the reference measure pz. We now address this by establishing principled criteria for μz selection, followed by presenting the SEINT metric definition.
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1D Wasserstein Distance
Φ Z
Φ
X
C
CY
Distance Matrix
Distance Mapping
Distance Matrix
Zz
X
Y
Polar
Polar
2 Unit
X E R
YERM
the input data from two spaces, X and Y. their respective norms are first calculated. Then, for each norm, a PTD is computed through a randomly generated reference 2. PTD is then convolved with distance matrix to obtain DePTD. Finally, the ID Wasserstein distance between the DePTDs yields the corresponding distance.
Figure 2:
An illustration the numerical implementation of SEINT.
2.3 SPECIAL EUCLIDEAN GROUP-INVARIANT TRANSPORT (SEINT) METRIC
To simplify the choice of optimal coupling 7 πÝ.2 in them directly connect to the data measures and jy. Firstly, we construct the candidate set of Z as the family of distributions conditioned on X, i.e.,
I we restrict the selection of μz and make
x [R)
{uz|x {X P(R)} = xx s.t. x [xp2ix
(5)
This guarantees that every is associated with a corresponding optimal coupling x E 11 x z. Given reference measure μzz E Pxx (IR) and its associated optimal coupling Tx,z, for any coupling TTxxx, nY), the optimal coupling on the other side (मÝ.2 E (T must satisfy
प..2 argmin my Ell Ex (KG (x) Coy (w)))
(6)
where S and Say' are the PTDs Definition T By restricting μz Px and adding condition 6 we ensure that the other optimal coupling मÝ.z is also associated.
Similarly, we define (IR), and consider their union PX, = Px 6 symmetrically on the opposite side. With this construction, selecting any reference measure μz specifies pair of optimal couplings(=xx T..)). mb.z). Thus, the optimization reduces to searching over μl within Px,y (R), and identifying single least favorable reference measure (4z, is adequate for comparing the distributions on the two metric spaces. The distance obtained under this formulation defines proposed SEINT metric, formally stated follows.
U (R)), where condi-
Definition ISnapi-Invrrian Transport (SEINT) Distance) Consider two measured Banach spaces (X, I1-Ilx ,x) and (Y, 11-1l- ,yY)ly )Y) with induced metrics dx and dy. Let Px, (R) Px PR)UUP (R) denoted equation For a given reference measure pz Px,,, let x E I and Tÿ. E Il be the corresponding optimal couplings. Let Φu and Ox;, be the DcPTD maps associated with couplings æx,z and Z.. The SEINT distance defined by finding an optimal coupling I[(xx. [P)) that minimizes the least favorable measure μzz over all choices Px, (IR):
CSEINT (X.Y,xx,xx) := inf xellixx y
sup
(Enr [10x,,2 (z) Φng 2 (w))))
(7)
The pipeline for calculating SEINT is outlined in Fig. 2 We now provide the following theorem, which demonstrates that SEINT a well-defined distance metric and provides a characterization of isometry across different spaces.

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Theorem (Metric Property on Isometry Classes) ESEINT defines metric the space of isom- etry classes of metric measure spaces. Specifically, for any measured Banach spaces (X, 11-Ilx ,x), (Y, I-llly (y) and (2-11-llz Hz):
1) Identity of Indiscernibles: LSEINT J, HYY 2 O, and (X, only if (X, dxxxxx, ,xx) and (Y,dy, ,,,y,uy) (UY ure isometric.
and
2) Symmetry: LSEINT( X,Y,
LSEINT (Y, KY,YHHX
3) Triangle Inequality: LSEINT X, Y, ,Y,, S LSEINT X, Z. (z) + LSEINT (Y, E-yY,uu
The proof of the theorem is detailed the Appendix B metric measure spaces equipped with a well-defined base point; see the Appendix [B3 for additional theoretical extensions and related results. In addition, recall that the Special Euclidean group SE(p) represents the orientation-preserving isometries (rigid transformations) of R", Given SEINT's explicit design for invariance under such transformations, we immediately obtain the following corollary regarding its invariance property.
Notably, SEINT extends naturally arbitrary
Corollary (SE(p) Invariance) Let (X, II Hx) and (Y, I1-Illy (y) be two measured Banach spaces embedded in Euclidean spaces, where X RD. For any transformation 9 (R. t) e SE(p) := (O()) x IR" its action on X is defined as 9(X)= == R.X + tt. Define the transformed space as (g( (X), II-I1 9(X)) 9(x))9#xx). }. Then SEINT is invariant under the SE(p) transformations, i.e.,
ESEINT (X, ,,x, ( 9
(8)
2.4 NUMERICAL IMPLEMENTATION
In the discrete real-valued setting, let X = (æI..... vn) T E Rxx be a sample matrix. The associated empirical measure is μr Σn =i= Pw; Swg, where px (Pa ..... Pa. E An-1 is probability vector in the standard (n 1)-simplex. To enable efficient optimization of equation 7 in the discrete setting, we require only the following constraint on the space of reference measures, which leads to the corollary presented below.
Corollary 2 (Metric Property for Discrete Distributions) Let two discrete measures be defined by their supports E [Rax and Y Rmxx,, with corresponding probability vectors Px and py: And their norms are bounded by M. Further, consider the space of discrete measures P(R) = {Hz = E 1 15, {2j "j]ısssk E [0, M) Then, under the assumption that values of the norm || (i=1..... m.) are distinct and l is large enough, SEINT defines a metric on this space of measures.
The proof is detailed in Appendix B.4, Note that the distinctness assumption is generally satisfied in empirical distributions. Consider reference distribution μzz uniformly drawn from P(RR) in Corollary?? 2 Under this distinctness condition, the set of optimal transport plans ITx, in equation contains a unique element, which we denote II* E Rax=, In this scenario, the values of the PTD from equation 2 depend only on X and and can be expressed as:
11
i= i= 1,..,,,
Pa:
The algorithm for the numerical implementation of SEINT is given below.
Complexity Analysis. The time complexity of Algorithm I involves several steps. Computing the distance matrices x and CY takes O(n² + m²) time. According the North-West corner rule(Peyré et al. 2 computing G.. and cy2 takes O(n log n + m log m + k log k) time. The matrix-vector multiplications Cx
S2, and CY S2 require O(n² + m²) time. Computing the final distance between feature vectors requires (n log n + m log m) time. Assuming is constant, the overall time complexity is thus Onn" m²). Note that the computational complexity of Algorithm primarily stems from two operations: (1) distance matrix computation and (2) subsequent matrix- vector multiplications. For structured distance functions, e.g., squared Euclidean distances d(x, y) = exploit matrix decomposition techniques accelerate the overall computation,
||æ - y 2. resulting in an overall complexity of O(n log n + m log m) when n > p, Implementation details and accelerated algorithm provided in the Appendix C and Algorithm

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Algorithm 1 SEINT Computation
Require:
Two datasets X R=xP and Y E Rmxx probability vectors px E An-1, Py A Am-1 number of reference distributions a, support size for reference distributions k Initialize the data X and Y.
2: Generate random reference distributions Pzz PZze. ................................ P
with l samples.
3: Calculate the distance matrix: C* <- (| - زÿ by; Cx + (1lv - y Jilly)i O(n² + m²)
4: for i 4- 1 to do
5: Compute 2z using X, Px, HZZ via equation
>
On log n + log k)
6: Compute cY using ,y, Hz, via equation 9
0 log m + k log k)
7: 2% C* S ΦY + C Cz.
On + m²)
8: Calculate the ID-W2 loss: C, W2 ( #xx, (0%)
D On log n m log m)
9: end for
Set the final loss LSEINT + maxisisa Li
11:
return LSEINT
2.5 INTEGRAL- SEINT
SW distance (Bonneel al. 2015) and Max-SW distance (Deshpande al.) 2019) are two popular variants of the standard Wasserstein distance. The key difference between them is that, SW integrates ID Wasserstein distances with respect to all random projections. while Max-SW selects the projection that leads to the largest ID Wasserstein distance. Drawing parallels to our SEINT framework, which adopts worst-case perspective by maximizing over reference measures, we can similarly construct an integrated variant called Integral-SEINT (ISEINT):
Ex (ExxD(Px.x [ (x) - Pny, (WP)) (10)
LISEINT (X, Y,Kx, := inf
*Ell(ux.Pr) .Y)
where the expectation is taken over a suitable distribution D(Px,y (R) of ID reference measures Fzz Px.y (R.. numerical approximation for ISEINT can be obtained by simply modifying Line
10 in Algorithm D we compute the average value instead of taking the maximum value.
3 EXPERIMENTS
We conduct a systematic investigation of our method from two key perspectives: (1) a comprehensive analysis of its metric properties, and (2) a practical assessment of its performance as regularizer in downstream applications. All experiments were conducted on a server with 256 GB RAM and 64 cores Intel @ Xeon Gold 5218 CPU, and standard workstation equipped with a single NVIDIA RTX 4090 GPU. Representative experiments are shown below, and more implementation details and results are in the Supplementary Material.
3.1 AS A DISTANCE METRIC
(Elp)-Invariann To empirically validate the SE(p)-invariance property established in Corollary I we conduct experiments on 3D point cloud classification using specially constructed ModeINet4O- SE(3) dataset. This benchmark is derived from ModelNet40 (Wu et al. 2015) through the following augmentation protocol: (1) We randomly select an object from the original pre-aligned 3D models in each category. (2) Each base model undergoes 25 random SE(3) transformations (rigid rotations and translations); (3) We further add random Gaussian noise to models increase variability. The final dataset contains 1000 transformed models. We selected eight SEn(p)-inarrian metrics for comparison, including the GW distance (Mémoli) (0TT), the Entropic-GW (EGW) distance (Peyré al. (016)), the Rotation- Invariant Sliced GW (RISGW) distance (Vayer et al. 2019). SW distance (Bonneel et al.) 2015). SGW distance (Vayer et al. 2019), and Wasserstein distance (W (Cuturii 2013). Notably, the first four metrics explicitly claim to possess rotation invariance. The regularization parameter for EGW is set to 0.1. The number of projections for RISGW, SW, and SGW, as well as the number of reference distributions for SEINT, are all set to 50.
