Slicing Wasserstein over Wasserstein via Functional Optimal Transport
Authors: Moritz Piening
Robert Beinert
Venue: ICLR 2026 Poster (2026)
Keywords: Optimal Transport
Sliced Wasserstein
Dataset Distances
Wasserstein
Function Spaces
Infinite-dimensional
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: We propose double-sliced Wasserstein: a scalable and stable optimal transport metric between measures over measures.

Abstract: Wasserstein distances define a metric between probability measures on arbitrary metric spaces, 
including *meta-measures* (measures over measures).
The resulting *Wasserstein over Wasserstein* (WoW) distance is a powerful, but computationally costly tool for comparing datasets or distributions over images and shapes.
Existing sliced WoW accelerations rely on parametric meta-measures or the existence of high-order moments, leading to numerical instability. As an alternative, we propose to leverage the isometry between the 1d Wasserstein space and the quantile functions in the function space $L_2([0,1])$.
For this purpose, we introduce a general sliced Wasserstein framework for arbitrary Banach spaces. 
Due to the 1d Wasserstein isometry, 
this framework defines a sliced distance between 1d meta-measures via infinite-dimensional $L_2$-projections, 
parametrized by Gaussian processes. 
Combining this 1d construction with classical integration over the Euclidean unit sphere yields the *double-sliced Wasserstein* (DSW) metric for general meta-measures. We show that DSW minimization is equivalent to WoW minimization for discretized meta-measures, while avoiding unstable higher-order moments and  computational savings. Numerical experiments on datasets, shapes, and images validate DSW as a scalable substitute for the WoW distance.

Published as a conference paper at ICLR 2026
SLICING WASSERSTEIN OVER WASSERSTEIN VIA FUNCTIONAL OPTIMAL TRANSPORT
Moritz Piening and Robert Beinert Institut für Mathematik, Technische Universität Berlin, Germany (piening, bei @ttrrtetuubbe 1 in. de
ABSTRACT
Wasserstein distances define a metric between probability measures on arbitrary metric spaces, including meta-measures (measures over measures). The resulting Wasserstein over Wasserstein (WoW) distance is powerful, but computation- ally costly tool for comparing datasets or distributions over images and shapes. Existing sliced WoW accelerations rely on parametric meta-measures or the exis- tence of high-order moments, leading to numerical instability. As an alternative, we propose leverage the isometry between the 1d Wasserstein space and the quantile functions in the function space (g( 1]). For this purpose, we intro- duce a general sliced Wasserstein framework for arbitrary Banach spaces. Due to the 1d Wasserstein isometry, this framework defines a sliced distance between ld meta-measures via infinitediimesiional proojejecss,ns, parametrized by Gaus- sian processes. Combining this Id construction with classical integration over the Euclidean unit sphere yields the double-sliced Wasserstein (DSW) metric for general meaa-messsrss. We show that DSW minimization is equivalent to WoW minimization for discretized moments and computational savings. Numerical experiments on datasets, shapes, and images validate DSW as scalable substitute for the WoW distance.
measures, while avoiding unstable higher-order
INTRODUCTION
Optimal transport (OT) enables geometrically meaningful Wasserstein distances between probability measures on arbitrary Polish spaces X' (Villani 2003), while remaining computationally tractable. This has had profound impact on machine learning. where Wasserstein distances are used to train neural networks (Tong et al., 2024) and to compare data distributions (Yang et al.] 2019). A key fea- ture of OT is its applicability to non-Euclidean spaces, even allowing the definition of Wasserstein distances on Wasserstein spaces (2) (Bonet et al.) 2025b). This is particularly useful for com- paring distributions over non-Euclidean objects. For example, Euclidean distances ground metric between two images often yield poor results (Stanezuk et al. (202I). whereas Wasserstein distances robust small image perturbations (Beckmann al. 2025). Similarly, comparing point clouds (Nguyen et al.) 20215) is natural with but not even dell-defined with Fuclidean distances. While most OT applications focus either on comparing pairwise objects or distributions over Euclidean spaces, recent work leverages Wasserstein distances on Wasserstein spaces for non-Euclidean do- mains, such as image (Dukler al. 2019 or point cloud (Haviy et al. 2025) distributions.
The underlying concept of multilevel OT has been introduced under various names, including hier- archical OT (Schmitzer & Schnörr 2013 Lee al. 2019), mixture Wasserstein (Chen et al. 2018
2019 Delon & Desolneux (2020). and Wasserstein over Wasserstein (WoW) (Bonet et al. (025b). It
has applications beyond images and shapes, including domain adaptation (Lee et al. 2019 EEl Hamri
et al. (022), single-cell analysis (Lin et al. 2023), point cloud registration (Steuernagel et al. 2023),
Bayesian inference (Nguyen & Mueller] 2024), generative modelling (Atanackovic et al. 2025 Ha-
viv et al. 2025), document analysis (Yurochkin et al. 2019),
approximations
(Mémoli 2011 Piening & Beinert 2025a), and reinforcement learning (Ziesche & Rozo 2023). Ex-
tending this framework with another Polish space y for dataset labels yields the OT dataset distance
(OTDD), defined on P2()) x (2)))) (Alvarez-Medl & Fusi 2020). However, all these approaches incur high computational cost due repeated pairwise Wasserstein evaluations.

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Published as a conference paper at ICLR 2026
Due to the complexity of OT, sliced Wasserstein distances (Bonneel et al. 2013 Nguyen 2025
provide efficient OT based alternatives to standard Wasserstein distances. Initially developed for probability measures Euclidean spaces, they have since been extended to the sphere (Bonet al.
2023a Quellmalz et al., 2023), manifolds (Bonet et al., 2025a), functions (Garrett et al. 2024), hyperbolic spaces (Bonet et al. 2023b), the rotation group (Quellmalz et al. (024). and matrices (Bonet et al. (023c). Repeated sliced Wasserstein distances can be linearized by mapping mea- sures onto function spaces Naderializadeh et al.) 2021 Nguyen 2025). For WoW-type distances, sliced accelerations have been proposed for Gaussian mixtures (Nguyen & Mueller 2024, Piening & via sliced Wasser- stein Busemann Gaussian (SWBDG/SWBI) IDG distances (Bonet et al. 2025c) and the sliced OTDD (s-OTDD) Nguyen et al., 2025). The SWBDG and SWBIDG, which appeared after the original submission of our work, are based on Busemann functions whose level sets act as natural gener- alization of affine hyperplanes in the space of meaa--masurre. The numerical implementation relies on Gaussian approximations and closed-form of the Busemann function. As alternative, the s-OTDD employs a hierarchical slicing approach based on the method of moments. However, is only well-defined for finite moments, and practical implementations are limited to the first few moments because of numerical instability - originally, the first five.
Beinent 2025b) and more generally for measures over measures
In this paper, we aim circumvent the issues of the s-OTDD. Therefore, we build on theoretical ideas originally proposed in (Han) 2023) to develop a computable sliced Wasserstein metric on gen- eral Banach spaces. Employing the isometry between 1d probability measures in the Wasserstein space and quantile functions embedded in the space of gguurr-inteerabbe functions L2 (00, 1J), we uti- lize this define a sliced Wasserstein metric on the space of 1d meta-measures via L-zprrjections. Due to the lack of uniform distribution on the unit ball of dimenio-dimensional function spaces, we parametrize our projection directions as Gaussian processes (Kanagawa al., 2018). To extend this idea to nulti-dimensional meta-measures, we combine this approach with a classical slicing ap- proach, mapping these meta-measures to meta-measures. Lastly, we prove that the minimization of our sliced distance results in WoW minimization. This leads to the following contributions:
We generalize the sliced Wasserstein distance to arbitrary Banach spaces. Moreover, we show how two distinct parameterizations of the random projections may result in equivalent metrics. As special case, this allows for a sliced distance between 1d meta-measures.
Beyond 1d metaa-messres, we extend our approach to the multivariate case by introducing the double-sliced Wasserstein (DSW/)) metric between meta- mata-mmassees. Illustrating the use- fulness of our DSW metric as a WoW replacement, we prove a form of topological metric equivalence between the two for discretized meta-measures.
Lastly, we present various numerical experiments showcasing the advantages of our ap- proach, allowing for meaningful distribution comparisons for datasets, shapes, and images.
2 WASSERSTEI DISTANCES
The so-called Wasserstein distance or Kantorovich-Rubinsterr metric is an optimal transport-based similarity gauge between probabilities a common Polish space. To this end, let X be a Pol- ish space, let P(X') be the space of Borel probability measures on X' with respect to the Borel o-algebra induced by the underlying metric, and let Pp(X) = {u P(X) | 3x1 fx d21 du(xz)) < 5o p E [1, be the subset of measures with finite pth moment. For a E P(X)) and a second Polish space Y, the push-forward by a measurable map T: X + y is defined by 1 == T E P(Y). The set of transport plans between P(X), E P(Y) is given by
X
T(u,v) (r P(X x)) 7 -,
where denotes the canonical projection onto the ith component. For complete, separable metric space (X' d), the (2-JWasserstein distance
inf (Jeye & (2) dy
W(,,x;
(EL'
Note that the acronym DSW is also used for the distributional sliced Wasserstein distance Nguuen et al. 2021a which modifies the slicing distribution of the classical sliced Wasserstein distance.
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defines metric P2(X). More precisely, (P2 W) is again complete separable metric space, allowing the construction of the so-called Wasserstein over Wasserstein (WoW) distance ((.:;)) == :P(x)))))))) which is studied in (Bonet et al. 2025b).
In difference to other similarity gauges like the Wasserstein distances leverage the underlying geometry, allowing for meaningful comparisons be- tween empirical measures. Although the Wasserstein distance relies on linear program, the ac- tual calculation is computationally costly. For two empirical measures supported at points in R equipped with the Euclidean metric, the exact computation has complexity O( logn). The approximate computation based on entropic regularization still has complexity O(n" log n), see (Peyré & Cuturi] 2019). Notably, this computational burden becomes even more involved for non- Euclidean metric spaces, where the computation of the underlying distance itself challenging. For instance, the computation of the WoW distance relies on the pointwise evaluations of Wasserstein distances. If the empirical meta-measures in (P2(((())) are supported on N empirical measures on (RRd). each with support points, then the approximate calculation of the required distance matrix already has complexity ON222² logn).
divergence or the total variation,
From computational point of view, the Wasserstein distance on (R. 11) is notable excep- tion since this may be evaluated analytically. To this end, for μ E P(R). 3. its quantile function
Qa: (0, 1) + R is given by (u(s) = inf fi R #((-30, 2 }) The Wasserstein distance now becomes
W(n vs RR) = (f - (x()) -u()))
- ds) n. E P2(R),
meaning that the quantile mapping
q: P2
T.2 (,2 1]), p Q₂
is an isometric embedding into the space of square- integrable functions, see (Villani 2003). For empirical measures, the quantile functions are piecewise constant and can be efficiently computed by sorting the support points.
