Fast Estimation of Wasserstein Distances via Regression on Sliced Wasserstein Distances
Authors: Khai Nguyen
Hai Nguyen
Nhat Ho
Venue: ICLR 2026 Poster (2026)
Keywords: Optimal Transport
Wasserstein distance
Sliced Wasserstein distance
Regression
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: We propose a fast method for estimating Wasserstein distances across multiple pairs of distributions by formulating a regression problem, using variants of sliced Wasserstein distances as predictors.

Abstract: We address the problem of efficiently computing Wasserstein distances for multiple pairs of distributions drawn from a meta-distribution. To this end, we propose a fast estimation method based on regressing Wasserstein distance on sliced Wasserstein (SW) distances. Specifically, we leverage both standard SW distances, which provide lower bounds, and lifted SW distances, which provide upper bounds, as predictors of the true Wasserstein distance. To ensure parsimony, we introduce two linear models: an unconstrained model with a closed-form least-squares solution, and a constrained model that uses only half as many parameters. We show that accurate models can be learned from a small number of distribution pairs. Once estimated, the model can predict the Wasserstein distance for any pair of distributions via a linear combination of SW distances, making it highly efficient.  Empirically, we validate our approach on diverse tasks, including Gaussian mixtures, point-cloud classification, and Wasserstein-space visualizations for 3D point clouds. Across various datasets such as MNIST point clouds, ShapeNetV2, MERFISH Cell Niches, and scRNA-seq, our method consistently provides a better approximation of Wasserstein than the state-of-the-art method, Wasserstein Wormhole, and classical methods, particularly in low-data regimes. To illustrate its robustness, we also experiment the method with intra- and inter-class settings. Finally, we demonstrate that \emph{RG} can accelerate Wasserstein Wormhole training, yielding \emph{RG-Wormhole}.

Published as a conference paper at ICLR 2026
FAST ESTIMATION OF WASSERSTEIN DISTANCES VIA REGRESSION ON SLICED WASSERSTEIN DISTANCES
Khai Nguyen*
Hai Nguyen Independent Researcher
Department of Statistics and Data Sciences University of Texas Austin Austin, TX 78713, USA
namha a 8 @ggma
khainb@utexas.c
Nhat Ho
Department of Statistics and Data Sciences University of Texas Austin Austin, TX 78713, USA
A.nhnhattuteexen edu
ABSTRACT
We address the problem of efficiently computing Wasserstein distances for multiple pairs of distributions drawn from a meta-distribution. To this end, we propose a fast estimation method based regressing Wasserstein distance on sliced Wasserstein (SW) distances. Specifically, we leverage both standard SW distances, which provide lower bounds, and lifted SW distances, which provide upper bounds, predictors of the true Wasserstein distance. To ensure parsimony, we introduce two linear models: an unconstrained model with closed-form least-squares solution, and constrained model that uses only half as many parameters. We show that accurate models can be learned from a small number of distribution pairs. Once es- timated, the model can predict the Wasserstein distance for any pair of distributions via linear combination SW distances, making it highly efficient. Empirically, we validate approach on diverse tasks, including Gaussian mixtures, point- cloud classification, and Wasserstein-spaae Across various datasets such as MNIST point clouds, ShapeNetV2, MERFISH Cell Niches, and scRNA-seq. our method consistently provides better approximation of Wasserstein than the state-of-the-ard method, Wasserstein Wormhole, and classi- cal methods, particularly in low-data regimes. To illustrate its robustness, we also experiment the method with intra- and inter-class settings. Finally, we demonstrate that RG can accelerate Wasserstein Wormhole training, yielding RG- (GG-Womhhoee
visualizations for 3D point clouds.