As shown in Table 2 our method consistently achieves 100% accuracy, confirming that SEINT is a strictly SEEp)-invariant metric. Moreover, SEINT attains the highest computational efficiency among
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Table 2: Comparisons of classification accuracy (%) across different numbers of neighbors (k)
Method
(cC(E=1)
5c
(cc(
Time (h)
-E
GW
2655.9
EGW
2644.3
RISGW
74.9439
59.5
53.0
400.6
W2
52.5432
38.4423
31.3
113.4
X
SGW
59.1
46.214.
40.9
20.2
X
SW
56.7+4.1
41.644.11
37.2+4.0
16.6
X
SEINT
100.0....0 0.00
100.0...0
100.0
9.01
ISEINT
100.0.0.0
100.0+0.0
100.0
8.29
All reported times are CPU times.
2 For GW/EGW, entries marked "-" indicate results that are not reported due to their prohibitive computational cost.
all evaluated approaches: it is faster than competing methods while uniquely preserving perfect accuracy, whereas alternatives such as SW and SGW remain slower and fall below 60% accuracy. These results highlight SEINT's strong balance between efficiency and accuracy.
Cross-Space Capability. To evaluate the inter-space capability of our metric, we employ the horse- gallop dataset (Sumner & Popović (200)), which contains reference horse model and its dynamic galloping sequence. Our experimental protocol involves two steps: (1) Projecting
reference point cloud onto three orthogonal principal planes (XY, XZ, YZ) to create 2D variants, (2) Computing distances between each projected/referenee space and all gallop-sequence point clouds.
The results in Fig. reveal three key findings about SEINT's cross-space performance: (1) It precisely captures the gallop dynamics through three peaks that align with the horse's stride phases, outperforming EGW in terms of variation amplitude; (2) While GW and RISGW show larger fluc- tuations, their patterns deviate from the expected biomechanical sequence; (3) Most importantly, SEINT demon- strates strict metric consistency its loss values increase monotonically as the source-target distribution diver- gence grows, validating its theoret- ical properties comparisons. This systematic eval- uation demonstrates SEINT's ability to compare distributions across het- erogeneous ambient spaces (2D vs 3D) while maintaining meaningful ge- ometric relationships.
Orginal 3D Space
RISC
Finst Project Space
Space
Third 20 Projecion Space
in cross-space
Steps
Figure 3: Loss evolution between the dynamic horse sequence and four references. The SEINT distance exhibits three distinet and consecutive peaks, while EGW shows similar qualitative trends with a weaker response magnitude, and GW/RISGW fail to maintain consistent distance relationships.
Capability on High- liggenDimnnion Data. We compare SEINT with several OT-based methods, in- cluding GW, Oeoation-IIvarrinn Sliced GW (RISGW), Sliced GW, and Wasserstein Distance, under high-dimensional settings. Let {æ:)²_.
fe-1 and {3s 1 /= be i.i.d. samples generated from Na (0a, Ex) and Nº, (0., Ey), respectively. We set n 200, diag (3I, + 30 Ia-2), where Ia and B, are identity and backward identity matrices with size x d, respectively. We generate different samples with respect to each and calculate the distance between them under different metries and normalize these distances to ensure comparability. Fig, 4a) presents the average distance across ten replicates. While most of the distances considered here demonstrate the curse-of-ding ofus--of-immensionalii through either parameter insensitivity irregular fluctua-
50, Σx diag (3I2, Ia-2) and Ey
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tions, SEINT and ISEINT maintain smooth, consistent distance variations with respect to 8, indicating
superior dimensional stability.
Computational Efficiency. To further evaluate the efficiency of SEINT, evaluate the runtime of all metrics across varying sample sizes n. We retain the data-generation setting from the high- dimensional experiment, but fix the dimension at d = 4 and set 0 = 0.5. For each n, perform 10 independent runs and report the averaged results with error bars.
The computational efficiency analysis, presented in Fig. 4b)
(a)
(b)
reveals two key findings: (1)
1.00
Our Methods
While SEINT and ISEINT show
Ditance
0.75
SEINT
ISEINT
modest overhead for small n,
0.50
logbal
Other Methods
they achieve superior scalabil- ity as sample sizes grow, outper- forming alternatives by increas- ing margins; (2) The observed log(n) complexity confirms the complexity analysis in Sec- tion 2.4 and algorithm
RISGW
SGW
0.00
Wisser
-1.0
-0.5
0.0
0.5
1.0
0
(00::(0)
Figure 4: (a) Distances versus different 0. (b) CPU time (s) versus
different when d = 4.
3.2 AS A REGULARIZATION TERM
3D Molecular Generation. Recently, molecule generation (Zhou et al.
2023 Jin et al. 2020: Shi
ict al. 2021 Nichol Dhariwal 2021; Yue et al. 2025) has emerged as a prominent application of
diffusion-basse basodd generative models (Wu et al. 2022 Huang al. 2024 Ho et al. 2020p. Among these, the E(3) Equivariant Diffusion Model (EDM) (Hoogeboom et al. 2022) is the first diffusion model directly generate molecules in 3D space, showing strong performance across datasets. We also include UniGEM (Feng et al.) 2024),, an equivariant, geometry-awe awwre model that jointly generates atom types and 3D coordinates, and serves
competitive baseline for 3D molecular generation.
We integrate SEINT as a regularizer into the training objectives of both EDM and UniGEM, augment- ing the standard Lg loss (for coordinate denoising) with SEINT-based alignment terms; their relative weights are balanced via simple scale alignment. Following EDM and UniGEM, we also conduct experiments using the QM9 dataset (Ramakrishnan et al. 2014). a widely recognized benchmark containing approximately 1 130,000 small organic molecules, each composed of up to 9 heavy atoms. The dataset partitions follow Anderson et al.) (2019). consisting of 100K training, 18K validation, and 13K test molecules. We further include GEOM-Drugs, the "drugs" partition of the GEOM dataset (Axelrod Gomez-Bombarelli 2022), which provides large-scale, eneryy-annotated 3D conformational ensembles for trug-like molecules. The detailed experimental settings are provided in the Appendix D.2
Table 3: Pre-training results: Comparison of methods on atom stability (Atom.), molecular stability (Mol.), validity (Val.), and validity xuniqueness (V*U.), each drawing 10000 samples from the model.
Backbones
EDM
UniGEM
Metries
(%)
Mol.(%)
Val((%)
V=U.(%)
Atom (%)
Mol(
Val.
V*U.(%)
Data
99.0
95.2
97.7
97.7
99.0
95.2
97.7
97.7
Baseline
98.7+a.1
87.2+04
93.4+00
91.7+a.
98.9+a.1
89.4
94.640.
92,8
LorNorr
98.8
87.64a2
94.5402
92.84az
98.94a.
89.0
95.0402
93.1
L2-N L
98.7
87.2
93.740.3
92.1403
98.7
85.3204
93.940s
92.2.144
-Normas
98.7
86.0+a7
93.82.04
92.4
99.0+a.1
89.5
95.2
93.3
SEINT
98.9
89.04a.1
95.3
93.0
19.1
91.0=
95.5
92.8
SEINT,2
99.0=a.1
90.1
95.8402
92.6403
99.2+a.1
92.1 12.2
96.5402
92,4
SEINTa3
99.1
91.5
96.5
92.0+a3
99.3
93.5
96.9402
91.8403
The notation SEINT, indicates that the objective is L aC + (1 a.) CMSE.
Lz-Norm: loss computed only the coordinates.
Tables and report the performance of SEINT when applied as a regularization term in both pre-training and fine-tuning experiments. The results demonstrate that SEINT significantly enhances

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Published as a conference paper at ICLR 2026
chemical validity, atomic stability, and molecular stability. In particular, when the SEINT proportion set to 0.3 in UniGEM, the model achieves state-of-the-ar (SOTA) performance both atomic stability and molecular stability. Although slight reduction in uniqueness is observed, this trade-off justified by the substantial improvements in validity, which is arguably the most critical factor for generating chemically plausible molecules.
Table 4: Fine- Fine-tuning results on QM9 GEOM Drugg.
Datasets
QM9
GEOM- Drugs
Metries
Atom.
Mol.(%)
Val.
V*U.(%)
Atom.
Mol.(%)
Val.
V*U.(%)
UniGEM
98.9
89.4
94.6
92.8
84.8
1.2
98.3
98.3
Ly-Norm
99.1
91.2
95.7
93.5
84.4
1.3
99.2
99.2
SEINT Ta1
99.2
92.4
96.6
93.5
86.7
4.18
98.5
98.5
SEINTa3
99.4
94.9
97.8
88.6
90.7
13.0
97.7
97.4
4 CONCLUSION
In this paper, we have introduced two SE(p)-invarian representations: the Polar Transport Discrep- ancy (PTD) and the Distance-convoluted Polar Transport Discrepancy (DePTD). We have proven that optimizing DePTD via ID Optimal Transport formulation yields strict metric the space of isometry classes. Through a series of experiments, we have demonstrated the effectiveness of our proposed metric and its computational efficiency. Furthermore, it can be effectively employed regularization term in various complex shape learning tasks. Promising future directions include: (1) extending the loss function's 's regularization applications generative modeling, computer vision, and multi-modal domains; and (2) designing novel learning architectures that capitalize on its distinctive characteristics.
ACKNOWLEDGMENZ AND DISCLOSURE OF FUNDING
This work was supported by National Natural Science Foundation of China under Grant 922701 10 and Grant 12471244, and National Key R&D Plan of China under Grant 1016200
ETHICS STATEMENT
The datasets used in this paper are all publicly available and do not involve any ethical issues.