SLICED WASSERSTEIN DISTANCES
At their core, all sliced Wasserstein distances exploit casy-to-compute, 1d optimal transports to define efficient alternatives to the standard Wasserstein distance. Originally, the sliced Wasserstein distance has been studied for measures in (RR)) and is based the slicing operator
no R2d
(0.z). 2).
gd-1 (a Rd
|=||= = 1},
with respect to the Euclidean inner product and norm. The sliced (2-) Wasserstttt distance reads as
SW v) = (L WT u. Ta v; R) ds dsddd (0)
where integrate with respect the uniform probability on gd-1 Similar the Wasserstein distance, SW metricizes the weak convergence (Bonnotte 2013 Nadjahi et al. 2020). The spherical integral is usually approximated by Monte Carlo schemes (Bonneel et al. 2015 Nguyen al. 2024
Hertrich et al. 2025) or Gaussian approximation (Nadjahi et al. 2021).
3.1 SLICING INFINITE DIMENSIONAL BANACH SPACES
As preliminary step towards an implementable sliced WoW distance, we consider the slicing on an infinite dimensional, separable Banach space U with norm ||-I|- We denote the continuous dual by
U* and equip it with the dual norm, i.e., ||ا|| == Relying on the dual pairing, we generalize the slicing operator
E [|]] & 1) for all v E U
2
n U - R. u + (v. u)= v(u),
V U.
The crucial point in defining a sliced distance on Pu = {u P(U) I follu]] du(()) < oo). is that there exists uniform probability measure on the infinite dimensional sphere. As remedy,
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choose an arbitrary& & P2(U*) = is P(U*) So Ju.llol/2s(1 d((u) < oo) and define the g-based sliced Wasserstein distance as
1
SW Us = (/.. (T u4 Top R) de(u))
11 11 MUU)..
This approach extends the slicing on Hilbert spaces Haa 2023). However, unlike Han 2023 we do not construct specific measures on the sphere. This allows the use of masy-lo-sample slicing measures. If the support of & covers all directions in U*, the E-based SW distance becomes a metric. Here, the crucial point is the definiteness. For this, we show the Lipschitz continuity of the integrant in (4) and exploit the uniqueness of the characteristic function, see Appendix A
Theorem 3.1. For Pu()"), the E-based SW distance defines well-defined pseudo-metric S A Sssaan E {0, {0}} for all E U {0), then 4 desines a metric on P2(U).
If the slicing measure & has full support, then the assumption in Theorem 3.1 is fulfilled, and the É-based SW distance is a metric. Two measures fı, P2(U** are equivalent if they are mutually absolutely continuous. If their Radon-Noi Nikodým derivatives d51/ df2 and d dı1 are bounded, then the resulting SW distances are metrically equivalent. The proof is given in Appendix [A] Proposition 3.2. Let &1, 2 E P22(U*) be equivalent. If d dε2 and dC2/ dí1 are bounded, then find cı, 02 0 such that
C1 SW v; Gi) S SW( us Ea) S o2 SW EI) VU, U P2(U).
In the mimes-nimensional Euclidean setting, special cases of the E-based SW distance correspond, for instance, to so-called energy measures on the sphere (Nguyen & Ho 2023) and the standard Gaussian (Nadjahi et al. 2021). Relying on the latter, we obtain strong equivalence to original SW
g is equivalent the standard Gaussian. The short proof, which relies on Proposition 3.2 and the fact that the classical SW distance (3) and the reference- based SW distance (4) with the standard
Gaussian reference coincide on P2 P2( is given in Appendix A
Proposition 3.3. Let E E (2(RR be equivalent to rj ~ Ia). If dg/ dy and dn/ de are bounded, then we find c1, 02 > 0 such that
C1 SW(,, < SW(H, = SW v) S C2 ZW() Vp, E P2
Assuming that samples of & are available, the E-based SW distance on every separable Banach space can again be computed using the Monte Carlo scheme. E has a finite fourth moment, i.e. & E (U"),, the Monte Carlo scheme converges. The details are given in Proposition A.6
3.2 SLICING THE ID WASSERSTEIN SPACE
Exploiting the generalized SW distance in (4. we introduce a first slicing of the Wasserstein space (R2(R), W), which later builds the foundation of our sliced WoW distance on P2(P2(RR)). Recall that the quantile mapping ( is an isometric embedding and thus measurable. Therefore, we can push every meta-measure la E (2))2(()) to 9ut4 E P2(L2 1J)). In this manner, the WoW distance between μl, V (P))) becomes W(μ, v; IR) = W(ggu, 4:2; P2(L2((O 1))). Fixing & E (2 1))), we introduce the sliced quantile WoW (SQW) distance:
SQW (m, E) SW (q:u,, q:9; E), μ. E Pa (2 (R)).
(5)
If & is positive, i.c., E has full support, then the assumptions of Theorem 3.1 are satisfied. Corollary 3.3.1. Let & E (2 1])) be positive, then SQW is metric on (2))).
For the later implementation, we require an easy-to-sample slicing measure. To this end, we propose ( (0. 1]) such that n, is Gaussian for all E ((0, 1)).
to use Gaussians, i.e., measures & E (T2
see (Bogachev) 1998). On L2( 1)), there exists a one-to-one correspondence between Gaussian measures and Gaussian processes (Rajput & Cambanis 1972 Thm. 2). In our numerics, we restrict
ourselves to the Gaussian process G that is related to the covariance kernel
ko: : [0, 1] x j0, 1] I, (t, + exp(-It s|² 2/? This means that we consider the function-values random variable G with (G(t))...,,tt
(6)
~ ~N(0, (k(t,, t))),...........
Vt,...., tn E [0,
Since the kernel is smooth, the sample paths (realizations) of G are smooth too, i.e., G E C [0, 1) almost surely, see (Da Costa et al. 2026 Cor. 1). Since k, is universal (Steinwart) 2001.. the corresponding Gaussian measure has full support (Van Der Vaart et al. 2008
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4 DOUBLE-SLICING THE WASSERSTEIN SPACE
The slicing schemes in $ 3.1 and $ 3.2 cannot directly be generalized to the multidimensional Wasserstein space (P2(12), W) due to the lack of the Banach space structure and since do not have an adequate generalization of the quantile mapping. Instead of slicing the Wasserstein space directly, in first step, we therefore propose to slice the underlying domain using
Tp: (2 P2 (RR;
μ + T0
S
with To from Notice that Ta is continuous with respect to the Wasserstein distances and thus measurable. This allows us to define the sliced WoW distance via
SW(, = (LSm. (TO To. R) ds -0) $
where we again integrate with respect to the uniform measure. Based on finite-dimensional slicing, SW essentially reduces meta-measures μ and to series of 1d meta-measures T and ;
As the computation of the 1d WoW distance remains challenging, we resort to a hierarchical slicing approach, inspired by (Nguyen et al. 2025 Piening & Beinert 2025b). Using quantile slicing (5)
with slicing measure Ç E P22 I)), we introduce the double-sliced WoW distance
k; = ( STo Te, 1ss
1, E P2(22
By construction, the double-sliced WoW distance defines at least a pseudo-metric. If we restrict ourselves empirical To be more precise, we denote the subset of empirical measures by Pa and the Dirac measure by 5.. An empirical meta-measure μ (e((P((RR) has the form
DSW even becomes metric that is weakly equivalent to W.
N
with
F4,b E R and Sa
Th
for arbitrary N and n4. For N and n, = ì fixed, we denote E PNN (PO with E ( Theorem 4.1. For positive & E (20 1J)), DSW defines a metric on Pe (Po(K))) Moreover, for pas P. (Po (X)) with compact X C Rd and positive Gaussian &, it holds
DSW (pbr pc; 8) + = SW pc 5) + + W pc; R2) +0 as n + 00.
The detailed proofs are given in Appendix B There, a transport plan is constructed to rely on the metric properties of WoW. The equivalence then proven via a discretization the Gaussian pro- cess and the compactness of P.N. (P.t (x)). Numerically, DSW can be implemented combining several integration techniques, see Appendix C.T C.2 C.2 Here, we consider the Gaussian & related meta-meassres fl, V P. (IR as in 7 To approximate the
to the kernel in O and empirical outer integral over Sd-1 and the inner integral over & simultancously, we employ the Monte Carlo method. For sample path g of the corresponding Gaussian process and direction 0 Sd-1, we evaluate the integrand as follows: The first slicing gives Te.;/4 (1/N) Σ 1
and the quantile mapping F = (1/N) Σ 2 &ytoo.oo.. where the piecewise constant quantile func- tions can be determined by sorting the support points. For the slicing on L2((0, 1]), we employ a
quadrature approximate the inner product. To this end, for knots ti.....ta tR [0, 1] and weights W1, .... WR. we estimate
R
g To
N
(a(xo.pu:).o)
with (u:)),
Σ
wr ((To..n))tt)) g(t,).
1=1
Note that the samples of the process G satisfy (G(tz), .t.,, G(tR)) N k(tr, tp. 1) and can be easily generated. Finally, the double-sliced WoW distance is computed by
DSW (M, v) = - 8
Σ
2 2 (T To, 9 1R)
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The remaining Id Wasserstein distances can again be efficiently computed.
At its core, DSW requires the computation of the domain projection Te and a quantile projection q. As alternative approach, s-OTDD considers variation of Ty with general 1d feature projections, including convolutions, and a (finite) moment projection instead of quantiles (Nguyen et al. 2025). Another concurrent work constructs direct slicing of (P2 (Rd)) via so-called Busemann projec- tions onto geodesic rays and relies on Gaussian approximations (Bonet et al.] 2025c). Lastly, notice that the collection of these push-forward q;TT, can be interpreted as meta-measure version of the so-called -asserrsenn embedding (Kolouri et al. (2066) Naderializadeh et al. 2022] Nguyen 2025) that maps a classical measure in (2 to the quantile functions of its slices.