INTRODUCTION
Optimal Transport (OT) and Wasserstein distances (Villani 2009 Peyré & Cuturi 2019) have be- come essential tools in machine learning, widely used for quantifying the similarity or dissimilarity between probability distributions. Fundamentally, the Wasserstein distance measures the minimum cost required to "transport" mass from one distribution to another, effectively capturing the un- derlying geometry of the data. Thanks to their clear geometric interpretation and mathematical robustness, Wasserstein distances have found applications across various fields, such as generative modeling Genevay al.) (2018), computational biology Bunne et al. (2023). chemistry Wu al. (2023). and image processing Feydy et al. (2017). Despite its utility, computing the exact Wasserstein distance is computationally expensive. It typically requires solving a large-scale linear program to find an optimal transport plan, with time complexity of Ore log n) for discrete distributions of size n. This high cost severely limits its use in large-scale or real-time settings.
"Equal Contribution
Code published at httph:ttt///gt www.ggtttb....
kegressi wasserstein

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In many applications, Wasserstein distances computed (repeatedly) for many pairs of distributions instead of a single pair, e.g., dataset comparisons (Alvarez-Melis & Fusi (020). 3D point-cloud autoencoder (Achlioptas et al. 2018), point-cloud nearest neighbor classiicattonn/eerrssion (Rubner et al. 1998). learning embeddings for distributions (Kolouri et al.) 2021), density-density regres- sion CChen et al. | 2023), and so on. Therefore, the high computational complexities of the Wasserstein distance become the main bottleneck to scaling up these applications. As result, speeding up the computation of the Wasserstein distance has become a vital task in practice.
To address this bottleneck, a straightforward improvement is to speed up the computation of the Wasserstein distance. For example, entropic regularization (Cuturi 2013) enables fast approximation via Sinkhorn iterations, while other methods exploit the structure in the transport plan, such as low- rank approximations (Scethon et al. 202T). In addition, some approaches rely on strong structural assumptions, such as the Bures- Wasserstein metric (Dowson & Landau 1982) gives a closed-form solution for the exact 2- Wasserstein distance (W2) under the Gaussian assumption on distributions.
Another approach is to cast computing Wasserstein distances for many pairs of distributions learning problem, i.e., learning a model first to predict the Wasserstein distance given any pair of distributions, then use the model later for the mentioned downstream tasks. For example, Deep Wasserstein Embedding (DWE) (Courty et al. 2018) trains a Siamese convolutional network to match OT distances between 2D images, while Wasserstein Wormhole (Haviv al. 2024 employs transformer-bant -based architectures to learn embeddings of distributions, allowing Euclidean distances in the learned space to approximate Wasserstein distances efficiently. While effective, these deep learning-based methods require significant computational resources and time to train, and their performance may degrade when limited training data are available. Moreover, these approaches are limited to empirical distributions because of the use of neural networks.
In this work, we propose novel approach to predict the Wasserstein distance without relying on any neural networks or learned embeddings. Moreover, the proposed approach relies on parsimonious model and can handle both continuous and discrete distributions. In particular, we propose to regress the Wasserstein distance on sliced Wasserstein (SW) distances (Rabin et al., 2010 Mahey et al. 2023 Nguyen Ho 2023] Liu et al.) 2025) Deshpande et al.) 2019; Rowland et al.] 2019). greater detail, we introduce linear models with Wasserstein distances as the response and SW distances the predictors. We provide estimates of the models via efficient least-squares estimates. In addition, since sliced Wasserstein distances have low computational complexity, the resulting Wasserstein regressor is computationally efficient.
Contribution: In summary, main contributions are three-fold:
We introduce the first regression framework where the Wasserstein distance serves as the response variable and various sliced Wasserstein (SW) distances act as predictors, in the setting of random pairs of distributions. This framework not only uncovers the relationship between the Wasserstein distance and its SW-based approximations but also enables efficient estimation of the Wasserstein distance. Specifically, we use SW distance (Bonneel et al., 2015).. Max-SW (Deshpande et al. 2019), and energy-based SW (Nguyen & Ho| 2023), all of which provide lower bounds on the Wasserstein distance, as predictors. In addition, we incorporate lifted SW distances, which provide upper bounds. including projected Wasserstein (Rowland et 2019), minimum SW generalized geodesics (Mahey et al. 2023). and expected sliced distance (Liu et al. 2025).