REPRODUCIBILITY STATEMENT
We are committed to ensuring the reproducibility of our research. The source code for all experiments provided in the supplementary materials, which can used to reproduce the results presented in this paper. For the molecule generation experiment, we have included our key evaluation checkpoint. This checkpoint can also be reproduced using the provided training code. Furthermore, we have explicitly stated all key assumptions for the theorems presented. The sources or preprocessing scripts for datasets used in this work are also provided.
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A ADDITIONAL BACKGROUND
Wasserstein Distance. Let (Lt, d) be a Polish metric space, and let μxx, jy be two Borel probability measures on (52, B(S)). The p- Wasserstein distance (p 1) between μx and jy is defined via the optimal transport problem:
Wo(xx,1x)) = -1ee inf (FEI(ux wy) Jaxx d(z, dr dz
(11)
When the space is the real line ( 1). let μxx be probability measures (R, B(R)). Let Fx(x) - px (x(((oo,at) ai)) and [yyy = jay ((=00, be their respective cumulative distribution functions (CDFs), and let Fx1 and Fr1 be the corresponding quantile functions. The p- Wasserstein distance admits the closed-form expression:
Wp(px, = ( (F (w))
(u)l du
(12)
Metric Measure Space. Following the work of Mémoli 2011 Sturm 2023) ) we define mer- ric measure (mm) space via the triplet (X, dx, B(X)), where (X, dx) is Polish (complete and separable) metric space and (X) be its Borel o-algebra. Let μax be probability measure on the measurable space (()). The support of px, denoted supp( ), is the smallest closed set C S X such that x (C) 1.
Distance. Consider two metric measure spaces (X, dx, px) and (Y,dy, (.). The Gromov- Gromov-Wassersten (GW) distance compares their intrinsic metric-measure structures. For 2 1, the p- Gromov- Wasserstein distance is defined as:
Fromov-Wassersiee
S [xx(x, dy dy(,.'') dx v) dn .u) GWp(Hx.py) +Flll(x (p) ( XxX JxxY
inf
(13)
where the pairs (z, y) and (z', are drawn independently from the coupling measure æ. The term |dx (xxxxxx)) - dy (v. (U,'') 1 measures the distortion of pairwise distances under the coupling m
Computational Costs. When px and jy are empirical distributions supported on m and samples respectively, computing the Wasserstein distance (1T) corresponds to solving a linear program with un variables and m + constraints, incurring a time complexity of at least ((m + n2))) (van den Brand et al. 2020). Applying entropy regularization (Cuturi 2013) transforms the problem, enabling the Sinkhorn algorithm with a significantly lower per-iteration complexity of (mm).. Further acceleration techniques include those presented in (Altschuler al. 2019 Li et al. 2023a), among others. In the specific case of the one-dimensional Wasserstein distance (12) computation is much faster via sorting, requiring only O(m log + log n) time. This effciency in one dimension has motivated several fast OT algorithms that leverage projections onto ID subspaces (Bonneel al. 2015 Li al. 2024), achieving considerable success. Computing the
distance 13 involves solving a non-convex quadratic program, potentially requiring up to O(m²) time complexity for exact solutions. While recent fast algorithms (Li et al. 2023b Sceebon et al.) 2022] Ouyang et al., (226)) offer significant computational speedups, this accelcration often comes the cost of preserving the metric structure. The overall computation for the Gromov- Wasserstein distance remains significantly more demanding compared to the Wasserstein distance.
Discussion on convexity. Extrinsic SE(p) -nvarraat approaches such as Softassign Procrustes Matching (SPM) (Rangarajan et al. 1997). Rotational Invariant Sliced Frrmov--Wsseeste (RISGW) (Vayer et al. 2019). and Wasserstein Procrustes (Grave et al., 2019: Alvarez- et al., 2019) fundamentally rely the joint optimization of transport couplings and orthogonal transformations.
This formulation leads to an inherently non-convex problem. Recent advances such as the Normalized
Radon CDT (Beckmann al. 2025) reduce multi-diment multi-dimensional data into sets of one-dimensional projections through the Radon transform, which completely avoids the non-convex orthogonal alignment step. For intrinsic methods, representative metrics such as achieve strong empirical performance 1981) and
(GH) distance (Gromov
(GW) distance (Mémoli 2011)
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in applications such shape matching. However, both metries require solving highly non-convex optimization problems, which results in substantial computational complexity. Several accelerations have been proposed, including SGW et al. 2019), EGW(Peyr) et al., 2016), and Low-Rank (Scebon et a 2022), although none of these approaches change the fundamental non-convex nature of the original formulation. Another line of work focuses on optimizing convex upper (Li et al.) 2023a or lower bounds of GW (Sato et al. 2020). These relaxations avoid non-convex optimization, the cost of losing the metric property.
In contrast, SEINT achieves SEAp)-invariance, the metric property and linear computational complex- ity, while retaining convexity its discrete formulation according Eq. [5] The continuous version requires further theoretical investigation, which we consider a direction for future research.
B PROOF OF THEOREM 1
Before presenting the proof, we first state the following lemma.
Lemma 1 (Gluing Lemma) For given measure spaces (x,))), (Y, and (Z, and given couplings xx22 E M[xx, Pz) and Ty,z E [Tuy, (zz), there exists joint coupling xx,YZ E Mpx, PY, (zz) such that ity marginals satisfy
( = xx22 2)) = mY
and its (X, Y--Y)mgrmnnl,,, (proj(x,r) # is coupling in l(
Proof. Given the disintegrations da dxx,z(xx,2)
(x) and (v,
dry z= d (z)duz(z), we define the joint coupling Tx,Y via its disintegration: dxx, Y,z (a, y, da x (1)) This xx,Y, satisfies the required marginal conditions by construc- tion.
For distributions on the real line (R), we obtain important corollary:
Corollary Consider three probability distributions pxx Juy, uyz on R. Let xx and ny, z be the optimal couplings for the Wi distance between (px,Pz) and (ny, ,,z), respectively. Let uxY be the joint distribution obtained via the Gluing Lemma construction from xx and Ty,z- Then, its (X, Y) marrginal,, ( XXY is optimal coupling for the W1 distance between μxx and μy.
Proof.
Let Fx, Fy, denote their respective cumulative distribution functions (CDFs), and F 1.F..FFF the corresponding quantile functions. The optimal couplings xx,2, Ty,z can be represented pushforward measures:
x = (Fx F2,Id)
Ty.2 (Fy' F2,Id)
Applying the Gluing Lemma construction (effectively using Z as the common random variable whose quantile Fz (Z) drives the others), we obtain the joint distribution
xx,YZ (Fx Fz, Fy'
Fz, Id)
The (X, Y)- -mmaginaal is (r) = (FF Fy oz) mzz. Let U = Fz(Z). IF continuous, U is uniform on [o. 1]. and the marginal corresponds to the law of (U), which is the optimal quantile coupling between ya and ny. Thus, ) Z is optimal coupling between μx and μay
To prove that SEINT defines a well-defined metric, we will demonstrate that it satisfies the Identity of Indiscernibles. Symmetry, and the Triangle Incquality properties.
B. PROOF OF IDENTITY OF INDISCERNIBLES
Drawing inspiration from the characterization of isometry in the context of Wassertesssnsttein distances, such as (Sturm) 2023 Lemma 1.10), suffices to demonstrate the equivalence of the following two conditions:
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(i) CSEINT(X, Y, Mx, O.
iii There exists a coupling E [(xx,,y) such that dx (x0, dy (yo, U1) holds for т @ r--almost every pair points ((wo, No), (x1,31)).
We first prove the implication (i) (ii). By definition, the condition LSEINT X, Y, ,xx, Myy) implies that there exists an optimal coupling π E Cpl(ux, juy), ), for any reference measure Jaz and the corresponding optimal couplings xx.2 E x2 and TÝ.2 Hī satisfying condition in equation 6 the equality Φ (x) = Pry, (0) holds for all (z, y) E Supp(T).
The proof will proceed in three parts.
Part 1
For measured Banach space (X, II- ,,x), let px X Rz0 denote the norm function, px(x)) == || We denote the distribution of this norm (its pushforward measure) by Max (px)=t #xx- We first aim to prove that if CSEINT (X,, Nx. py) = 0, then these distributions of norms must be equal, i... Ppx Ppy
Proof. Assume for contradiction that the distributions μpx and ppy not identical.
Let the reference measure pzz = Mpx Consequently, it follows that Sozz 0 almost everywhere
(a.c.).
Regarding Say.e* argue that there must exist some 0 such that the set {y E ; Suz. (v) > E} has positive juy-mn measure. Otherwise, if S (g) a.e., then we would have:
0 = E((ny.z) )= = == // (pr(v)) 2 dr ( (2)dur
/ [pY (v) 2/d7y, (z.
/ Ipy (v) Px dr x
2 Wi (Hpxx Hpy).
which implies Wi (ppx: Mpy) = O, and thus Mpx Hpy a.e.. This contradiets assumption. Let
V. = (y E/: Smz (1) c). We know jy (V:) > 0.
From SF (x) = 0 a.e. we infer that Φ=x,2 2 (x) = 0 a.e. By the premise LSEINT (X, Y, Y,,,, ,xxx =
0, there exists coupling Cpl(ux, (Y)) such that Φxū,z (v) Φa 2 (z) 0 for all (z. v) Supp(r). This implies that &xv, (y) for pa-malmst every y. Therefore, for a.e. y E V we have:
Onūz (v) - / dr
dr (y, (L, dur(p') 0.
Furthermore, from this, we have:
0 / fdy /dy(v, dy. Sax z (w))
(/) [ dy( (4)))(()))))))) 2 yr
/ edy dd v/) dpy(s/).
Since e and assuming (v, 2 O, this yields that for p.ea.e Sy. dy (v. (v) O. If d metric and py (Y.) > 0, this implics
V. the integral
(y. = for a.e E y and py- y' YY.
Consequently, for a.c. F1, 1/2 E V. we can find s/ Y. (more accurately, from subset of Y, with full measure within Y.) such that (21,,177 0 and dy = 0. By the triangle inequality (if d a metric or pseudometric)::
dy (y(1 U2) S dy (11, v') + dy dr =
0 a.e.