NUMERICAL EXPERIMENTS
In this section, we aim to showcase the numerical properties and benefits of our sliced distances. We start with the 1 case. Drawing a connection between meta-measures and the so-called Gromov- Wasserstein (GW) distance, we consider shape classification experiment from (Piening & Beinert 2025a). Next, we compare our multidimensional sliced distance to the s-OTDD (Nguyen et al,, 2025j. Finally, we present applications from the evaluation of point cloud distributions and percep- tual image analysis. For all these experiments, we employ trapezoidal integration weights W,,. We refer Appendix C for further experiments and detail?"
5.1 SHAPE CLASSIFICATION VIA LOCAL DISTANCE DISTRIBUTIONS
First, we repeat shape classification experiment from (Piening & Beinert 2025a based on parametrizing shapes as so-called metric measure (mm-)sspacs. mm-space is a tuple (X, dx, μi consisting of compact metric space (X', dx)) and measure μa E (2(X). Modelling data (finite) metric spaces allows for invariance to isometric transformations such as rotations, often desirable for shapes. Particularly, distances and 3d shapes as triangular meshes with pairwise surface distances (Beier et al. 2022).
may parametrize 2d shapes point clouds with pairwise Euclidean
While Gromov- iassersteen (GW) distances define metric between mm-spaces (Mémoli) 2011 computation is costly and relies on inexact non-convex minimization. As remedy, alternatives em- ploy pseudo-metries via local distance distributions. Namely, they map a finite, weiihhee mm-space X (Z)...., IN) with E (X) to P. (IE)
via the (non-injective) mapping
5dx(zur ddx
N
Now, we can represent two mm sum-spaces as μI, BY E (P)( and compare them using the Id WoW distance (Mémoli 2011 "Third Lower Bound' (TLB')), another SW-based distance (Piening
& Beinert 2025a| "Sliced Third Lower Bound' (STLB)) or so-called energy distance (Sato et al. 2020 'Anchor Energy" (AE)). Alternatively, we may use our SQW distance.
Repeating experiments from (Piening & Beinert 2025a), we precompute pairwise distance matri- with respect our SQW distance, TLB, STLB, AF, and the GW distance. Then, we estimate the accuracy of a k-nearest neighbor (KNN) classification by assigning each test point to the ma- jority class among its three nearest neighbors. We average the classification accuracy over 1000 random 25%/75% training/test splits on four datasets. Based on preprocessing from (Piening & Beinert 2025a), we employ the '2D Shapes' dataset (Beier et al. (2022). N 50), the 'Animals' dataset (Sumner& & Popović| (2004). N = 50) and the "FAUST" dataset (Bogo et al., (0144) with 500 (TFAUST-500' and 1000 (FFAST--100") vertices per shape. Additionally, we report results on synthetic version of MNIST (LeCun et al. 1998) denoted 'MNIST-2000" (N = 100, 5 classes, 2000 points), whose details are given in Appendix C.4.I. We set the kernel parameter or to 0.01, R 10 and S 100. We use the same integration grid and projection number for STLB. We display the results in Table D where we additionally report the mean runtime of single distance computation. We observe comparable performance across all distances and runtime advantage of SQW and STLB for the large-scale FAUST- 1000 and MNIST-2000 dataset, in particular against the GW distance. See Appendix C.4.2 for further studies.
Code: htt.a//ttp//...aaaaaccc/////aaaa

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Distance
2D shopes
Animals
FAUST-500
FAUST-1000
MNIST-2000
Acc. (%)
Time (ms)
Acc. (%)
Time (ms)
Acc. (8)
Time (mx)
Acc. (%)
Time (ms)
Acc. (%)
Time (ms)
SQW (Ours)
99.5:112
0,940.3
99.1+11.
0.940.11
38.67-577
9.8+5,77
42.74:5.9
13.8:1551
84.8-44.7
48.6+2288
TLB
100.0::0.3
0.5400
100.0=0000
0.440.1
36,74:566
27.2+158
40.2
60.149.6
88.74:4.5
298.7=25.8
STLB
99.5.11.2
0.8.110
99.3.11.8
0,8.1..1
37.61.566
9.815.77
39.4115.6
14.0:114.9
84.11.5.0
45.1
AE
99.7::099
0.4:0.0
9..811.88
0.410.1
37.711566
8.122.1
41.815.3
25.2
88.114455
16.
GW
99.7::0.6
1.345.7
100.0:000
2.51111
29:24:44
266.4494.7
33.0::.3
1048.24:357.3
-timoout
Table 1: Shape classification with KNN: Mean accuracy (Acc., 1) and runtime (Time).
S
R
Acc (%)
01400
1100 0011
10 10
0.14
84,8
Speernon p 3.50
Specrman on 0.99
Sprarran a 2,20
10a 10
0.12
84.7
01000
10a
10
0.12
84.7
10a
10
0.1²
84.8
0:007
10a
0.1
85.0
10a
0.1
0.000
USR 1d WOW
U2C UDR via WoW
TLB via
Figure I: Left: Impact of projection number S. grid size R. and kernel parameter a on MNIST-2000 SQW classification. Right: Scatter plots and correlation between TLB (Id WoW, horizontal axis) and SQW (vertical axis) for different parameters on MNIST-2000.
Based MNIST-2000, we additionally perform a parameter study to analyze the impact of the pro- jection number S, grid size R, and kernel parameter a. We again classify our MNIST-2000 dataset via SQW, but with varying parameters. The accuracies are displayed in the table in Figure [] where we observe only slight variations of the resulting accuracies. Additionally, we analyze the correla- tion between TLB and SQW in Figure The SQW distance is meant to emulate the WoW distance on (P))) Since TLB is exactly the WoW distance between μr and jay, we would therefore hope for linear relation between SQW and TLB. To analyze this relation, we produce a scatter plot displaying all SQW-TLB pairs on MNIST-2000 for three different parameter configurations. In addition, we compute the Pearson and Spearman correlation coefficients. Overall, we see a clear linear relation with correlations coefficients of 0.99 more. Changing the kernel parameter does not seem to impact this relation, but increasing the number of projections reduces the variance and raises the correlation from 0.99 1.0.
5.2 OPTIMAL TRANSPORT DATASET DISTANCE
Next, we consider comparison with the OTDD (Alvarez-Melis & Fusi 2020) and the s-OTDD (Nguyen et al. 2025). These metrics have been developed to quantify the similarities between la- belled datasets in model-agnostic manner. Such similarity metrics are especially important for applications in transfer learning. In this area, it has been shown empirically that the OTDD and s- OTDD display a strong correlation with the performance gap in transfer learning and classification accuracy in data augmentation, see (Alvarez-Melis & Fusi 2020). To assess the suitability of our DSW distance as a drop-in replacement for the computationally costly OTDD, we repeat an experi- ment from (Nguyen el al. 2025). We randomly split MNIST (LeCun et al., 1998), FashionMNIST (Xiao al. 2017). 7), and CIFARTO (Krizhevsky 2009) create subdataset pairs, each ranging in size from 500 to 1000, and compute the OTDD, the s- OTDD, and our DSW between subdataset pairs.
We plot the results of our DSW distance and the s-OTDD against OTDD for 100 dataset splits in Figure 2 where we include the Pearson and the Spearman correlation coefficients between both sliced distances and the OTDD. As our DSW distance is originally defined on P2((P((RR)) and OTDD and s-OTDD on P( x ))R we compute OTDD and s-ODDD with the label metric on y' set to zero for comparability, effectively representing each dataset as an empirial measure in (2 )) and computing the OTDD via WoW. To be precise, each class-conditional distribution is modeled as b4 E (R and the distribution over class clusscooniitioal distributions becomes our meta-measure μ (P.P.(RR)) Similar to (Nguyen et al. 2025). ) we estimate both DSW (R 10, 10, 000 projections. Based on this
= 0.1) and the S-OTDD (with Radon features) with setting, we employ the original default implementation for the s-OTDD and the 'exact" OTDD.
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833
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Figure 2: Scatter plots and correlations (1) between the s-OTDD and the OTDD (top) and our DSW
('Ours") and the OTDD (bottom) for MNIST (2a), FashionMNIST (2b). and CIFAR- 10 22
Treating OTDD as our ground truth, we clearly see a stronger correlation between DSW and OTDD for all datasets. Additional experiments can be found in Appendix C.5
5.3 COMPARING DISTRIBUTIONS OF POINT CLOUDS
For our next experiment, we assess the potential of DSW for evaluating point cloud generative mod- els, which aim to generate 3D shapes such as chairs planes. Evaluating such models is challenging since common quality metrics are insensitive to mode collapse (e.g.. 'coverage")
tolerate low- quality samples (c.g.. 'minimum matching distances') )) see Yang al. (2019) for details. A common remedy is the OT nearest-neighhor accuracy ('OT-NNA') test, which uses I-nearest-neighbor classi- fication based on pairwise Wasserstein distances between real and generated point clouds. However, for N real and M generated shapes, this requires (N + M 1)2/2 OT computations, without defining a true metric.
As a natural alternative, one might instead represent batches of real and generated point clouds N4 E Pa (TR)) ) as empirical meta-measures in P.PP (TR))) and compute the WoW, respectively, the DSW distance between real and generated batch. To assess the suitability of our resulting quality metric, we consider shapes from ModelNet-10 (Wu et al. 2015)) and construct meta-measures
M
N Σ Jn (P(P(RR)) V
on, E Po(P.(R))),
M
where the support points are again Euclidean empirical measures
m o3 Po(RR)) e e~ ~~,,00, Is).