2. We propose two linear models for the regression problem and describe their estimation via least-squares. The first model is unconstrained and admits a closed-fone least-squares solution. The second model incorporates constraints that leverage the known bounds between SW distances and the Wasserstein distance, thereby reducing the number of parameters by half. Based on these estimations, we obtain a fast method to approximate the Wasserstein distance for any pair of distributions, with the same computational complexity that of computing SW distances.
3. Empirically, we demonstrate that our approach yields accurate estimates of the Wasserstein distance, particularly in low-data regimes. We first evaluate its accuracy through simulations with Gaussian mixtures. We then apply the estimated distances to visualize distributional data and to perform k-NN classification on ShapeNetV2 point clouds. Next, we benchmark our method against Wasserstein Wormhole, the state-of-the-arr Wasserstein embedding model, across four datasets of increasing dimensionality: MNIST point clouds, ShapeNetV2, MERFISH cell niches, and scRNA-seg,
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Finally, we propose RG- CGG-oomhhle, variant of Wasserstein Wormhole that replaces its Wasserstein computations with our estimates, preserving accuracy while substantially reducing training time.
Organization. Section reviews preliminaries on the Wasserstein distance, its sliced variants, and their computation. Section introduces our regression framework for approximating Wasserstein dis- tances from sliced variants, together with both constrained and unconstrained linear models. Section reports the experimental results. The appendices provide supplementary experiments (mixtures of Gaussians and distributional space visualizations), detailed experimental settings, theoretical proofs, and additional related work.
4
Notations. For any d 2, let 8 = {0 Rd |10 }} denote the unit sphere in IR and let 1 denote the uniform distribution on it. For p 2 1, we write Pp(2)) for the set of all probability measures on X' with the finite p th moment. Given two sequences an and b,, the notation an (br,, means that On I Cb, for all 2 1, for some universal constant C' > 0. For measurable map P, the notation Přp denotes the push-forward of μ through P. Additional notation
will be introduced as needed.
PRELIMINARIES
We first review definitions and computational aspects of the Wasserstein distance and its related properties in one dimension.
Wasserstein distance. Wasserstein-p (p 2 1) distance Villani (2008) : Peyré et al. (2019) between two distributions μ E (RR") and v (IR)) (dimension d 1) is defined as:
J f | v1l.dd((.1)) WB xETI(a.))
inf
where [1 f E P(R4 x R2)} SRe dr di (x, = =xx)) S dr (2x, y) vvv) } is the set of all transportation plans i.e., joint distributions which have marginals be two comparing distributions. When and discrete distributions i.c., Ei= 0gd2, (n 2 1) and v = Σラ= B,dy, (m 1) where Σ N a; Σグ= B, = and a; 2 0,3, 2 0 for all = 1, :,...; and j ......
Wasserstein distance between μ and defined as:
n
W wF(av)- = min yeT(a.3)
1x Mi Wilb
where [(a, = (T E 12*"" 1 B}. Without loss of generality, we assume that > m. Therefore, the time complexity for solving this linear programming is (n log n) Peyré & Cuturi (201)) and O(n²), which are expensive.
One-dimensional Case. When d 1, the Wasserstein distance can be efficiently calculated. For the continuous case, Wasserstein-2 distance has the following form: WP (1,, v) = So [F) F- (t)( Pt),, where Fi- and F,-1 denote the quantile functions of μ and v respectively. Here, the transportation plan is ((H,)) = (F F () 1]). When μ and v are discrete distributions, i.. En agd2, (n 2 1) and v >y j==1 Br0yg quantile functions of μ and v are:
i-1
FT ( Σ I i=1
Σ
an) <
<
ag) (1) MG <t=
Σ
B
<
Σ
B())
j=1
where (01) < < (() and D(1) .... S M(m) are the sorted supports (or order statistics). Therefore, the one-dimensional Wasserstein distance can be computed in O(n logn) in time and (n)) in space (assuming that
m).