This forces dy (v1. v2) for a.c. V1 1/2-
similar argument (potentially by choosing Pzz = Mpr) leads to d(x), = 0 a.e. for all ,1,42 X. This implies that both μx and hy supported on sets where all points have zero
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distance from each other. Consequently, their push-forwand measures under the radical map, Mpx and Hpy, must be Dirac measures (point masses), i.., Apx = dex and Hpy = dey for some constants C 0. However, if Mpx and Pay are Dirac measures, we can consider Mpx = Mpy in the sense of translation invariance.
This contradiets initial assumption that Ppx and Pay are different. Therefore, the assumption must be false, and we conclude that Mpx = Ppy.
Part 2 Next, we show that the coupling must satisfy BT [[]11x(()) py (v)l] 0.
Proof. In the previous part, we established that Mpx = Hay a.e.. Consider two specific choices
for the reference measure Z. First, let Z1 = px (2) + 1, then S=k,2, (z) 1, (v) 1. This
choice is valid and satisfies condition in equation 6 for any pair of optimal couplings (T zy
Px(x)) Px (x) +x px(x) > 0 then Crk,2, (1) = (xx(xx) > 0} and Gr (1)
Second, let Z2
1 (v) > 03. This choice also holds for any optimal couplings T "x, T satisfying equation 6 Since X and Y are assumed to be centered, the optimal coupling и satisfying equation
6 must map
the origin to the origin (i.e., I = +
0 for π-a.c. (æt, v)). Consequently, we obtain
E E1 2,, (x) Sny, 2, [v)ll E 1I] = 0 and E, 1151 22
(x) S.y. (y)ll E,, || (px (z) >
- 1px (v) > 0}] =.
Consequently, for any reference distribution juz and corresponding optimal couplings मेद
satisfying equation 6 from the condition dx (I) 2ny,2 (y) for all (z. y) Supp(r). we have the equality of the transport maps:
ddx(x, (xx(x,z)sax,, (2) dpx))x / dr(w. v/ v' (v') duy (3').
(14)
This equality can be rewritten by integrating over the coupling æ. Since Sax (2') Cmzzz for (v')
(a', 3²) E Supp(T), equation 14 implless:
/ (dx(z, dy (y. v\) Sux,
(x') d = 0.
Substituting the potentials corresponding to Z1 and Zg into this relation, we obtain the following two equations:
/ [(xxle,2') - dy (v.8)) dT(' (an((''))=== =0
/ (xx(e,x')1(xxx() 0} - dy dr > 0} d'( = 0
/ ((x( -rr0)dr(r,.111 0)) = dx - dy 00
=0
00
Since (x, 0) px (x) and dy 0) py (v), this implies px (x) pr (v) for r-almost every (x, v). Consequently, E, ||px (x) py (v)l = 0.
Part 3 Based on the above conclusion, we can readily prove that dx (z0, = dy (No, v1) for any pairs (no, (I1, E Supp(=).
Proof. For any given (zo, Uo) E Supp(z), we can define a reference measure Z:3 = (v), py (v) I py (10) and y * yo
For the given Polar transport discrepancy (1) +1 1, (y > pr (y6) or y = yo
Su (I) = (x) > (z)) or x = Io), we can setsa; (v) = (p) > py (30) or y = = 0, it can be readily shown that EE, I|5 (x) (x)- (w)l = 0, thus T..2, satisfies equation is corresponded to Tx22, Therefore, for any given (I1,81) 31) Supp(#),
(0). As E. (x)
substituting Srx,29 into equation 14 yields:
/ [(dx(en,=)) dy 1(px > px( or x' = =z) do v/) 0.
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Similarly, we can define Z4- { py(y): py pv) I py (yu)
(v) +1, 1, Pv( > pr In this case, the Polar transport
discrepancys Sax (x) 1 (x) px ( and Eni.2, (v) = 1 (p) > py (yo)} also satisfy
En (x) - Sazie (v) = O and equation 6 Substituting this Cxx into equation 14 gives:
/ (dx( ,') - dr (v). (xx(xx) > px
d .v) =
Subtracting the second resulting equation from the first, we obtain:
/ (dx(=1,x - dy (v1. 1 = xo) dī (/)xx//))= ./)= =0
f dx((x,=') = mo) dux((')= = f dy (U1, = yo d1)
(0) = dy (91180)
Since the previous derivation holds for any pair (21, V1) E Supp(T) and any pair (Tos Mo) Supp(#). we conclude that dx (211 = dy (11, 3o) for all (æ1, V1). (Zo. yo) Supp(T))
We next prove the implication (ii) (i). Assume (ii) holds, meaning (X,dx, px.xxx) and (Y, p
are isometric. According to (Sturm 2023 Lemma 1.10), there exists a measurable map f f:Xxx X + Y such that f1x = juy and dy (J (x), ((x
dx (xr, x²) for jix px every x'). Assume without loss of generality that X and Y are centered, and then the isometry I maps 0 E X to 0 E E. Consequently, the distributions of the norms Hpx and ppr are identical. Let x be the coupling induced by the isometry f. i.e. x (Id, f) Since y y= fz) for (I, y) E Supp(z) and f is an isometry preserving the origin (implying Py (y) = Px (x)). we have E, IPIx (x) py (v)l =. O.
Now, consider arbitrary reference measure Pz P(R) and associated optimal coupling Tx,z E x- By applying Lemma] [ we can construct joint distribution xx,Y.2 on X x Z whose marginals on X x Y and X x Z are π and xx,2- respectively. Let Ty,z = (projy.z) TXY denote its marginal distribution on Y x Z.
Since x is an optimal coupling between Apx and Hpy, and π x is an optimal coupling between Hpx and μzz, it follows from Corollary that the constructed marginal Ty, is also an optimal coupling between Mp and pzz That is, TY,7 E ITy,z.
f(x). The construction via the gluing lemma ensures that the conditional distribution of Z given (X,Y) = (x, y) depends only on This implies that the conditional distributions satisfy Tz Y- = z x Since the Polar transport
Furthermore, for any (x,y) E Supp(T), we have
discrepancys 1 (Definition 1) depend on both the point (z or y) and the conditional distribution of 7, the conditions px (x) pr (y) and =z
z ensure that Grx (1) = CCr,2 (V).
Consequently, 1=l1srx, (x) STV. (v)l] = 0. Thus, the chosen optimal coupling Ty.z E Hl,2 together satisfy the constraint condition in equation 6
Therefore, we have:
2 (v) (rv,2(((x))
/ dy (I(())) (v) duy
I (dr()())) (f(x') dux(x')
/ ddx(=, (x') dpx((a)
(x)
Since Orx, 2 (x) = Sni,z (v) holds for all (a, v) E Supp(=), the term E., [1003, (1) Un (v) P is zero for any choice of reference measure jzz and its corresponding couplings π xx 2 Then the infimum over π in equation 7 is zero, which means LSEINT (X, H, = 0. This completes the proof of (ii) + (i).
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B.2 PROOF OF SYMMETRY AND TRIANGLE INEQUALITY
The symmetry of LSEINT follows directly from the symmetry of the Wasserstein distance involved in its definition.
We now prove the Triangle Inequality. Consider three measured Banach spaces (x, Il.1Ix px)) ).
(Y, : and (2, ||| z (uz), we can find corresponding SEINT optimal couplings л 2 HZ) and Ty, E Li pz). According to the Lemma XX,Y 11(xxxx My, such that
| we can find a joint coupling
(proj(x.2)) x # Tx,2 (projcy,)) TT,
Let xx,Y == ( Then we have XXY E I(x
Furthermore, we can find reference measure Jnn and corresponding couplings *x, ". that satisfy condition in equation (relative to xx,Y), such that
(BTx. Il (z) Φn (w))1)) LSEINT(X,, (X,Y,
For the given xx PR and the corresponding coupling Tx R we can find a corresponding coupling Z. satisfies condition in 6 And similarlily to xx,2; MR, R. Therefore, we have:
[ (x) 2xtER (1))) %
CSEINT (X,Y)
(Exx.
1 (x) Pxy.n (w)pp)
(1 (2) - 1 (=) + (x) - O (w))))))))))) 7
F
1 (x) - - (=)pp) + (Enxax [1orzn (z) - Dny,n (w))pp)
Exx,2.
= Exxx, | (z) 2az, A (2))²² " + (Erax [|0 (2)- (v)p))
S LSEINT (X, Z) + LSEINT ()
B.3 EXTENSIONS OF THE THEOREM
is evident from the preceding proofs that our theorem can be extended more general settings.
Extension to Metric Spaces. Specifically, the proofs primarily rely on the properties that the norm non-negative and is zero only the origin (æ: = 0). This suggests that the norm function II-IIx could potentially be replaced by any function px X - R>o that is positive definite (i.e., px(x) 2
with (x = if and only if x = O, assuming 0 is uniquely defined). For instance, consider a metric measure space (X, dx ,xx) If we designate base point O E X, can define the function Px : X + T2 as the distance to this base point, Px (z) == dx (x, 0). In this setting, it becomes apparent that if the metric measure spaces under comparison also satisfy the condition that relevant isometries are trigin-preserving method remains valid metric (or divergence) that respects these isometry classes within the context of metric measure spaces.
(mapping the base point 0 to the base point 0), then our proposed
While extending from normed spaces general metric measure spaces might seem straightforward without imposing further conditions, in reality, there is subtle difference. For normed vector spaces, according to (Narici Beckenstein) 2010 Theorem 9.1.2), a surjective isometry between real normed spaces that maps the origin to the origin, such an isometry also maps the mean of distribution to the mean the transformed distribution. Furthermore, the preservation of the mean by isometries between normed spaces provides crucial simplification: it guarantees that centering the distributions is sufficient for achieving a meaningful alignment. Consequently. we do not need separate procedure select the base point 0 beyond this standard centering process. However, for general metric measure spaces, which lack inherent vector space structure, the choice of the base point ( becomes more critical and might require explicit selection.