J 8Ry, (.(R), Ug
m
Given a certain shape class, e.g.. 'chair', we initialize μ as our fixed reference meta- measure with (downsampled) shapes from the ModelNet-10 training set and v as our varying target meta-measure with shapes from the ModelNet- 10 test set. To compare OT-NNA, WoW, and our DSW metric, we independently vary the number of target shapes M, the level of Gaussian noise Noisee and the point cloud discretization m while fixing the remaining parameters according to the reference μ (Default parameters: N M, Noise 0, m n). The results of our experiment where we display the average result of 5 runs with varying M, Noisse and n, and the reference (' ('ground truth') parameter of μ is marked with dotted red line. Lower better for all metrics.
visualized in Figure
Looking at the number of target shapes M in Subfigure 3a (Class: 'bed', N = 10, NNoise = 0, n 50), see that all metrics successfully capture mode collapse, i.e., M = 1, and decrease for
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larger number of target shapes M. Unlike the plateauing WoW and DSW metrics, OT-NNA displays an undesired behavior by increasing for M 2 N 10, however. As for the random Gaussian perturbations in Subfigure 3b (Class: 'sofa', N M 10. n
50), all three metrics increase with increasing noise. Whereas OT-NNA is more sensitive to small noise levels, the WoW sensitive high noise levels. Considering the point cloud resolution m in Subfigure 3c (Class: 'monitor', N = M = 10, = 500), OT-NNA seems inconsistent regarding the resolution. In contrast, the WoW and DSW metrics are higher for m 100 and plateau after seemingly sufficient resolution has been reached. Overall, we see that WoW and DSW display similar behavior the OT-NNA and offer the advantage of being unbounded metries. Additionally, DSW (S 10, 000, R = 50, and m = n 500, where WoW takes about 4.5 seconds and OT-NNA about 8.5 seconds (on our CPU). This makes it especially suitable for high-resolution point clouds and large point cloud batches. Further studies are contained in Appendix C.6
and DSW metrics
0.1) offers computational advantages as it takes around 0.25
seconds for M = N
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Gaussian Noise in Targe
# Points in Target
(a) Number Target Shapes
(b) Noise Target Points
(c) Number of Target Points
Figure 3: OT-NNA, WoW, and our DSW ('Ours' >eetween target and reference point cloud batches for varying numbers shapes M in target batch (3ab, for Gaussian noise Noise for target points (3b) and varying target point cloud resolution m Bch. Fixed reference values are marked in red.
5.4 COMPARING IMAGE DISTRIBUTIONS VIA THEIR PATCH DISTRIBUTIONS
Given the importance of OT for imaging, we conclude with an imaging experiment. Interestingly, OT is utilized on two levels in this area. On the one hand, it is used to compare two butches of images using pairwise Fuclidean distances (Genevay et al.) 2018). On the other hand, using OT as distance between two individual images remains of relevance due the disadvantages of Euclidean distances. Those methods represent images as 2D histograms (Beier et al. 2023 Geuter et al. 2025)
patch distributions (Hertrich et al. 2022 Elnekave & Weiss 2022 Flotho et al. 2025). Thus,
a natural combination is the comparison of image batches using WoW (Dukler et al. 2019).
Since patch-based OT distances serve a perceptual metric between images (He et al. 2024). in- corporate patch-based image representations into the WoW framework. This is based on parametriz- ing images via their distribution of localized features. More concretely, map each (grayscale)

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image to the empirical measure over all contained (overlapping) square-shapel image regions of size p p. see (Piening et al. 2024) for an in-depth description. Thus, we may represent each image as an empirical measure E Pe ((())) supported on vectorized patches and a batch of images as P. (P. In this experiment, establish quantitative com-
an empirical meta-measure parison of image batches using our hierarchical OT framework as an alternative to comparing two image batches using standard OT neural-network-bassee perceptual metric, such the 'Kernel Inception Distance" (Sutherland et al.) 2018
'KID").
To validate this approach, consider synthetic 64 x texture images based random Perlin noise (Perlin 1985). This texture synthesis model is controlled by several parameters, among them the Similar our previous experiment, initialize
lacunarity and the persistence, see Appendix C.7 reference meta-measure μ over images represented as patch distributions according to some ref- erence parameters and compare it to a target metaame v with varying lacunarity, respectively, persistence. For batch size 32 and patch size p = 8, both meta-measures are initialized according random images, where each image is represented by (64 - x (64 7) = 3249 uniformly weighted patches of dimension 8 X 8 64. In Figure 4 compare the average behavior over five runs of the standard Wasserstein distance with Euclidean cost between our reference and target images and our patch-based DSW distance (a 0.1, S 10, 000 projections, R 10), where the 'ground truth' lacunarity and persistence reference parameters are again marked red.
We observe that both distances minimized for the 'true' reference parameters. Still, patch- based approach is more sensitive to parameter variation and better at discriminating between dif- ferent batches, also supplementary comparisons. Note that patch-based WoW calculation takes about 40 seconds, whereas our DSW distance merely requires about one second.
Wasserstein
1000
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Moans z Sid
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Lacunarity in Target
Persistence in Target
Persistence in Target
(a) Varying Lacunarity in Perlin Noise
(b) Varying Persistence in Perlin Noise
Figure 4: Comparing synthetic texture image batches via Euclidean Wasserstein and our sliced patch-based distance based on varying 'lacunarity" (4a) and 'persistence" (4b). Both distances are minimized for 'true' parameters (red), but our DSW ("Ours") distance leads to clearer discrimination.
CONCLUSION
We introduce a general sliced OT framework for measures on arbitrary Banach spaces. Lever- aging the isometry between 1d Wasserstein and (22 1)), Gaussian-process: praiiannppoeess-ppaarrriiie L-z- projections, and classical spherical slicing, we define the DSW distance between meta- meta-measures, well-posed, computationally efficient substitute for WoW. We prove that DSW minimization corre- sponds to WoW minimization for discretized meta-measures and demonstrate practical effectiveness on datasets, shapes, and images.
On the practical side. future work could align DSW with the original OTDD by employing hybrid slicing (Nguyen & Ho 2024) to extend DSW P2()) P2(R)) or integrate convolutional projec- tions (Nguyen & Ho 2022) similar to the S-OTDD. Also, one might employ our Banach slicing for infiiiteddimensioall generative models (Hagemann al., 2025) or integrate dynamic transport into framework enable applications in generative flow matching similar to (Chapel et al.) 2025).
interest to analyze further topological properties similar to Han 2023). Another interesting direction is the question of sample complexity: In high-dimensional Euclidean spaces, sliced Wasserstein distances require fewer samples to approximate a continuous probability measure than the Wasserstein distance (Nadjahi al. 2020). As WoW distances require many samples to approximate a meta- measure (Catalano & Lavenant] (202), would interesting analyze if our DSW distance displays better sample complexity properties.
On the theoretical side, it would
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ACKNOWLEDGMENTS
MP gratefully acknowledges the financial support by the German Research Foundation (DFG), GRK2260 BIOQIC project 289347353. Moreover, gratefully acknowledge fruitful discussions with Gabriele Steidl, Nicolaj Rux, and Gregor Kornhardt that helped improve the ideas in this work.
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Maria Zontak and Michal Irani. Internal statistics of single natural image. In Proceedings of the CVPR' /1, pp- 977-984. IEEE, 2011.
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A NON-SPHERICAL SLICED WASSERSTEIN DISTANCE ON BANACH SPACES
Here, we present proofs for Section main paper as smaller statements.
For clarity, we restate and prove our statements from the
Lemma A.1. For μ. E P2(U), U > W(To,; ; n. no.; vs IR) is Lipschitz continuous.
Proof. We generalize the proof of (Han 2023
Lem. 2.3). For this, 81, 02 E U be arbitrary. Using the triangle inequality and its reverse, we have
IW (To, 14, = v; R) W(T0 M4 T v; IR)]
$ [ (ToTi aoa. pa IR) + (( To.. M; R)]
[W No vs R W(T0, , T02.t IR)]
= W(Te, P4, Te,, R) + W (Te,. v, mo.. Mi 1R).
For the first term the left-hand side, it follows
W (To, a ,4 #0..t si R.
<
f1 tzl² d(TO, To.) u(t1, Kr,, 01 (62) du(x))
R2
< 110 - 02llē- / / du(x)) 110 - 02112. M2(u)),
where M2(μ) = forlla | du(x) is the second moment of Using analogous estimate for the
second term, we obtain
[ μ4 Te,, v3 IR)
W (TTe μ. mo. pi R)) S ||0 - 02||u- (M (v) + A2 (v)).
This allows us to prove the first part of Theorem 3.1
Proposition A.2. For & E P2(U*), the E-based SW distance is well-defined.
Proof. The Lipschitz continuity in Lemma A. implies that the integrand in the formulation of the E-based SW 4 is measurable. To show that SW (p, (; {) is finite for μ, E PU))), let y T(4, realize W(n, v; U). Because of
W Te R) - [ ((21, - (x2, (d d1
UUX
< |1011..
|| x2 ( x2) = -1011.. (2(n
(8)
Xxx
the E-based SW distance is bounded by SW g) E) W(H,v:) 22)
Now, we prove the second part of Theorem 3.1
Theorem A.3. Let & PP(UU)) be such that supp O span 0 & {0, (0) } for any 0 E U*, then SW(. {) defines a metric on P2(U). Otherwise, SW(- defines at least pseudo-metric.
Proof. Positivity, symmetry, and triangle inequality follow from the corresponding properties of the Wasserstein distance. For the definiteness, assume that SW (W,, v; {) O, which implies (Tee μ. ne, 1/; R) for almost all e U* with respect to &. The Lipschitz continuity in Lemma A. limplies W W(TT, To, R) = 0 and thus Te. p = Te, for all 0 e supp Now, let 0' E U arbitrary. By assumption, we find E \ (0) such that t0' E supp Furthermore, we have
[ dhaer du(x)= =
/ Maneo0 du(x)) = / d dm uss)
int dow ((s) )
Since every measure on has a unique characteristic function, see Ledouxx & Talagrand [991 $ 2),,, conclude
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We continue with Proposition 3.2
Proposition A.4. Let C1. P2(U*) be equivalent. If d&i/ dE2 and d52/ dıi are bounded, then we find Ci2 02 such that
SW i; E1) SW(,, (2) Sw(, VILUE P2(U).
Proof. Exploiting the bounded Radon-Nikodým derivatives, we obtain
dēl (u; E2) = [.- ( 14, No, vs IR) déa (0) dei S || (0) (0) ( SW2 (1, G1).
(u,
We employ this to prove Proposition 3.3
Proposition A.5. For & E P(RR)) absolutely continuous and de/ dn, dn/ de bounded, for ~ (0, Ia), there exist cı, C2 0 such that
e1 (W U 5) SW v; SW(,, 02 SW(G E) Vj, E P2(R)..
Proof. This follows directly from Proposition A.4 and the identity SW(,,v) = SW(,, (Nadjahi et al. 2021 Prop. 1).
We extend our findings from the main paper with a statement about the computational approximation via Monte Carlo. In particular, the & g-based SW distance can be numerically approximated by
1
SW (M, v; E)
w (T R), 0g E iid.