Random Projection. A key technique that plays a vital role in later discussion is random projection. We consider function Pg Rdd + I where 0 aff) (a(() P( 1)) is random variable. For simplicity, we consider the traditional setup where 8 ~ U and P (x(x) = (8,x) (Bonneel al. 2015: Rabin al. 2012). However, the following discussion holds for any other types of
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projections (Kolouri et al. 2019; Bonet et al. 2023b 2025 2023c). For Pp(RR") and E Pp(RR).
one-dimensional projected Wasserstein distance with Pe Is defined as:
WP M: Pa) (Poduu Potu) = L. | (t) FP dt)
(2)
The second approach to construct Wasserstein-type discrepancy from one-dimensional projection is using lifted transportation plan. There are many ways to construct such lifted plan using disintegration of measures (Muzellec & Cuturi 2019 Tanguy et al. 2025). In practice, the most used way (Liu
et al. 2025 Tanguy et al. (025) is:
JAXRR {
W Wi,, Po)
| y u1ldd (1
(3)
1 - yladur, @ ves Vas (ro (21,
(4)
JRR (p)
where x² E 11(, v) the lifted transportation plan, TiPota.Popo is the optimal transport plan between Paij and Patv, Me, and Mt, are disintegration of μ and v at t and l2 the function Pe. and denotes the product of measures. When dealing with discrete measures μ and v, W p(, ( ug Po) still be computed efficiently (Mahey al. 2023) Liu et al. 2025) i.e., O log n.) in time and (nn) space (assumed that n > m). The quantity W ( 1/; Po known as lifted cost (Tanguy et al. 2025) or sliced Wasserstein generalized geodesic (Mahey et al. 2023 Liu et al.) (2025). From
previous work (Nguyen & Ho. 2023 Mahey al..
2023
Tanguy 2023). we know the following
relationship W,,,,, V; Po) S W (p, W Wp(,, M; Po).
3 REGRESSION OF WASSERSTEIN DISTANCE ONTO SLICED OPTIMAL TRANSPORT DISTANCES
In this section, we present framework for regressing the Wasserstein distance onto sliced Wasserstein distances, propose some models, and discuss related computational properties.
3.1 SLICED WASSERSTEIN AND LIFTED SLICED WASSERSTEIN
Sliced Wasserstein distances. Given μ E P. (Rd) and v E Pp (RR), sliced Wasserstein-p distance
can be defined as follows (Rabin al. 2012 Nguyen 2025
SW (1, 120) = Eg WP v; Pa)]
(5)
where Po Rd + IR is the projection function, W v; Po) is the one-dimensional projected Wasserstein distance (equation 2). and a P(Sd-1) is the slicing distribution. By changing the slicing distribution, we can obtain variants of SW. There are three main ways:
1. Fixed prior: The simplest way is to choose to be fixed and known distribution, e.g., the uniform distribution (S as in the conventional SW (Rabin et al. 2012).
2. Optimizaaionn-basel We can also find a that prioritizes some realizations of that satisfies notion of informativeness. For example, a can put more masses to realizations where (P,,, v; Po) have high value, i.e., setting informativeness discrininativeness. For example, can find by solving [Nguyen et al. 2021):
sup Engo WP us Po)], MM
1)
where 1( P(S be set of probability measures on 1d-11 When (( {ōo E gd-1 I). 11, max sliced Wasserstein distance (Deshpande et al. 2019 is obtained: Max-SW (H, v) maxges*-: W,,,, P.)).
Energy-based An optimization-fee free way to select a is to design it as an energy energy-based based distribution with the unnormalized density: po f(WT (1, Po)), where is often chosen to be increasing function on the positive real line, i.c. an exponential function. This choice slicing distribution leads to energy-based SW (EBSW) (Nguyen & Ho 2023
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Empirical estimation. For SW, Monte Carlo estimation is used to approximate the distance:
SW p, On, ...... 0r.) 1 Σ 1;
where 01,..., Oz ii.d u(s (L 0) are projecting directions (other sampling techniques can
also be used Nguyen al. we can use Or which is the solution of an optimization algorithm with T 0 iterations, e.g..