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Extension of the Distance Function in Normed Spaces. The distance function used in proofs naturally the one induced by the norm of the underlying Banach space. However, we are not restricted to using this specific metric. We can replace the original distance dx (1) = 1| x|x with a powered version:
Vp O. We examine how this change affects the key steps of our previous proofs:
dx (2o, 01) ||000 x1 I1
In the proof establishing Mpx = Mpy (previously referred
Part 1), the step involving the integral
over Y, becomes Jy, < dy (V.1)) (/)d duy (v') = 0. Since dz (v, v/) = lly - v/l 2 0 and c > 0, this still implies dy (y. = O for a.e
Y and jy-a.e In the proofs of the previous Part 2, the derivation relied on showing (x( 0) = (y. 0). If we instead obtain d& (2r, 0) = dy (y, 0) (i.e., ||==1l& |lyl|y) this directly implies (x, 0) (y, 0) since > 0 and norms are non- negative. Similarly, in the proofs of Part 3, the derivation relied on showing dx (I1, zo) = dy (0). Obtaining d zo) dyy (y1, yo) directly implies dx (21, dy y0) for the same reason.
The proof the triangle inequality for the SEINT distance itself operates on the extracted features Φnz.2 and the Lp structure of the SEINT definition (e.g., equation 7. not directly the properties of the function dx. Therefore, this part of the argument is unaffected by whether dxx itself satisfies the triangle inequality.
Consequently, regardless of the chosen value p > 0, even when p > 1 (for which de (zo, 11) is generally not metric due to violating the triangle inequality), the resulting SEINT formulation constructed using d& to define the DcPTD still satisfies the metric properties stated in Theorem
B.4 PROOF OF COROLLARY
Consider the discrete setting with inputs X (æ), TT,) Rexp and Y (YI..... vm) ) Rmxq, with corresponding probability vectors px and py. For simplicity, we can assume m n and uniform weights, assume that the norms of the points within each distribution are distinct, i.c.,
any discrete measure can be resampled fit this structure. We further
[ * |æ,| for i < [m |l and
This allows us to define a unique ordering of the norms: || ||<
|. Under this assumption, the set of optimal couplings for the ID norm distributions in equation 1 becomes a singleton, simplifying the analysis.
|y yr2)
< ||y(n)||
As the norms of X and Y bounded by M, there exists >> s.t. || ll. |y,||+
M. Following
the construction of the reference measures jzz detailed in Appendix [.. we can define a specific, minimal set of reference measures, which is denoted as P(R)) This set can be constructed from a matrix Px (R), where each column defines the support of a reference measure with uniform weights:
0
0
0
0
0
...
0
0
0
Px (K)
1
0
0
0
+
|
1
1 0
0
||
1
1
0
Similarly, we can construct (R). Let P(R) be the stacked matrix (Px Pr). As established in Appendix B.I. the following quantity defines a metric on this discrete space:
ÉSEINT (X,Y, :=
inf =Ell(ex.A))
sup (xEP(R)
Ex
[ 2 (x) - 2x; 2r9 (w)))
-
Now, consider the larger space of reference measures P(R) [0, M defined in the theorem. The corresponding dissimilarity is:
Luz ΣØ= #6s, Layhissa
(x);)))))))) (w))))
ÊSEINT (X,Y H... TEll
inf sup pz (
Ex
[0rk 2 (x) ,xū
Proof of corollary 2 Given the condition that ||>;|| + 1 < M and || + S M for all i, is clear that our deterministically constructed set of measures is subset of the larger space, i.e.,
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P(R) S P(R). This implies that the supremum over the larger must be greater than equal to the supremum the smaller set, leading to the inequality LSEINT LSEINT.
We can now prove the identity of indiscernibles for LSEINT. If LSEINT(X, ,,,,, NV) then from the inequality, must be that ÉSEINT (X, Y, (xx MY) this implies that (X, ll-lix jxx) and |l-1I-l/y, ,YY) are isometric. The reverse direction, along with symmetry and the triangle inequality, follows by the same logic as the proof of Theorem [ Therefore, we have shown that LSEINT well-defined metric comparing discrete distributions in this space.
Since LSEINT is valid metric,
C IMPLEMENTATION DETAILS
In this section, we primarily discuss some implementation details of Algorithm associated acceler- ation techniques, and the implementation of the expected algorithm.
C.I DETAILS OF ALGORITHM|
Whether to Standardize. In the proof of Theorem we discussed that standardizing in normed space avoids the need for explicit base point selection. Therefore, our standardization procedure first involves centering the data, Additionally, if the two distributions reside in the same ambient space, no further adjustments are necessary. Otherwise, we need to perform further scaling adjustments on both distributions, for example, by dividing by their respective standard deviations or a pooled standard deviation.
Selection of jzz. Following the result of Corollary 2 the support points for the reference measure μzz are drawn from the space [0, Mj Specifically, we construct jiz as a discrete uniform measure supported on k points sampled i.i.d. from (n, M). In the special case where measures μa and Jly are discrete and uniform, the optimal coupling between their sorted norm distributions and the sorted support of μzz can be obtained directly by matching sorted indices. In the more general case, the DcPTD can be efficiently computed using the North- West Corner Rule in conjunction with the formulation in equation 9 Regarding the choice of the upper bound M, our theoretical analysis in Corollary Saaieryy@eeee establisshs a requirement on the norms ||>||| for x E supp(ux). This suggests that M should be related the maximum observed norm (c.g., M max [[]])). However, to mitigate the potential impact of outliers that could unduly inflate this value, we typically set M based on the statistics of the observed norms. For instance, robust choice is: M mcan(||<]] + std( [|=]]].
and the
Additional Scaling L. I. Since our reference measure μz is generated randomly, when the input point clouds themselves exhibit significant scale differences, the specific values sampled for jzz can potentially influence the resulting loss magnitude. For example, if a sample drawn for jzz has value larger than all norms ||y|| for y E supp(ux). then the larger this z is, the larger the corresponding Polar transport discrepancy values < (y) (and consequently the DcPTD o (y)) will become. This inflates the overall loss value. However, this scaling effect might not significantly alter the choice of the optimal coupling P. Therefore, handle such potential scaling sensitivities related to extreme values in μzz, we consider applying an additional standardization step to the Polar transport discrepancy vectors X and CY.
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It is worth noting that in the discrete setting, our loss can be expressed as follows:
"
m
SSENT
min PE
max
C 2 C. P.
pz
m
min ΣΣ
Soo Seo Cj + C. GZ CY
C.
-2C; 2Cxx X
S70 S2o C2)
PELl(xx
j=l
max
20, Zo
Cj
Pij + Const
XX Zoo
4Zo SZo
x
T
ΣΣΣΣ Fi/ly'
XY C (Soo )ay Const.
max PEIl(xX
(15)
Simultaneously, we note that the discrete Seenov-Wasseestein (GW) loss can be expressed as:
GW2 = min
(C. ) P.P,Prr
Pe
min
CC** C 20 P.pP.r
PEll(ax.HY)
11
m
n
m1
max ΣΣEX j=1''11=
2C Jj P.pP,y Const.
)Y)
is readily apparent that the computational structures of these two objective functions share sim- ilarities. A primary difference lies in the term involving the coupling matrix P: our loss (in the final derived form) involves a term linear in weighted by elements derived from 2%, while the GW loss involves term quadratic in P. Motivated by this structural comparison and the goal of
normalizing the influence of Pzz we naturally consider scaling the matrix <% = C% S% such that its elements
1, This is conceptually equivalent to appropriately normalizing the individual
vectors 2 and 2 derived from each
C.2 ACCELERATED COMPUTATION FOR ALGORITHM
In the preceding discussion, we mentioned that for specific distance functions, algorithm be accelerated, Let us assume our distance function the squared Euclidean distance, d(x, y) = -|ø-y/l y As established previously, using the squared distance does alter the fundamental metric properties of the resulting loss. Furthermore, for data matrix X E _R= representing points in Euclidean space, the matrix of squared pairwise Euclidean distances C can be expressed as:
C C== = -1 x, 112
(z) x, 2 x 2x, x 2)),;
| 1 1 1n now 2XX
where ||2 1/ denotes the vector whose i-th component is the squared Euclidean norm of the i-th row T;, Based this explicit decomposition, we can derive the accelerated version, presented follows.
As shown in Algorithm 2. by leveraging the explicit decomposition of the squared Euclidean distance matrix, we bypass the computationally intensive step of explicitly forming the full distance matrices. Provided the condition n. >>> and m >> holds, this reduces the time complexity bottleneck from On + m²2) to complexity dominated mainly by sorting of O(n log n + log m). This algorithmic approach can be extended to any distance function whose corresponding distance matrix admits efficient decomposition or allows for fast matrix-vectis products.
and m m
D ADDITIONAL EXPERIMENTS
D.1 EFFECT OF NUMBER OF REFERENCE DISTRIBUTIONS ON SEINT.
We perform sensitivity analysis of SEINT and ISEINT for the number of reference distributions. To this end, use the horse and flamingo datasets from Section D.4 and evaluate how the loss
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Algorithm 2 SEINT Computation (Accelerated)
Require:
Two datasets X R=xP and E Rm probability vectors px E An Py A Am-1 number of reference distributions a, support size for reference distributions k Initialize the data and Y.
2: Generate random reference distributions
.Z, with samples
3: for i + to a do
4: Compute 2z. using X, Px, jiz, via equation
O(n log n log k) (m m + log k)
5: Compute C2. using Y, PY, Pz,
equation
6: 27
| 1 (1 47 + 1,, (Ixx X
(X C
7: 27.
Iyy kun S2, + 1m I SZ,
Y
2Y (%
> Onp mq) D O log m log m)
8: Calculate the ID-W2 loss: L We (10).