It is well-known that such Monte Carlo estimates have convergence rate of O 5) for S random projections (Nadjahi al.] 2020). Given suitable conditions, similiar results holds for this estimate. Proposition A.6. For & E Pa(U*), it holds
1 std, (TT. VS
Eg SW (1, { SW E) I
Te. R).
Proof. Using Hölder's inequality, and exploiting that the directions 0, are independent and identi- cally distributed, we have
Rg ..... Uy ISW (M, v; 8)
SW2
(W, E)
S
(L. 1/.13 Σ
( To R) SW2 $) dE(O.) ... dE(os))
8=1
(To To,. i4 R) - SW E) (((0)) $/
I2
VS
aml
VS std, W ( μ4 no. R),
where the standard deviation exists due to W (To Top:: R) s 1111. UV,
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B DOUBLE-SLICED WASSERSTEIN DISTANCE
B. METRIC PROPERTIES
To prove the positive definiteness of our double-sliced metric for empirical meta-measures, we uti- lize extension of the "Cramer-Wold" theorem by Cueeta-Albertos et al. (2007) about the set of projections required to separate measures on RR. The statement is based on the so-called Carleman condition. A measure μ E (R fulfills this condition if all moments
M2(1)= = / | du, p2l. 1,
are finite and it holds that
Σ p=1
Mp 1/p
Mp
This condition is fulfilled for compactly supported measures (Schmüdgen al. 2017 Ch. 14) and,
in particular, empirical measures. We also refer to (Heppes) 1956 Tanguy et al. 2024) for similar results targeted at empirical measures.
Lemma B.1. Alberts al. 2007
Corr. 3.2) Given measures μt, v P2( that fulfill positive surface measure with
the Carleman condition and a set C S Sa-1
W(TorH Top;; R) = 0 for all 0 E S.
holds that = V.
This allows us to prove the metric properties presented in Theorem 3.1 Proposition B.2. Given positive E P2(L2a(Y)), DSW defines metric on P. (P(P)((R))
Proof. 'eseudo-metrie:: The symmetry and triangle inequality are trivial and follow directly from the ambient spaces of the embedded measures and the properties of the Wasserstein distance.
'Positive Definiteness': We aim to prove that
DSW v; {) =0 + 1= V.
for empirical meta- measures..... Therefore, assume that DSW (1,, {) 0 for μ. E (P(P.(R.))
Due to DSW (1,, v: {) 0, we know that
To E) = 0 for all 0 E g except
for a zero measure set Z. Since SQW a metric on (R))), we thus know that Tp.1/4 = nØ v for
every 0 E Sd-1 Z. Now, this means that there exists % [(TT, TT C Raxm such that 20
(70 Co) = 0,
(10)
where Cs (W2 ( Te Te) C Faxm and (.. denotes the Frobenius inner product. Here, our costs and transport plans take matrix form due the empirical measure structure.
We now want to find a suitable transport plan from this set of transport plans to construct an upper bound for the meta-measure metric W(,, v; Rd). Consider the set of index pairs Idx = {ii,]) 14 * Fy}. If this set is empty, then we are done. Otherwise, we know from Lemma B. that there exists no set S c gd-1 of positive measure for (H4, 1/)), (i,j) E Idx, such that is fulfilled for all 0 E S. Conversely, for (i, 1dx, it holds that (Ce) 0 for every 0 outside zero measure set Zz,-- Subsequently, it has to hold that (r))s, = 0 outside the zero measure set Zr,j due to (I0)
Now, we have (70);) or Cr,; W vy) = 0 for some 0 E sd-1 outside the zero measure set (Z U (U(iJ)eEdx 2.))) Thus, it holds for almost every 0 that
(T C)
Since this expression is an upper bound W (M, v; Rd). it follows that W W(n, R; = 0. This concludes the proof since the Wasserstein distance is positive definite metric.
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Remark. Although stated our statement for the empirical measures employed in our experi- ments, our proof merely requires that all measures satisfy the Carleman condition. As an exam- ple, this would be fulfilled for mixtures of compactly supported measures, ie.. for P. (P2(())) X' C Rd compact. Note that (P)()(()) is dense in (P2(()) (Villani 2003] Ch. 6). Moreover, Lemma A.1 allows this with statements on continuous extensions of metrics on topological spaces [Enellknn 1989) can expect DSW be metric on P2)P)(2)).
show the Lipschitz continuity of DSW on (2))) Combining all of
B.2 RELATIONSHIPS BETWEEN METRIC
In this section, we aim to prove our convergence result. We point out that (Han) 2023 Thm. 3.4) contains a proof of weak convergence for measures on general Hilbert spaces. However, the un- derlying argument appears to rely settings whose validity in this context is, to the best of our understanding, not fully clear.
application of convergence result in infinitediimensional
To prove our convergence statement, separate the proof into couple of smaller statements. As a first step, we show that our sliced metrics produce lower bounds for W. We continue with a lemma that relates the subset of (2( 1J)) supported Euclidean measures. Based on this, we prove a statement about the equivalence between DSW and SW. Lastly, we prove statement about the equivalence between SW and W, which utilizes the compactness of the support.
piecewise constant step functions to
First, we state a proposition that allows us bound W from below via DSW and SW. Proposition B.3. It holds Ce DSW (n, M; E) SW E) s W(,, R2) for u, (IRP)((R)) and & (2) 1))).
Proof. DSW is essentially a double integral. For E L2(10, IJ), 0 E sd-1 and y E P(u,, v), the DSW integrand be estimated using the Cauchy-Schwarz inequality by
Te. (9: TR) < (aare)rree)) (a(Teau) - ((TTau), o)? da v)
f (1((T0.11) - ((TT dR(u.))
Subsequently, it holds
(T no.a (o.a(U:(TT,;")) IR) < |||| W Ta Ta. R) S [ Wu, v; Rd).
The last inequality follows from (TT T R) < W(,, Rd) for u. E P2( and 0 E gd-1, Because of & E (2, 1])), we know that SL Integration with respect to g and the uniform measure on gd-1 gives the statement with C3 Mz()..
= M2(E) < cc.
From this relation, we can easily see that W(mn la; Rd) - 0 results in SW (pmn pa; 8) 0 and that SWWnn.Pe po <) - results in DSW (pm μt; { -
In the following statement, we use indicator functions Lees that take the value 1 for E S and 0 otherwise.
Lemma B.4. For uos p. E P2(L2((0, 1))) only supported on fixed-length step functions, i.e., sums of indicator functions of the form
Sp EATeR : S
for a fixed partition
U.-.P P. = = 1}, and a positive Gaussian measure E (2(L2((0,, 1))) we have SW SW(nn, /4; 8) ++ > W(ps μr T=([0, 1)) 0.
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Proof. To prove the statement, want to leverage the established metric equivalence between the Wasserstein and the classical sliced Wasserstein distance on P2( Therefore, we construct an isometric mapping p Sp R similar to (Piening & Beinert 2025a). Instead of consid- ering probability measures on the infinite-dimensional space (2, 1), this allows us to consider Euclidean probability measures. We define Sp-isometric mapping the space [=2 1]) via
IMp : (20, 11) + RR,
f+ [PP) /
pp. f(z) dr) 8
ER"
im1
This is an isometry on Sp because for 8 E Sp holds that
1 PAU0 -9.)²
Now, this means that
W (pas is (2( 1))) W (IM Pny IM 15 RR).
(11)
Now, we aim construct a similar relation for the g-based sliced Wasserstein distance. We require E E P2 (R") based on & E (22(2 1])). Therefore, we again employ our mapping to link the L2- projection projection on R. For Sp, g L2((0, 1].. have (f. g) (MMr(ff),MM(()
Hence, we define & 1M As sliced counterpart of (1). we get that
SW (un, () = SW (IM, (n, IM₃ (PEP).
Now, we want to utilize Proposition 3.3 to connect our g-based sliced Wasserstein to the classi- cal sliced Wasserstein distance. Indeed, since & is Gaussian measure and IMp is linear, & is Gaussian by definition of Gaussian measure. Moreover, it is a nondegenerate Gaussian since is positive. Thus, SW(IM, (4n, IM; (r) is topologically equivalent to the classical sliced dis- tance SW(MM, M4, IM, (1) by Proposition 3.3 Since SW and W induce the same weak topology on P2 (2) Nadjahi et al. 2020), we overall conclude that
SW E) > 0
14 {P) 0 SW(MM, Un: IM, js;
SW(MM, J4n 11) +
W(IM; pm: IMg jr; R²) - 0
W(en, us (2 1))) - 0.
Proposition B.5. Given positive Gaussian E P2 P2(TT2((0, 1))) and empirical meta-measures
Mny / E P.N (Pa (X)), X' C IRd compact, we have that
DSW(pn è) +0 + SW {) + 0.
Proof. We assume that
(s {) J SW2 (q:(TT,: (: qT(TT, ga--
{) dg) 0.
follows that
SW(g 2 To {)
for almost any 0 E gd-1, Since we are dealing with fixed, uniform weights, all quantile functions piecewise constant step functions with fixed step length. By Lemma B.4 it thus follows that
W (q: Ta, 9 L2 ([0 1]) W(Te, :.. 8 +
for almost every 0 E gd-1, Since the compact support results in boundedness, it thus follows from the dominated convergence theorem that
SW pe; E)
WT To: R) '(0)
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Now, state the last result. Note that our proof is inspired by proof in (Piening & Beinert 2025b). Proposition B.6, Given pox p E P (Po (2)), X c R compact, it holds that
(W μi E) - W(nn pc; R$) -> 0.
Proof. We write
Un
pn.s
8 8pg
Now, we assume that
N
SW un; E)
Yo.k.t W (ma Mn Tass Mei R) (B) (0) - 0,
where 7o E I(TT, TT, E gd-1 Because of L2(Sd-1) convergence, for any subsequence of pas we find a further
R220 denotes the optimal projected WoW plan for fixed
subsequence (pmm) IEEN such that
ΣΣ 1=1==1
Am. W2( Momkks Tap Pei R) for almost every 0 E gd-1 V.R.
(12)
pointwisely. Since X' is compact, (Hnm)meN can be chosen such that Pnm. converges to some Ai E )2 Moreover, since Un and μ are both empirical meta-measures with N sup- assume without loss of generality that ke,k,t is a permutation matrix, i.e., Vokk. E {0, NNN with only one positive value per row or column (Peyré Cuturi 2019 Prop. 2.1).
port points, n,
in the case of 0 otherwise. It follows that either ToRke + 0 or W (J md) + Due to the compactness assumption, we know that To and W (K bounded and thus
For any E gd-1 such that 12 holds true, it follows that Fons
Te. ja = Te. Me and
Pa Ka
K
K
W u R
W ( ds Mei Rd) + 0 as
n - 00,
Because this holds true for any subsequence of (Jn)ENN the statement follows.