2024; Nguyen & Ho 2024 Sisouk et al. 2025)). For Max-SW,
projected gradient ascent (Nietert et al. 2022 or Riemannian gradient ascent Lin et al. (2020):
Max-SW p (1,, v; Ūr) W5(n, us P. For EBSW, one simple way to estimate the distance is to use importance sampling:
EBSW 2 (I...,, = Σ w. W
(p.....) Σ/ (WWgaers and 01....,00. 02 ~
where di
Lower bounds. We summarize the connection between SW, Max-SW, EBSW, and Wasserstein distance in the following remark. The detail of the proof can be found in Nguyen & Ho (002)).
Remark 1. Given any (RR()) and (IR))), we have:
(a) SW,(,,)) S EBSW, (U, S Max-SW, p(u, v) S Wp(u,v).
(b) SW 21 ....,... ) S EBSW ((, v; .
) S Wp(,,) v) for any 0,....,,0r 0g gd-1,
(c) Max-SW" ((, v, S Wp(,, v) for any 02 E Sd-1
Lifted sliced Wasserstein distances. Given Pp (R( and E (Rdd), lifted sliced Wasserstein- p distance can be defined as follows (Rowland et 2019):
LSWP Egno
W po)
(6)
where Po Rdd - R is the projection function, W W (1, Vi Po) is the SWGG (equation 3 and P((á-1)) is the slicing distribution. Similar SW, we can obtain variants of PW by choosing ø.
1. Fixed prior: The original LSW is introduced as in projected Wasserstein (PW) in Rowland et al. 2019) which uses the uniform distribution 21 (Sd-1).
2. iiimizzation-bseed In contrast to the case of one-dimensional projected Wasserstein, which is always a lower bound of Wasserstein distance, SWGG is always an upper bound of Wasserstein distance. Therefore, it is desirable to select 0 that can minimize the corresponding lifted cost, that leads to min SWGG distance: Min-SW (G,(H,,))) min W pp vz Po)
06S--1
3. Energy-based: Similar to the case of EBSW, authors in Liu et al. 2025 proposes to choose a as an energy-based distribution with the unnormalized density: Pa (=(() (f)
J( (p, v; Po)), where
is often chosen to be an exponential function with temperature. The authors name the distance as expected sliced transport (EST).
For PW, Monte Carlo samples are used to approximate the distance: PW" (1,, v; (y,,..,,,)) = + Σt Woln, v; Po.), where 0,, 0,,...,0,. i.d u((d-1). For Min- oy which is the solution of optimization algorithm with T 0 iterations, e.g.. simulated annealing (Mahey et al., 2023), gradient ascent with a surrogate objective (Mahey et al. 2023), and differentiable approximation (Chapel et al. 2025): Min-SWGG, (14, v; P For EST, importance sampling estimation is used:
Empirical estimation.
SWGG,
EST v: 01 (.....,.
02)) Σ wi W (H, Po.),
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f(- W. Po.) where aiy Σb S(-W (a,e:Po
and 01....,,0 Or ~ (((-11))
Upper bounds. We summarize the connection between PW, Min-SWGG, EST, and Wasserstein distance the following remark. The connection between Min-SWGG, EST, and Wasserstein distance is discussed in Mahey al. (2023); Liu et al. (2025). The connection between EST and PW can be generalized from the connection between EBSW and SW iguyen & Ho (2023).
Remark 2. Given any μ E (R and v E (IR)), we have;
(a) W v) S Min-SWGGp(u..0 EST,,,, PW,(,,
(b) (,(,, v) S EST plu, ,01) 01)) u; for any .1....,0r E gd-1, (c) ( S Min-SWGG, GG (1t, v; 0r) for any Or g
3.2 REGRESSION OF WASSERSTEIN DISTANCE ON SLICED WASSERSTEIN DISTANCES
We consider the setting where we observe pairs of distributions (μ) v1).... (uN, UN) ~ P(u, Here, P(u v) is the meta distribution, and we are interested in relating W, Wp(n,, vj) with > OSW distances Sp (14, v4), SS
{) (Hi 241) for ,. N. We first start with general model.