#Xx, (0%)
9: end for
Set the final loss LSEINT maxisisa bi
11:
return LSEINT
varies under different distribution numbers. From the figure observe that the loss for both SEINT and ISFINT gradually stabilizes as the number of reference distributions increases. This trend suggests that, given finite reference distributions, it feasible to capture the maximum distinction between two input distributions. This observation not only demonstrates that SEINT is a well-defined metric but also indicates that its performance under finite number of reference distributions closely approximates that obtained with an infinite number of references. In addition, we evaluated SEINT with the constructed reference measure Jzz (see Appendix [B.T]. The results indicate that, when the sample size is sufficiently large, SEINT with uniform reference distribution gradually exceeds the former, thereby providing further support for Corollary 2
ISEINT
SEINT
SEINT deterministic references
Losk
100
200
300
400
500
Number distributions
Figure 5: Effects of number of reference distributions SEINT and ISEINT.
D.2 MOLECULE GENERATION IN 3D
Hyperparameter Configuration. In the molecule generation experiments, we uniformly set the scale of the reference distributions to 3 and the number of reference distributions 50. All other parameters were kept consistent with the settings in Hoogeboom et al. (2022). For the pretraining task on QM9, all methods were trained for 3000 epochs. For the fine-tuning experiments, our models were initialized from baseline checkpoint. Specifically, for the QM9 dataset, started from checkpoint obtained after 2000 epochs of training and continued training for an additional 1000 epochs. For the GEOM- -rogs dataset, models initialized from checkpoint at 13 epochs and further trained for three additional epochs. The learning rate was set to 10 for QM9 and 10-5 for GEOM-Drugs.
Additional results. For the molecule generation task, we conducted additional experiments by adjusting the proportion of SEINT to 0.4 and 0.5. As shown in Table both proportions significantly
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outperform the baseline across validity, atom stability, and molecular stability metrics. However, also observed a critical effect as the proportion SEINT continued to increase. This limitation arises because SEINT utilizes only the first thrce-dimensional coordinate information, thus restricting the amount of information can extract.
Table 5: Comparison of three regularization terms validity, atom stability, molecular stability, and uniqueness, each drawing 10000 samples from the model.
# Metrics
Validity (%)
Atom stable (%)
Mol stable (%)
Uniqueness (%)
EDM
93.4++0.4
98.7 +0.1
87.2 +00.4
98.2++0.2
SEINT 0.4 (ours)
96.6+-0.1
99.2++0.0
91.1+00.4
93.2++0.5
SEINT 0.5 (ours)
96.7++0.3
99.2++0.1
91.8 +-0.4
92.0++022
Furthermore, observed that despite the improvements in validity, atom stability, and molecular stability, the uniqueness metric gradually decreased. We argue that there exists a trade-off between validity and uniqueness, with the metrics exhibiting degree of complementarity. Among them, validity is relatively more critical, as it ensures the chemical plausibility of the generated molecules. Therefore, consider the decrease in uniqueness be acceptable it results from an improvement in validity.
D.3 MASKED AUTOENCODERS FOR POINT CLOUD
The success of Masked Autoencoding (MAE) HHe et al. 2022a Xie et al. 2022 Baevski et al. 2022) in NLP and computer vision has inspired its extension to 3D data, with Point-Min MAE (Pang al. 2022) emerging as the pioneering approach for point cloud reconstruction. Building on this, we augment Point-MAE by integrating SEINT and ISEINT regularizers with the standard Chamfer Distance loss. Our evaluation protocol involves: (1) pretraining on ShapeNet (Chang et al. 2015).
followed by (2) finetuning on ModelNet40 (1k), ModelNet40 (8k) (Wu et al. (015), Scan-Objonly and ScanObjectNN's 's most difficult configuration (PB-T50-B) (Uy al. 2019).
The results shown in Table connfrmm that SEINT and ISEINT regularization consistently enhance classification accuracy over the baseline.
Table 6: Comparisons for various methods their accuracy (%).
Methods
PB-T50-RS ModelNet40(1k) Scan-Objonly ModelNet40(Sk)
Point-MAE
84.67
93.07
87.60
93.52
SEINTa1 (ours)
84.87 85.39
93.52 93.68
88.47
93.68
ISEINTo, 0.1 (ours)
88.64
93.92
The notation SEINT01 indicates that the objective loss function L = 0.1
0.9.rvim-MAE:
Hyperparameter Configuration. In the point cloud reconstruction task, we consistently set scale of reference distributions to 3 and the number of reference distributions 100. All other parameters were kept the same as those used in Pang et al. (2022).
D.4 EMPIRICAL EXPERIMENTS OF CONTINUITY OF METRIC
Continuity one of fundamental properties of metric. Although we have not yet established still empirically examine its behavior under gradually varying distributions. We selected three distinct 3D shapes from the mesh data (Sumner Popović, 2004): elephant, horse, and flamingo. Given shape X, we generate progressively perturbed versions by linearly interpolating with Gaussian noise & ~ N/ (0,.03) o²), with a = 0.2. At each step f, the interpolated shape is defined as:
formal proof that SEINT satisfies continuity,
X, (1 -a)) .X + 00 6, where ar 0.005l.
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Published as a conference paper at ICLR 2026
Source shape: Elepant
Sicp 20
Step - 40
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Source shape Horve
Step 20
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shape:
Step
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Loss Curve for Elepant
Loss Curve for Horse
Loss Curve for Flam
2.0
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Figure 6: Loss across different levels of noise.
Figure 6 illustrates the SEINT and ISEINT loss values computed between the original and noisy point clouds throughout the noise accumulation process.
As shown Figure 6. loss between the source shapes and their corresponding noisy versions increases smoothly the noise level grows, providing preliminary evidence that SEINT responds continuously to distribution changes.
D.5 ADDITIONAL RESULTS FOR SE(P)-INVARIANS
To further validate the novelty and performance advantages of SEINT, added additional base- lines for comparison, including LRGW LRGW(Scetboog (Scetbon et al. 2022), PGW[Kerdoncus et al. (021). SFGW(Piening & Beinert, 2025). AE(Sato et al. 2020). and OChowduyy al. 2022 We set the number of points in each point cloud to 1O00, and the other experimental setup follows the same configuration as the (SE(p)-Invarian)) section in the main text. Table 7 reports the KNN
classification accuracy under different neighborhood sizes, along with the corresponding runtime for each method.
Table 7: Classification accuracy (%) across different numbers of neighbors (k).
Method
Acca-1)
Acca-s)
Acc
Time (ms)
SE(p)-Inv
Complexity
LRGW
2.34
2. 2.17
2,0
1350.36
Onrr)
PGW
45.641336
42.9
40.2
693.47
/
O(n²)
S(F)GW
100.0.0.00
100.0
100.0
485.19
O(n log m)
AE
2.5
2.5
2.3
406.25
On log n)
QGW
5.0
5.0
5.0
242.50
O(m² log m)
SEINT (Ours)
100.0
100.0
100.0....0
4.46
O(n logn)
All reported times are CPU times and denote the per-run runtime of each method. n denotes sample size, and m represents the number quantization blocks used
QGW, and r specifics the rank parameter in IRGW. The number of SEINT references and the rank of LRGW are both set to 50, whereas all other methods
with their default parameters.
From Table 7 SEINT achieves the best overall performance in both accuracy and runtime. Among the additional baselines, only SFGW attains comparable accuracy. It is crucial to clarify that SEINT and SFGW employ fundamentally different representation mechanisms. Specifically, SEINT first utilizes polar length information to compute the optimal transport against reference, yielding the PTDs; it subsequently convolves the PTD with the distance metric to derive the DePTDs as
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the final representation. In contrast, SFGW directly extracts information from the sorted distance matrix. This latter approach inevitably imposes a time complexity of O( logn). Consequently, in practical scenarios, SEINT is significantly faster, demonstrating distinct advantage in computational efficiency.
D.6 ADDITIONAL RESULTS FOR CAPABILITY ON HIGII-DIMENSIONAL DATA
To further evaluate the ability of SEINT to handle more complex high-dimensional data, we extend the previous Gaussian setting a mixture-Gaussian framework. Specifically, define
Vg 0.5.V 0.5. (0055((1 (1-
Lald
# AO(O La),
+ (1.50))) 0)
where Ia denotes the all-ones vector and 0 E [0, 1]. We generate samples over range of values and examine how the estimated distances (p,, ve) vary with 0, which enables systematic comparison of how different methods respond the target distribution gradually departs from the baseline distribution.
We consider two representative high-dimensional regimes, n 200, p 50 and n 50, p 200. Figure 7 presents the corresponding results.
(a) n= 200, 5=
(b) n= d= 200
1.00
1.00
Our Methods SEINT
0.75
0.75
- ISEINT
Distance
Distance
0.50
0.50
Other Methods
GW
0.25
0.25
RISGW
SGW
0.00
0.00
Wasserstein
0.00
0.25
0.50
0.75
1.00
0.00
0.25
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1.00
0
0
Figure 7: Behavior of SEINT under high-dimensional settings: (n 200, p 50) and (n 50, 200).
As shown in Fig. 7 SEINT exhibits a consistent and behl-be trend both settings: the loss increases smoothly as the discrepancy between the distributions grows. This suggests that SEINT capable of accurately capturing distributional differences even in challenging high-dimensional scenarios.
D.7 EFFECT OF NUMBER OF REFERENCE POINTS ON SEINT.
ISET
SENT
1.6
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1.0
0.6
0.2
300
2000
300
400
500
Number of Points in Referune: Distibutions
Figure 8: Effect of Number of Reference Points SEINT.
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To further examine how the number of reference points affects SEINT, fix the sample size n = 2000 and set the number of references to 100, while varying the number of points within each reference distribution. Fig. reports the resulting loss values of SEINT under this setting.
From the Fig. 8 we observe that as the number of reference points increases, the SEINT loss gradually stabilizes. This indicates that SEINT requires only relatively small number of reference points to achieve stable performance, demonstrating its robustness with respect to the choice of the scale the distribution reference.