Combining Proposition B.5 and Proposition B.6 gives the second statement of Theorem 4.1
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C ADDITIONAL DETAILS AND EXPERIMENTS
C.1 IMPLEMENTATION DETAILS
All experiments were conducted with Python on a system equipped with 13th Gen Intel Core i5- 13-3300K CPU and an NVIDIA GeForce RTX 3060 GPU with 12 GB of memory. For all exper- iments concerning the ) OTDD, and STLE" we employ the official implementations for algorithms and experiments based on the corresponding public GitHub repositories. Unless stated the experimental description, we use the default hyperparameters for algorithms, including the entropic regularization parameter for OTDD computation. For all other Euclidean Wasserstein com- putations, we employ the geomloss package (Feydy et al. 2019) to estimate entropically regularized Wasserstein distances, where we set the entropic 'blur' parameter to 0.01. All WoW distances calculated by estimating a pairwise Wasserstein cost matrix using this geomloss geomloss and finally solving the unregularized OT problem given this cost matrix using the POT package.
For our own implementation, we employ linear interpolation between the two closest support points for quadrature grid points. Moreover, instead of sampling the Gaussian processes and the unit directions completely independently, we sample 10 or 100 random Gaussian processes for each sampled unit direction 9 reduce the number of quantile computations. For our low- dimensional point cloud experiment and our mid-dimensional patch experiment, we employ 100 random 'outer projections' and 100 random 'inner projections' e per 0 (10, 000 in total). For the high-dimensional (s-OODDD experiments, we employ 1000 random 'outer projections' I and 10 random 'inner projections' per 0 (also 10, 000 in total).
C.2 PRACTICAL GUIDE ON PARAMETER CHOICE
The numerical implementation depends multiple parameters, namely the kernel parameter o, the grid size R. and the projection number S. We conduct multiple parameter studies that can be found Sections C.3
C.5
and C.6
For the parameters S and R. we note that both of them are essentially integration parameters of a Monte Carlo integration. The parameter R is the size of the quantile integration grid, and S is the number of Monte Carlo steps. Thus, we would generally expect better results for higher choices of R and S. However, the runtime scales linearly with the number of Monte Carlo steps and Monte Carlo typically converges with rate of O(S-1/2). Thus, choosing S is mainly question of balancing performance and runtime, where the marginal value of an additional Monte Carlo projection decreases for high S. As for the grid size R, we note that the quantile functions we aim to integrate are rather simple. In particular, they are monotonically increasing step functions. In line with this observation, we only found small marginal performance increases for 10 in most experiments. Moreover, we note that high choices of R result in sampling from highly correlated, high-dimensional Gaussians, which might become numerically unstable for some implementations.
we only found small impact of a. This is line with Proposition 3.2 showing that variations of a result in an equivalent metric. While 0 works for our discretization in practice and essentially reduces to the slicing approach discussed (Piening & Beinert 2025a). corresponds to sampling white 0 is not covered by theory. The limit o - oo leads to an increasing correlation and to sampling constant test functions. However, constant functions
During our experiments regarding SQW for the shape experiment in Section 5.1
noise process that is not contained in L2( [0, 1]). Therefore,
not dense in T.2 (22 1]), the conditions of Theorem 3.1 would not be met anymore. Hence, SQW might lose its metric properties. Thus, we generally recommend choosing o as value from [0.001, 0.1].
'Code: ht http://ss//.tttbb...c.//aaaaaaaa *Code: httppp//ggghhhoscccc////aaaaaaaa http:///.w.....ooooooooooooooon.
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C.3 SLICED FUNCTIONAL OPTIMAL TRANSPORT ON L2( 1)
Both of our introduced sliced distances, SQW and DSW, rely on the {-based SW distance on ((,, 1})) and its numerical implementation. We study the impact of parameter choices on this distance by looking at increasingly finer function discretizations
for different kernel pa- rameters o. We consider two empirical measure pair in 22 (10 11)) defined via
( (z) cos(iz), hy (x)) sin(jæ+jn)
(x) cos +1) sin(t)) (x) sin(jx))
and
((0)
(3) k 1,2.
10
In Figure 6 we plot our Monte Carlo estimate for the g-based SW between (((1) and p(1) resp. y((2) and y(2) estimates for different Gaussian bandwidth parameters and equispaced discretization grids with varying size R (20 < R S 100). To investigate the limit case for a + 0, we also include the case of isotropic Gaussian slicing directions 0j NO,, Iv) due (kg (yt, W1)l,,,e)s While this limit case is well-defined for finite discretization, the limit process would be white noise process, i.c., a process with sample paths outside of (2 1J). On the one hand, we observe that the SW estimate depends heavily on the grid resolution R. especially for isotropic Gaussian directions and for small. On the other hand, we see that our numerical estimate less sensitive to the discretization for larger and remains stable given sufficiently large grid. To get accurate estimates for our comparison, we employ = 10, 000. plot employed functions Figure
IN-
In the main paper, only employed the radial basis function (RBF) kernel k, However, we point out that we might use other kernels, such as the Brownian motion (BM) kernel k(s, = min( t)). As example, we repeat the experiment portrayed in Figure 6 with the BM kernel and plot the results in the last column of Figure 6 Similar to an RBF kernel with high o, we see that the resulting sliced distance is invariant the discretization.
10
0.0
-1.3
-10
3.00
0.5
0630
1.00
2.00
0.35
6.50
335
1.00
320
1.5
3.50
175
130
1.00
6.55
8.28
8.75
100
(1) and (1)
(2) and (2)
Figure 5: Plots of the support functions of the two empirical measure pairs from Section C.3
Gaussian Steing 1
Ours 6 0.00))
Ours (r - 0.02)
Oues (MM)
Diseretiztion
Discretization
scretizatoon
Discretization
Exsrrtization
Ours (a 0.001)
Ours (ir 0.02)
Ones (BM)
Cansiin
Durst (e
Oscretication
Discretization
Diccesiatioo
Discretization
Decretization
Figure 6: Dependence between function discretization and SW estimates between ((()) and (1) (upper row) and (2 and ((2) (bottom row) for different kernel choice. Smaller a values lead to
less correlated Gaussian slicing directions for the ho kernel, where include the limit case of fully uncorrelated Gaussian slicing directions in the first column. For such small a, the numerical estimator depends heavily the discretization.
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C.4 EXTENSION OF SECTION ON SHAPE CLASSIHICATION
ON
C.4.1 MNIST-2000
MNIST-2000 used in Section 5.1 is synthetic dataset, which we build based on the training samples of MNIST. For each of the five digits 0-4, we first compute the mean image. From each digit, we generate 20 empirical point clouds consisting each of 2000 points. For this, sample the mean image proportionally to pixel intensity. Afterwards, we disturb the point clouds by Gaussian noise and apply a random rotation. The points clouds equipped with Euclidean distance, which represented by matrices. total, this yields 20 almost isometric shapes per digit. We show one
sample per class in Figure 7
Class
Class
Class
Class
Class
Figure 7: MNIST-2000 Samples.
C.4.2 STABILITY ANALYSIS OF CLASSIFICATION PIPELINE
The shape classification experiment presented in Table I depends on random initialization deter- mined by a random seed and on the choice of the t-nearest-neighhor (k-NN) algorithm. Since each dataset includes only about 100 shapes, cross-validation offers too little statistical support for select- ing k. For this reason, we fix & a priori. We perform an additional experiment to demonstrate the impact of k and the random seed.
For our main experiment, we choose k = instead of to mitigate the impact of small outliers. Additionally, we aim for small & due the limited number of samples per class, e.g., 10 samples per class for the '2D Shapes' dataset. To illustrate the impact the choice of & and the random initialization, rerun classification pipeline for the '2D Shapes' and the 'Animals' dataset with a different initialization and report the resulting accuracies for h = 1, 3, 5. The results are presented in Table 2 and in line with those presented in Table in the main text. We observe drop in performance for k = 5, presumably due to limited data per class.
I-NN
3-NN
5-NN Acc.
I-NN
3-NN (8)
5-NN
SQW (Ours)
99.9 U.6
99.113
98.6 2.1 2.1
99.5 = 0.9
99.1 193
68.5
TLB
100.0 0.1
100.0 U.3
99.9 : 0.4
100.0 = 0.0
100.0 0.0
70.9 0.0
STLB
99.8 1 0.6
99.4 1: 1.3
98.3 1 2.2
99.8 0.7
99.3 1 1.1
69.2 1 1.7
AE
90.9 + 0.4
99.7 +: 1.0
99.0
+ 1.9
98.9 1.4
97.8 1.9
67.6 2.3
GW
100.0 - 0.2
99.8 0.6
98.9
100.0 0.0
100.0 0.0
70.9 0.0
(a) 2D Shapes
(b) Animals
Table 2: Classification accuracies for the nearest-neighhor algorithm (ck-NN Acc.') for varying neighborhood definitions and random seed that is different from the used in Table [
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C.5 EXTENSION OF SECTION 5.2 ON DATASET COMPARISON
C.5.1 RELATION BETWEEN CLASSIFICATION ACCURACY AND DATASET DISTANCES
Originally, the OTDD and the s-OTDD have been introduced to estimate the difficulty of transfer learning. Therefore, extend the experiment presented in Figure 2 to consider the connection between the different dataset distances and classifier accuracy. As a simple experiment, we consider the accuracy of a S-nearest-neighhor (5-NN) MNIST classifier our target. In particular, for each of our 100 MNIST splits, we average the two accuracies of classifying the first split based on the second split and vice versa. Now, this averaged accuracy is target quantity.
Again, we estimate the correlation between this target accuracy and various dataset distances. Here, would expect negative correlation since the dataset splits that are further away from each other should lead to worse training results. Beyond the previously reported OTDD, s-OTDD (S = 10%), and our DSW (S 10 1000 outer projections. 10 inner ones). we additionally report results for the Gaussian OTDD, the s-OTDD with S = 10*). our DSW with S = 1000 (100 outer projections, 10 inner ones). Note that the Gaussian OTDD approximates all inner pairwise Wasserstein distances via a Gaussian approximation Melvarez-Melis & Fusi 2020). Essentially. this distance reduces to the mixture Wasserstein distance between Gaussian mixtures (Delon & Desolneux 2020), since we compute all distances without feature costs, i.e, on ((P(R)) instead of P() x P (R))).