Definition 1 Regression of Wasserstein distance onto SW distances). Given meta distribution P(U,v) E (RPP(R.) Pp (R())) K > SW distances 5(), Sp a regression model of Wasserstein distance onto SW distances is defined as follows:
wp(nuv) [(5 (1, ( (100 +
where (1, ~ F(u, v), J E the regression function, and is a noise model such that EE(s)
To estimate f. one natural estimator the least square estimate:
(1(S)) (H, ,,, SKK -1,,))-W,(,,, w)))
FLSE - arg min E fEF
is worth noting that the function f can be constructed in both parametric ways (e.g., deep neural networks) or non-parametric ways (e.g., using kernels). However, in order to have a simple and explainable model, we consider linear functions in this work.
Linear Regression of Wasserstein distance onto SW distances. We now propose linear estimations of Wasserstein distances from SW distances.
Definition 2 (Linear Regression of Wasserstein distance onto SW distances). Given a meta distri- bution P(u, v) E ((P,(R4) x Pp (R.)), K > SW distances 5(1) Sp s(K), the linear regression model of Wasserstein distance onto SW distances is defined as follows:
Wy(m,v) Σ
wh
S(h) (H, +
(9)
where v) ~ P(u, v) and e a noise model such that B[e] O.
Again, we use least-squares estimation to obtain an estimate of Remark 3. The least square estimator admits the following closed form:
WT.S E (pp(,,)S P) E [Sp(u, D) W),, v)),
(10)
where Sp(H,v) (S (1, ..... st (H,z))
The detail of Remark in given in Appendix A.1 In practice, we can sample
(1I,)))..... (uM, P(u, v) approximate the expectation in equation 10 Let Š E B be the SW distances matrix i.c., S.tt s(k) (m, v;) for i = I,..., M, and W E RM be the Wasserstein distances vector i.e., W: Wp(ms, v) for 1,..., M, we have the sample-based least-squares

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W
w
(1,
W,(a)))
SU,())))
wSL,(un) (1-))((())
Span(Si)).. ....,."h g10)
Figure 1:
Linear regression of the Wasserstein distance vector W on sliced Wasserstein (SW) distances , S(K), distance onto the linear span of the SW distances. The right figure depicts a special case of constrained linear model with only two SW distances as predictors, which can be seen midpoint method.
S(K), The left figure illustrates a linear model, interpreted as the L2 projection of the Wasserstein
estimate: WLSF S - W, which is an unbiased estimate of w. is well-known that the linear model can be seen as L2 projection of the Wasserstein distances vector W onto the linear span of the SW distances vectors 51)) SKK) We illustrate the idea in the left figure in Figure
we know that SW distances are either lower bounds or upper bounds Wasserstein distance. Therefore, natural estimation can be formed using midpoint method. In particular, given lower bound pL 1/) and upper bound (1, v). we can predict the Wasserstein distance as w1 STpSST, 2) S2SSUp( v) with w1 S 1 and w2 Definition 3 (Constrained Linear Regression of Wasserstein distance onto SW distances), Given meta distribution P(u, v) P(P, x (R.))), K > 0 SW distances SLP SL. (K) which are lower bounds of Wp and K > O SW distances SU SU., which are lower bounds of W the constrained linear regression model defined as follows:
From Section 3.1
w1-
Wy(H,
1 K
Σ
SWSL (H,
E(1- (x)w,) (p,z) tE,
(11)
where 0 0. W/: S 1, (H, P( and a noise model such that Ela] = 0.
To estimate w = (wi,..., wx) under the constrained model, we again form the least square estimate, which can be solved using quadratic programming and Monte Carlo estimation. In a special case where K = 1, i.e., having one lower bound and one upper bound, can have closed-form.