D.8 MDS VISUALIZATION
To further validate the performance of SEINT, we included visualization experiment (Vayer et al. 2019) based on the distance matrices of the horse running sequence. The experiment consists of two settings: one utilizing original (unrotated) point clouds and another where random rotations are applied.
GW
SEINT
ORIGIAL
SGW
ROTATED
Figure 9: MDS result on horse gallop data.
As shown in the Fig. 9 for the unrotated data, SW, SGW, AE, GW, and SEINT all exhibit regular, closed-loop pattern. However, after applying random rotations, only AE, GW, and SEINT preserve this original pattern. This result empirically demonstrates that SW and SGW lack rotation invariance, whereas SEINT preserves this crucial property.
THE USE OF LARGE LANGUAGE MODELS
We acknowledge the use of Large Language Models in the preparation of this manuscript. Their role was strictly limited to functioning as a tool for language polishing and improving readability. We affirm that all definitions and theorems presented in this paper the original intellectual contributions of the authors. The core code intricately linked to the algorithm we propose, and the experiments were designed specifically investigate the properties of method. We take full responsibility for the integrity of this work, which is free from any form of academic misconduct.
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Reviews:
[{"review_id": "uf9QxEUFRy", "reviewer": "Reviewer_omzW", "summary": "The paper proposes SEINT as a Special Euclidean group-invariant distance between probability measures supported on normed vector spaces. The method builds 1D, rotation/translation-invariant features in two steps: (i) Polar Transport Discrepancy (PTD), which couples the norm distribution of data with a 1D reference measure; (ii) a Distance-convoluted PTD (DcPTD) that computes the convolution of pairwise distances against PTD to encode intrinsic geometry. The SEINT distance is then defined as the optimal transport (OT) distance between DcPTD representations, with a min-max structure that adversarially chooses a “least favorable” reference measure. Theoretically, SEINT is a proved to be a metric on isometry classes, which respects translational and rotational invariances. A discrete variant of SEINT provides metric guarantees under distinct-norm and support-size assumptions. Algorithmically, given sample size $n$, the proposed implementation with a single reference measure achieves $O(n^2)$ computational complexity in general and $O(n \\log n)$ when the ground distance admits additional structure (leveraging structure specific matrix decompsoition technqiues for accelerated computation). Empirically, the paper reports perfect rotation/translation robustness on a benchmark 3D point cloud classification task, cross-space (2D - 3D) comparisons, favorable perfomance scaling with dimension, and improved 3D molecule generation based on SEINT regularization.", "questions": "Please see the Weakness section, particularly the point about the translation invariance of SEINT.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper proposes a rotation invariant metric capable of comparing distributions across different ambient dimensions, with theoretical guarantees that establish the metric properties and rotation invariance of SEINT.", "weaknesses": "1. Although the paper claims both rotatioanal and translational invariances as its target invariance choices, the definition of SEINT as well as the applications of SEINT in experiment rely on explicit centering of the data points in the source and target domains, so that both source and target has their respective origins as the center of their corresponding distributions. The definition of PTD in Equation 2 implicitly assumes this, sincethe norm is centered around 0, and not around any arbitrary constant. Therefore, it seems that the claimed translation invariance of SEINT follows as a consequence if this origin-based centering approach, and not as a derived property that holds in greater generality. The paper should either clarify whether the definition of SEINT still works without origin-based centering, or restrict the claims to only rotational invariance.\n2. The discussion at the beginning of Section 2.3 (Lines 222-241) is dense and not well-written. Given the importance of the role of the reference measure, it is worth improving the presentation of this section of the text and being clear about the existence and selection fo the reference measure.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "nZfkGdYqIv", "reviewer": "Reviewer_mdhA", "summary": "This paper introduces SEINT (SE(p)-Invariant Transport), a novel metric for comparing structured data that maintains invariance under Special Euclidean transformations. The authors develop two unsupervised representation techniques—Polar Transport Discrepancy (PTD) and Distance-convoluted PTD (DcPTD)—that generate 1D SE(p)-invariant features through optimal transport. They rigorously prove SEINT satisfies metric properties (identity, symmetry, triangle inequality) on isometry classes. Experiments demonstrate SEINT achieves 100% accuracy in transformation recognition while maintaining computational efficiency, and improves molecule generation validity by 3-4% over EDM baselines. The approach effectively balances theoretical soundness with practical utility across multiple domains.", "questions": "1. In Corollary 2, you assume \"values of the norm ||x_i|| are distinct\"—how would SEINT behave when this assumption is violated in practice (e.g., symmetric molecules with identical atom distances)? Could you provide a mathematical bound on the error when norms aren't perfectly distinct?\n\n2. The complexity analysis mentions \"n log(n)\" scaling, but how does SEINT perform in high-dimensional settings (p > 100)? Have you tested scalability to non-geometric data where p is large but structure exists?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "**Theoretical Innovation and Rigor**: The paper makes a significant theoretical contribution by establishing SEINT as a proper metric with formal proofs (Section 2.3, Theorem 1). Unlike prior work that \"fail to simultaneously optimize for computational efficiency, rigorous metric properties, and general applicability,\" SEINT achieves this crucial balance. The authors cleverly bridge optimal transport theory with SE(p)-invariance requirements, providing Corollary 2 that establishes metric properties for discrete distributions under realistic assumptions. This theoretical foundation elevates the work beyond heuristic approaches common in the field.\n\n**Practical Impact and Validation**: The paper demonstrates compelling real-world utility across diverse domains. In molecule generation (Table 5), SEINT regularization improves validity from 93.4% to 96.7% while maintaining atom stability. For point cloud classification (Table 6), it consistently enhances accuracy across multiple datasets (e.g., +0.61% on Scan-Objonly). Notably, SEINT achieves perfect (100%) transformation recognition accuracy while being computationally efficient—outperforming alternatives like SW and SGW that \"fall below 60% accuracy.\" This practical validation across multiple challenging tasks underscores the method's significance.", "weaknesses": "**Limited Analysis of the Validity-Uniqueness Trade-off**: While Table 5 shows SEINT improves validity but decreases uniqueness (98.2% → 92.0-93.2%), the authors only briefly mention this trade-off as \"acceptable.\" A more thorough investigation would strengthen the work—specifically, quantifying how much uniqueness degradation is attributable to the SEINT regularization versus other factors, and whether certain molecular properties correlate with this trade-off. Including visual examples of molecules where validity improved at the cost of uniqueness would make this analysis more concrete.\n\n**Insufficient Comparison to Recent SOTA Methods**: The evaluation compares against GW, EGW, RISGW, SW, and SGW, but misses several relevant recent approaches like UniGEM (Feng et al., 2024) mentioned in the introduction. Given the rapid evolution in this field, the authors should benchmark against at least 2-3 additional recent methods (e.g., from ICLR/NeurIPS 2024) to better position SEINT's novelty and performance advantages.\n\n\n\n**Ambiguity in Reference Measure Selection**: While Section 2.3 discusses reference measure selection, the practical implications of different choices aren't fully explored. The paper states \"the support points for µZ are drawn from [0,M]^k\" but doesn't analyze how sensitive results are to k (number of reference points) or distribution type. Figure 5 shows some effects, but a more systematic ablation study would strengthen the method's practical guidance.\n\nWhile the weaknesses I identified (limited analysis of validity-uniqueness trade-off, insufficient comparison to very recent SOTA methods, and reference measure sensitivity) should be addressed fairly easy none fundamentally undermine the contribution. The authors' argument that the validity-uniqueness trade-off is acceptable (since validity is more critical for chemical plausibility) appears reasonable.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "0hqzSXT5ha", "reviewer": "Reviewer_4D2S", "summary": "This paper proposes a new variant of an Optimal Transport (OT) metric that is Special Euclidean group-Invariant (SE(p)). The main idea of the authors is to proceed in two steps. The first one is to align the original raw samples with a certain reference measure, where the cost function is defined as a difference between a given vector and a coordinate of the vector from the reference measure in 1D. The second step is to convolve the obtained embedding with the raw cost matrix and to use the obtained embedding to define a cost function for the resulting 1D OT problem. This allows the metric to be SE(p), yet it is to be computed efficiently in practice. The algorithmic implementation then takes the sup over the reference measures to calculate the final value of the metric. The experiments, both qualitative and quantitative, showcase the desirable properties of the proposed metric.", "questions": "1.\tWhat is the intuition of defining PTD? This is a crucial step in the paper, yet it is not explained very well. Why do we use the cost function defined as the difference between the norm of a vector in X and a real value of the reference measure?\n2.\tWhy are other more efficient GW variants not considered in the evaluations? I can move my score to 6 if a reasonable justification/additional experiments are provided here. \n3.\tI would like to understand better the algorithmic implementation in the sense of my sup inf remark above. Can authors elaborate more on this? \n4.\tI had to go to the appendix (page 22) to see how the reference measure \\mu_Z is defined. They seem to be uniform measures over the hypercube. Is there any motivation for this?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1.\tThe proposed metric seems to be novel\n2.\tProposed metric has metric properties and is cheap to compute\n3.\tExperimental evaluations provide a multi-faceted view on the proposed metric.", "weaknesses": "1.\tThe introduction of PTD and SEINT is very hard to parse in the main paper. In this sense, I found the way SEINT was presented in the Appendix very helpful. Indeed, on page 22, we see that SEINT is a GW variation where the second transposed coupling matrix is replaced with a product of the two convoluted embeddings depending on it. The text makes the introduction of this metric overly complicated, at least for me. Once we see it through this lens, a natural question is why SEINT outperforms GW? GW is theoretically a SE(p) metric too. It would be great to see some intuitions for why it fails in shape matching tasks, in addition to the favorable computational complexity of SEINT.