Looking at the results in Figure we observe the strongest correlation for the OTDD and the Gaus- sian OTDD with values of x 0.5. Regarding the sliced metrics, observe stronger correlation for our DSW ( -0.45) compared to the s-OTDD (Re -0.35). As for the number of projections, both distances seem to be stable with respect to the number of random projections S, and we observe only small differences between S = 10 and S = 10
Generally, observe clearly negative correlation of approximately -0.5 across metrics. While these correlations are overall weaker than in other related experiments Ailaree--elli & Fusi 2020 Nguyen al., 2025), we attribute this to the small variability of the 5-NN classifier accuracy (ranging from 88% to 94%) and the omission of feature costs. Thus, we hypothesize that the inclusion of the feature cost via hybrid hierarchical slicing (Nguyen & Ho 2024) would further strengthen these correlations.
22.5
Pearson = -6.1 Spearman p
Pearson
CorDe (1000 Proj)
Speermen -9.39 38
32.6
Spearmen
Ours (1000 Prol)
0TOD
23.6
22.6
1:2
18.
0.00 1.00 Co)
0.30
1.3 Bic
232 060 03
5-NN Classifier Accuracy
5 NN Clessifier Accuracy
5N Classifier Accuracy
OTDD
(c) s-ODD (S 10
(e) Ours (S = 10t)
Pearson -- -0.4
Pearson 0.36 Spearman p 0.36
Pearzon 0.46
Gaussien OTDD
3x0
Speermen - 49
S-OTDO (10000 Proj)
Ours (10000 Pro))
Spearman p -.48
22.2
220
32.0
11.6
230
600
0.99
693
CM
0.66
330
ES0
038
833
604
340
0.50
350
658
001
332 030 022
5.NN Classifeer Accurasy
5-NN Classifier Accuracy
5-NN Classifier Accuracy
(b) Gaussian OTDD
(d) s-OTDD (S = 108)
(1) Ours (S = 103)
Figure 8: Scatter plots and correlations (f) between the accuracy of a 5-NN MNIST classifier and the
OTDD (a). the Gaussian- OTDD (8b), the s-OTDD with S = 10 10a & &d and DSW ("Ours')
with S 10 10
Se 8f).
All distances are computed without feature cost on the labels.
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C.5.2 STABILITY ANALYSIS OF SLICING PARAMETERS
We investigate the impact of the projection number S, grid size and the kernel parameter a by considering the correlation between DSW and OTDD on MNIST displayed in Figure 2a As the projection number S is a product of the number of outer and inner projections, see Section fix the number of inner projections to 10 and only vary the number of outer projections. In particular, we simulate the DSW distance for all combinations of S 10 10", 10 R 10, 100, 200, and = 0.01, 0.1.
we
For each simulation, we compute the Pearson and Spearman correlation coefficients and present the results in Table 3 All coefficients are approximately in the range from 0.9 to 0.95, We see that the number of slices S is by far the most important parameter, whereas the impact of the grid size and the kernel parameter is more subtle. In particular, increasing S from 10x to 10 increases the correlation significantly (from 0.9 to 0.94). whereas the increase from 10 to 10y only has a slight impact (raising the correlation from 0.94 to 0.95).
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Table 3: Correlation between our DSW and OTDD (without feature cost) for various confi gurfttiationss on MNIST, extending the results in Figure 2a
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C.6 EXTENSION OF SECTION
5.3 ON POINT CLOUD COMPARISON
We extend the point cloud experiments from Section 5.3 by adding two experiments as an analysis of the projection number and runtime. In particular, we vary only the number of 'inner' 'outer' projections per experiment, see Section [C. In our point cloud experiments, the runtime hinges on the number of shapes (NV and M) and the discretization of the shapes and m). For this analysis, we sample only from the 'chair' class without Gaussian noise and R = 10. All results are averaged over five runs.
50 and vary only N . 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. For each pair of sampled shape sets, we then compute the WoW distance and our DSW distance. To analyze the impact of the projection number, we calculate with S 100 (10 outer, inner projections), S 1000 (10 outer, 100 inner p.). and 5 5000 (10 outer, 500 inner p.). The results are visualized in Figure g Note that we observed a rather high variance for WoW runtime in this experiment, generally. As result, observe the plotted WoW runtime estimates in Figure 9 vary rather drastically. Nevertheless,
For our first experiment, we set 71
m
a seemingly polynomial runtime increase for WoW in terms of the number of shapes N M, whereas only we observe quasi-linear runtime increase for DSW in terms of N M. As for the projection number, we observe linear runtime increase in terms of S. Moreover, we observe (small) reduction in the variance of the distance estimate for higher S.
10 and vary only Again, we compute WoW and DSW. For this experiment, we calculate DSW with = 1000 (100 outer, 10 inner projections), = 10, 000 (100 outer, 100 inner p.). and = 50, 000 (100 outer, 500 inner p.). The results are visualized in Figure [O] We observe similar results as before, i.e., polynomial runtime increase for WoW and quasi-linear runtime increase for DSW.
For our second experiment, we set 100, 200,300, 300, 400, 500, 600, 700, 800, 900, 1000.
M
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Figure 9: Averaged WoW and DSW ('Ours') estimates between sets of point clouds for 100 shapes and projection number S = 100, S = 1, 000, 5 = 5, 000.
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Figure 10: Averaged WoW and DSW ('Ours') estimates between sets of point clouds with 100 to 1000 points per shape and projection number S = 1000, 5 = 10, 000, S = 50, 000.
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C.7 EXTENSION OF SECTION 5.4 ON IMAGE COMPARISON VIA PATCHES
C.7.1 REPRESENTING IMAGES VIA PATCHES
We formalize the patch extraction. For grayscale image Img E Axx define the patch extractor
Patch g IR"
k = ;,...
with " (h- 1) p 1). Write
2 == Patch (Img) Rd²
so that the empirical patch distribution is
S2 E Po (R
Plng
n2p
Its support is and for a batch {Img, the meta-measure is
supp(Utmg) {Patchf (Img) ..... np} c R
B
6 Plmx, E Po(PP((I
B
C.7.2 THE RELEVANCE OF PATCH DISTRIBUTIONS
Many advances in image processing rely on the importance of local image features (Zontak & Irani 2011) Indeed, convolutional neural networks in computer vision repeatedly apply the same filter to a small receptive field, and this receptive field can be understood as a patch. Moreover, vision transformers decompose images into smaller patches (Dosovitskiy et al.) (02T). The advantage of this approach can be motivated by the relevance of small-range dependencies within images.
Notably, a key disadvantage of the standard MSE in imaging is its vulnerability to 'small' image operations. A small shift of all pixels can lead to the explosion of the MSE. However, such operations have only a small effect on the patch distributions. Indeed, the same goes for the translation of an object within an image, see visualizations in (He et al. 2024). In addition, the perceived style of an image seems to inherently linked to certain localized image features. As an example, style transfer algorithms successfully capture certain artistic aspects of painting via such features (Gatys) ct al., 2016). Moreover, texture images are characterized by certain type of stationarity, where model generate texture images by simply matching the patch distribution of single exemplary texture image (Houdard et al. (023). This could explain why patch-based WoW-type methods lead clearer discriminiation than Euclidean Wasserstein methods, ef. Figure 12
C.7.3 ADDITIONAL, EXPERIMENTAL DETAILS
In the experiment from Section 5.4 compare distributions over synthetic texture images. We visualize samples from our random Perlin texture model (Perlin) 1985) in Figure [T] Note that images with varying lacunarity (11a) are all generated with the following Perlin parameters: persistence of 1, scale of 100, 6 octaves. The generation model will be released with the code. For our images with varying persistence ([lb). we use different Perlin parameters: lacunarity of 2.5, scale of 100, 5 octaves. Note that while the resulting images in Figure IIa and Figure [Tb look rather similar, the ones from Tl a higher blur and less high-frequency artifacts.
Moreover, we extend Figure In addition to the Wasserstein distance between images represented as Euclidean points and our patch-based DSW distance plotted in the original Figure 4 we present the extended Figure by adding the patch-based WoW distance and the KKrnel Inception Distance' (KID) between the distributions of texture images. The patch- batch-based WoW distance computed on the same patch meta-measures our patch-based DSW distance. The KID is based on the latent space of a pretrained neural network, see (Sutherland et al., 20188). We see that the DSW and the WoW distance lead to similar results. Also, both are aligned with the KID.
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Figure 11: Samples from our Perlin texture noise for varying lacunarity (alab and 'persistence' 11b
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Figure 12: Comparing synthetic texture image batches via Fuclidean Wasserstein, patch-based DSW (*Ours'), ) patch-based WoW, and KID for varying 'lacunarity' 44) and "persistence" (126).
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D USE OF LLMS
L.MMs have been used to a limited extent to improve grammar and wording in this paper.