\n2.\tTo the best of my understanding, the metric is defined as an inf sup problem, yet it is solved algorithmically as a sup inf problem. This can be seen from Algorithm 1 (we solve inf inside the loop and then take max over the solutions). Shouldn’t we use minimax (von Neumann or Sion theorem) somewhere to justify doing this? In general, we know that sup inf <= inf sup, so we cannot use the majorization-minimization argument either, unless f is convex/concave in the first and second argument, which is unlikely given how the problem is defined.\n3.\tThe authors omit some very important GW variations in their literature survey. One such omitted reference that comes to my mind is the Quantized GW (Chowdhury et al.), which is extremely computationally efficient and was proposed, among other applications, for shape matching. Another one is the Sampled Gromov-Wasserstein paper (Kerdoncuff et al.). Its complexity is quadratic in N, and it approximates GW.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "BZqk3UeBgY", "reviewer": "Reviewer_SoMY", "summary": "The paper proposes a computationally efficient approach for a representation based metric  for the problem of comparing probability distributions that lie in different Banach spaces invariant to orthogonal transformations. The method involves looking at the optimal transport between one-dimensional measures over norms and each distributions  norm distribution. The contribution to the transport cost for each point in the distribution provides a non-negative value, and this is distance convolved to provide a representation that is isometric consistent and dimension independent. Solving an optimal transport between the two measures using this representation provides a way to align the two distributions.", "questions": "In the algorithm the choice of generating reference distributions as uniform (line 290) is not discussed. Corollary 2 does not mention how the supports $\\{z_j\\}_{j=1}^k$ are chosen. Is uniform the simplest, why not quantiles of of the empirical norms? \n\nLine 52 \"it may sacrifice the metric properties of the distance\"  but in Table 1 it is stated that it isn't a metric. Thus 'may' is not correct. \n\nLine 147 should state $\\lVert X \\rVert_X$ and $\\mu_Z$ \n\nLine 157, the notation with norm and $\\cdot$ breaking across the line is hard to parse. \n\nLine 278, it was not previously assumed that the space is $\\mathbb{R}^{n\\times p}$. While the special Euclidean invariant assumes this, at other times it would seem that the theory is applicable to complex-valued vector space (or possibly other field) which is not necessary in the discrete setting. This should be clarified. \n\nMinor extra space around the superscript for the footnote 1.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "The paper addresses a challenging problem that can be applied to point cloud registration and outperforms Gromov-Wasserstein distance. It is clearly written, original, and results are strong.", "weaknesses": "There are a few minor points to clarify.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "ouMdWemwud", "reviewer": "Reviewer_dTTa", "summary": "The authors propose a method to compare structured objects, e.g., graphs, shapes, etc. This is a hard task that is not solvable with standard optimal transport (OT) tools, but that requires tools such as the Gromov-Wasserstein (GW) distance (Memoli, 2019). Notably, the GW distance and its variants are computationally challenging. While more efficient methods exist, they often have severe shortcomings. As a remedy, the authors propose a method that relies on comparing two structure objects via pairwise distance matrices. In particular, they sample from a target reference 1D distribution to then map each distance matrix to 1D distribution. Then, they can compute the efficient 1d Wasserstein distance between the resulting 1D projected distance distributions. From that point of view, it can be considered as an interesting adaptation of sliced GW. \n\nI think that this is a really interesting idea. While slicing is an important method in the field of OT, the so-called 'sliced GW' (SGW, Vayer et al., 2019)  method has shortcomings. While there are other very recent 'sliced GW' approaches out there that have been missed by the authors (Piening et al., 2025), the authors certainly showed the advantages over the SGW.\n\nHowever, my main criticism is a lack of experimental validation. The experiments are very synthetic. As this is often the case with GW, this is not my main concern, however. More importantly, the authors failed to quantitatively compare their method against suitable alternatives. They mentioned GW (Memoli, 2011), entropic GW (Peyre et al., 2016), and low-rank GW (Scetbon, 2022), but did not really benchmark against them. The horse sequence is a nice visualization, but does not replace benchmarks on shape/graph classification.\n\nTherefore, I feel as if this is a really nice idea, but the paper lacks substance beyond the introduction of the new metric. In particular, I would have expected either more theoretical contributions or a solid benchmarking to vote for acceptance.", "questions": "- Why are some entries in Table 1 red and others black?\n- Why is (E)GW accuracy missing from Table 2? Almost all other reported distances are not isometry-invariant, and the one exception (RISGW) relies on solving a highly non-convex optimization problem.\n- Section D.4 ‘Metric Consistency’: I do not know what the term ‘metric consistency’ is supposed to mean here. I feel as if the reported experiments illustrate the *continuity* of the distance. While it is good to see that the empirical estimator displays continuity, I would be more interested in a proof of the continuity. Did you look into it, or did I miss it in the paper?\n- Sliced Wasserstein (SW) suffers from a curse of dependence. Can you comment on the dimensional dependence?\n- Often the quality of Monte Carlo methods, e.g., SW, scales with $\\frac{1}{\\sqrt(\\text{Projection Number}$ (Nadjahi et al., 2020). This is a simple result of Hölder’s inequality. Could you get similar results for your estimates?\n- Can you incorporate graphs describing geodesic distances similar to experiments in (Memoli, 2011), (Beier et al., 2022), or (Piening et al, 2025)?\n- Can you extend your setting to the setting of 'Fused Gromov--Wasserstein' (Vayer et al, 2020)?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 3, "strengths": "- A good and interesting idea that not everyone would have come up with.\n- A nice supplementary section on the OT background\n- A interesting discussion of possible extensions in the supplement\n- The code in the supplementary is accessible and mostly complete. That is very helpful!\n- A nice introduction and categorization of existing methods.\n- A well-motivated methodological section.\n- Well-done visualizations, especially Fig. 1 & 2\n- Proof section is well-structure. While the main theoretical contribution (Metric property) seems rather simple, the ‘identity of indiscernibles’ is not straightforward.", "weaknesses": "*Title*\n- I have my doubts about the title. Despite background knowledge, the lengthy title was more confusing than helpful for my initial understanding. I would recommend dropping ‘DRIVEN BY POLAR TRANSPORT DISCREPANCY-BASED REPRESENTATION’ since ‘polar transport’ is not even well-known in the OT community, the word ‘transport’ appears in the title twice, and it would make the title more understandable overall. This is only a minor remark and question of style, however.\n\n*Introduction*:\n\nThe introduction is well-written and relies on helpful categorization of existing methods. However, the authors did leave out a lot of modern methods.\n\n- While the broad classification visualized in Figure 1 gives a nice overview and I would mostly agree with it. However, I feel as if the authors missed key references:\n- - ‘Extrinsic Strategy’: While I acknowledge the authors’ efforts in citing key references from ‘90s, more recent research in the same direction has been overlooked, see papers on ‘Wasserstein Procrustes’ methods (Grave et al., 2019; Alvarez-Melis et al, 2019) and extensions beyond Euclidean spaces (Alaya et al, 2022; Beier et al., 2025). While these methods indeed rely on costly iterative, non-convex optimization iterations, another line of research (Beckmann et al., 2025) enables *efficient* comparison of Euclidean data via the ‘Normalized Radon CDT’.\n - - ‘Intrinsic Strategy’: Again, Memoli’s Gromov-Wasserstein (GW) paper from 2011 is a very important paper for comparing geometries. Nevertheless, a lot of effort has gone into efficient adaptations of this approach. While the paper mentions GW upper bounds (Li et al., 2023b; Peyré et al., 2016; Scetbon et al., 2022), GW lower bounds (Sato et al., 2020; Piening et al., 2025) have not been considered in the current submission, but they seem to enable rather efficient comparison of intrinsic geometries, while avoiding the non-convex optimization schemes of SGW/RISGW/GW/EGW/low-rank GW.\n\n- Table 1: While this Table gives a nice overview, I have a few comments (from less important to more important): \n- - 1.) The complexity of EMD solvers depends on the problem. I think that for $n \\neq m$ the complexity is not directly cubic, but in $\\mathcal{O}(n^3 log n)$.  \n- - 2.) The complexity of some metrics (especially sliced ones) is also dimension-dependent in a certain sense, whereas this is not the case for all of them. As an example, this should be the case of RISGW vs. GW. \n- - 3.) A key disadvantage of RISGW, GW, and other methods is that they rely on non-convex solvers. \n- - Conclusion: While the table is helpful for the reader, I feel as if further information should be given about convexity/convergence guarantees and dimensional dependence.\n\n- Please note the ‘Sliced Gromov-Wasserstein’ paper has been authored by ‘Titouan Vayer’ and should probably be cited as ‘Vayer et al.’. To my knowledge, the incorrect citation as ‘Titouan et al’ is a persistent error on Google Scholar.\n\n- I would like to point out that the aforementioned sliced GW variant (Piening et al, 2025) employs similar ideas based on projecting pairwise distance matrices to 1D distributions and computing 1D Wasserstein distances. While there are certainly differences, I think that they should be (briefly) discussed, maybe in the background section.\n\n*Experiments*:\nThe experiments are doing a good job of illustrating key properties. However, there is most certainly a large comparative gap.\n- The authors only compared against ‘standard’ OT metrics\t (which are not supposed to be isometry-invariant) and variations of sliced GW (SGW, Vayer et al, 2019). However: 1.) SGW is not isometry-invariant and 2.) the (RI)SGW algorithms has some theoretical problems (see the correction note on arXiv and (Beinert et al., 2022).\nI feel as if there should be a benchmarking against another ‘extrinsic’ metric, e.g., ‘Wasserstein Procrustes’ (Grave et al., 2019), against GW solvers (Memoli; Peyre; Scetbon; Sato; Kerdoncuff; Chowdhury …), and against another isometry-invariant slicing-based metric (Piening et al., 2025; Beckmann et al., 2025).\n- The presented experiments are highly synthetic. 1.) The horse-gallop experiment is a nice visualization, but not really suitable for benchmarking.  I would also recommend citing existing variations of this experiment, e.g., (Vayer et al., 2019). Moreover, the high-dimensional experiment is rather ‘easy’ as it only relies on normal distributions.\n- The experiment on graph generative models is interesting, but again, it is only a very simple ‘benchmarking’ (SEINT vs L2 regularization)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)