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Reviews:
[{"review_id": "61yuyTI2nt", "reviewer": "Reviewer_HPNJ", "summary": "The paper introduces a general sliced Wasserstein framework for arbitrary Banach spaces using quantile functions in the function space $L_2([0,1])$. Building on this, the authors present the sliced quantile WoW and double-sliced WoW distance.  Theoretical properties are presented to support the proposed framework.", "questions": "- How sensitive are results to the number of projections of DSW in the experiment of optimal transport dataset distance? For example, using 500 projects in s-OTDD varies from using 10000 projections. What would be the correlation in DSW using 500, 1000, or 5000 projections?\n- While it can be considered as concurrent work, it would be beneficial if the authors could compare their method conceptually with the work Busemann Functions in the Wasserstein Space: Existence, Closed-Forms, and Applications to Slicing. What are the key differences?\n- How does the proposed method perform in transfer learning tasks?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is well-written and easy to follow. The writing flow is clear and presents the relevant work in a nice and elegant way\n- The theoretical results are solid, and the proof in the appendix is organized\n- Extending sliced OT to general Banach spaces via quantile maps is interesting and potentially impactful", "weaknesses": "- The sliced Wasserstein distance defined in Eq.(4) - there are no experimental results on this distance. How does it compare to other sliced Wasserstein distance empirically?\n- SQW’s accuracy gains over competing methods appear modest, aside from FAUST-1000, performance is largely on par with STLB despite comparable runtime. It seems that the proposed method may be beneficial for large-scale datasets, but it is hard to judge based on merely one large dataset that is evaluated.", "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": "eDF38FB2Z6", "reviewer": "Reviewer_ir3L", "summary": "The objective of this work is to study Optimal Transport (OT) on measures over measures, so called meta-measures, aiming for computational efficiency. As noted by the authors, \"a key feature of OT is its applicability to non-Euclidean spaces, even allowing the definition of Wasserstein distances on Wasserstein spaces\". This leads to the Wasserstein over Wasserstein (WoW) distance, which is, however, computationally very costly.\nAnother important existing dissimilarity measure in this \"multilevel OT\" paradigm is the OT dataset distance (OTDD). OTDD was developed to quantify similarities between labeled datasets in a model-agnostic manner, making it particularly useful for transfer learning. However, OTDD is also highly expensive to compute, and slicing-based variants (s-OTDD) have been proposed to mitigate this. In addition, in applications such as evaluating point cloud generative models, the OT nearest-neighbor accuracy (OT-NNA) test is widely used, but it is computationally demanding too. Also, the Gromov-Wasserstein distance falls under this underlying framework, by treating metric measure spaces as measures over measures. However, one more time, its computation is contly and also it is a non-convex optimization.  \n\nIn response, the authors propose a general sliced Wasserstein framework for measures on Banach spaces. The theoretical foundation relies on the isometry between the one-dimensional Wasserstein distance and L2([0,1]) via the quantile mapping, as well as slicing techniques in Hilbert spaces (generalizing the inner product to Banach space by utilizing the pairing with linear functionals on its dual space). This allows them to introduce first the slice-quantile WoW (SQW) metric, which roughly speaking corresponds to 1d meta-measures, and then, more importantly, the double-sliced Wasserstein distance (DSW). On the practical side, for particularly sampling, they employ parameterized Gaussian-processes. They demonstrate that DSW can serve as an efficient surrogate for the previously mentioned meta-measures dissimilarities, such as,  WoW, OTDD, s-OTDD, OT-NNA, and GW.", "questions": "- No clousure in span(v) in Thm 3.1 is needed?\n- I would appreciate if the authors write the precise definition of $\\mathcal P_2(U^*)$.\n- Proposition 3.3: I am confused between $\\eta$ and the uniform measure $d\\mathbb S^{d-1}(\\theta)$.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- The paper is very well organized. \n- Previous works are clearly discussed. \n- The aims and problems are stated in a clear way both from the theoretical and practical perspectives. \n- The results, formulated as theorems, propositions, and corollaries, highlight important properties of the proposed dissimilarities. For the generalization of the sliced Wasserstein (SW) distance in Banach spaces, since the unit ball is no longer compact in the infinite-dimensional setting, arbitrary probability measures are considered over the dual space of linear functionals and comparisons are then provided: Proposition 3.2 shows that the definition of the new SW is independent of the choice of measure, while Proposition 3.3 establishes that it is a genuine generalization of the classical SW. Other results demonstrate that the proposed dissimilarities are in fact rigorous metrics. Finally, comparisons between the proposed DSW and WoW are presented, showing weak equivalence.\n- Comparisons/Experiments with existing meta-measure dissimilarities are presented across different modalities (datasets, shapes, images).", "weaknesses": "- A dynamical perspective on the proposed methods is not discussed.\n- Sketches of the proofs of the theorems, propositions, and corollaries are not included in the main text, which would provide the reader with at least a vague idea of the technicalities and main steps needed.\n- Section 5.1 could benefit from additional comparisons, for example by adding a few more columns in Table 1.\n- A discussion of possible reasons explaining the clear discrimination achieved by the proposed distance, as opposed to more traditional methods (Section 5.4, Figure 3), would be interesting.", "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": "5EJSnJyw60", "reviewer": "Reviewer_qqXT", "summary": "In this paper the authors propose an alternative to the Wasserstein\nover Wasserstein (WoW) distance between measures over measures. This amounts of\nslicing once in the original space all the distributions and then slicing a\nsecond time the resulting 1D distribution in the set of quantile functions using \ndirections corresponding to 1D Gaussian distributions. The resulting approximation\nis called the Double Sliced Wasserstein (DSW) distance. The authors also show \nthat DSW can be computed efficiently and converges to 0 similarly to WoW.\nNumerical experiments on shape classification, dataset distances, distributions\nof point clouds and image datasets seen as distributions of patches show the\npotential of the proposed approach.", "questions": "Please address the weakness comments above and the following questions:\n\n+ In Table 1, what does \"Time\" refer to? Is it the average time to compute the\n  distance between a pair of meta distributions? In this case could you also report\n  the variance?\n\n+ Could you be more consistent and use DSW everywhere instead of \"Ours\"?\n\n+ The proposed Gaussian slicing is important but could faster approach work well\n  (theoretically or in practice)? Could we use 0 variance, which is the step\n  function as a more efficient a basis not requiring quadrature ? \n\n+ On OTDD experiments It would be interesting to see how DSW compares to other\n  scalable OT methods such as OT on GMM where the components of the mixture can\n  be estimated on each of the sub-distributions (e.g. Delon 2020 \"A\n  Wasserstein-type distance in the space of Gaussian Mixture Models\").", "limitations": "", "rating": 6, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "+ The paper is well written and clear. \n\n+ It provides an efficient way to compute similarity between meta distributions,\n  and will scale much better to large distributions than existing approaches.\n\n+ Nice theoretical results and discussion on slicing Banach space and 1D\n  Wasserstein space.\n\n+ Interesting experiments on meta distribution learning tasks that suggests that the\n  proposed approach can be interesting in practice.\n\n+ The method can be directly used to approximate the Third Lower Bound (TLB) of\n  Memoli which is a very interesting application on challenging data.", "weaknesses": "+ The paper is missing a sensitivity analysis to important parameters such as S\n  (number of slices in the original space) and R (number of slices in the\n  quantile space) or even the variance $\\sigma$ in the Gaussian kernel. This is\n  important to understand the tradeoff between computational time and accuracy\n  of the approximation. Illustrating variances over multiple runs would also be\n  useful since the method is stochastic and Monte Carlo based approximation is\n  done here twice and in very large spaces.\n\n+ Application to shape distributions in 5.1 is very nice but could be made\n  stronger. Why is K=3 chosen for KNN? For a fair comparison, K should be\n  cross-validated for each metric. Also how correlated in DSW to TLB? to GW?\n  This is an efficient alternative so how close it is to other metrics is\n  important. Also the computational time is actually a bit underwhelming where\n  DSW is faster than TLB only on 2 of the 4 datasets. The authors should discuss\n  this.\n\n+ Experiments for OTDD in 5.2 are rather limited and do not focus on what is\n  very importante for OTDD, which is the estimation of the performance gap\n  between models trained on different datasets. It is possible that the proposed\n  approach actually works better than OTDD or s-OTDD in this context but this is\n  not investigated. Also R=10 is very small for slicing an infinite dimensional\n  space, and the method could be more correlated with larger R but this was not\n  investigated.", "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": "i9Y8HaR8Qr", "reviewer": "Reviewer_1W1P", "summary": "The paper introduces a new computationally efficient way to compare distributions of distributions using sliced optimal transport (SOT). Traditional Wasserstein over Wasserstein (WoW) distances are powerful but computationally expensive and unstable when applied to high-dimensional or hierarchical data. To overcome this, the authors propose the Double-Sliced Wasserstein (DSW) metric, which generalizes sliced Wasserstein distances to arbitrary Banach spaces. Leveraging the isometry between one-dimensional Wasserstein and \n$L_2([0,1])$ spaces, they define infinite-dimensional slicing using Gaussian-process–based projections, followed by classical Euclidean slicing. Theoretically, DSW is shown to be topologically equivalent to WoW for discretized meta-measures, offering similar accuracy with far less computational cost. Experiments on datasets, 3D shapes, point clouds, and image distributions demonstrate that DSW achieves comparable or superior results to existing WoW and sliced-OT methods, making it a scalable and robust replacement for hierarchical OT computations.", "questions": "1.  The approach is closely related to  sliced Wasserstein embedding (see Section 4.6 in [1]) which exploits the relationship between sliced Wasserstein space and $L_2$ function space. There are some related aspect should be discussed. For example, instead of double-slicing, I can define SW over SW embeddings (a function or vectors if quantization is used).\n\n2. Why $L_2([0,1])$ is used? As in SW embedding, we can map to $L_2(\\mathbb{R})$ using a fixed reference function. \n\n3. A minor comment is that the acronym “DSW” overlaps with “Distributional Sliced Wasserstein” in [2], which also involves modifying the slicing distribution.\n\n[1] An Introduction to Sliced Optimal Transport, Khai Nguyen.\n\n[2] Distributional Sliced-Wasserstein and Applications to Generative Modeling, Nguyen et al", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper rigorously builds upon optimal transport theory, generalizing sliced Wasserstein distances to arbitrary Banach spaces and proving metric properties and equivalence to the original WoW distance.\n\n2. The proposed Double-Sliced Wasserstein (DSW) metric is conceptually original, combining functional (infinite-dimensional) slicing via Gaussian processes with standard geometric slicing, which is both elegant and theoretically justified.\n\n3. DSW substantially reduces the cost of computing hierarchical OT distances compared to WoW and OTDD, making it scalable to large and high-dimensional datasets.\n\n4. By avoiding moment-based approximations (used in previous sliced WoW methods), DSW eliminates numerical instability issues that occur when high-order moments are poorly estimated.\n\n5. Demonstrations across domains (shape analysis, dataset comparison, point clouds, and image distributions) show that the method is general-purpose and effective for both geometric and perceptual tasks.\n\n6.  Experiments are well-chosen and show DSW’s performance to be comparable to or better than established baselines, often with much lower runtime.", "weaknesses": "1. While equivalence to WoW is shown for discretized measures, the paper lacks deeper analysis of approximation error, sample complexity, or convergence rates of Monte Carlo integration in DSW. However, these investigation seems to be challenging which might be quite beyond of the scope of the paper. \n\n2.  DSW requires choices such as the Gaussian process kernel width, number of projections (S, R), and quadrature parameters, yet sensitivity analysis or principled tuning guidelines are limited.\n\n3. Despite lower asymptotic cost, the implementation still involves multiple nested integrations (spherical and functional), which may be nontrivial to optimize or parallelize in practice.", "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)