Remark 4. For the case K = 1 with a lower bound SLp( v) and an upper bound SUp v/), closed-form of the least square estimate under the constrained model can be formed:
E (SS SL,((,,)) B[(SUp(H,V) - -0lp,(u,,)~LL((,,))))))
(p(,,u))
WCLSE
(12)
The detail of Remark in given in Appendix A.2
The corresponding sample-based estimator for the model is: WCLSE ISU, (SU,,,H -S, (e,,
We show the idea in the right figure in Figure T Compared to the unconstrained model, the constrained model has half of the parameters. addition, it adds inductive bias to the model, which often helpful when having
+ IM,(SU, (,SU,((m, (M...) (.S...))SL(p(....1
limited observed samples.
Wasserstein Distance Estimation with Few-Shot Regression. We recall that we observe (M1, vI)..... (uN, (NN P(u, v) in practice. It is not computationally efficient to compute the discussed least square estimates using all pairs of distributions since those estimates require evaluation of Wasserstein distances. We then sample subset (n', 1/),...... (M'y, v/4g) from the original set with M << N. After obtaining an estimate & from (H², 14),... (M'y, t/r)). we can form estimations of the Wasserstein distances for other pairs and any new pair of distributions given their SW distances.
Computational complexities. We assume that N pairs of distributions have the number of supports be at most n and in d dimensions. For fitting the estimate on M pairs, we need to compute M K SW distances (using L projecting directions) which costs O K In([og + d)) in time and M
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Table 1: K-NN accuracy on point- classification on ShapeNetV2 dataset.
Methods
nta
1=1
k=3
k=5
k==10
k=15
WD
88.6% + 0.0%
83.5%
+: 0.0%
84.2% +: 0.0%
82.9% + 0.0%
79.2%
+: 0.0%
RG-s
0.868 + 0.02
82.1%
+ 0.1%
81.7%
+ 0.1%
80.8%
0.1%
79.4% + 0.2%
75.5%
+ 0.2%
RG-e
0.926 4: 0.04
62.5%
0.1%
82.25
80.9%
0.2%
79.6%
0.3%
75.7%
+ 0.3%
RG-1
0.774 1 0.38
65.1%
0.3%
67.7%
0.3%
67.6%
0.5%
66.7%
1 0.5%
66.0%
1 0.5%
0.933
52.5%
0.4%
82.2%
82.6%
81.9%
0.5%
RG-sco
0.937 + 0.01
82.8%
+ 0.4%
83.3%
+ 0.5%
83.5%
0.7%
82.3%
+ 0.7%
77.9% + 0.7%
Wasserstein distances which costs (' log d)). Computing the least square estimate has
the time complexity of O(MK2 + k²). Then, we compute (N - M SW distances which costs D(N-M) Ln(log -d) and predict (N-M) Wasserstein distances which costs ((NNNMM)) Total time complexity is ONNLLn(log n + d)) + Mn² (n log n + d)) + M K + (N - k) compared to ONN (n log + d)) of computing Wasserstein distances for all N pairs.
Extensions on regression. In this work, we focus on regressing the Wasserstein-p distance. If other ground metrics are used e.g., geodesic distances manifolds, variants of SW distances such as spherical sliced Wasserstein distances (Bonet et al. [202aa] Tran et al. [024] Quellmalz al., 2023), hyperbolic sliced Wasserstein distances (Bonet al. 2023b), sliced Wasserstein for distributions over positive definite matrices (Bonet et al., 2023c), and other non-linear variants of sliced Wasserstein (Bonet al., 2025; Chapel [ [ Tanguy et al. [2025] Kolouri et al.,
2019). However, they might not be upper/lower bounds of the corresponding Wasserstein distances. Moreover, to incorporate uncertainty quantification, we can also perform Bayesian inference (Box &
Tiao 2011). e.g., putting prior on the regression function.
EXPERIMENTS
We define some specific model instances: RG-o uses Max-SW and Min-SWGG as predictors; RG-s uses SW and PWD as predictors; RG-e uses EBSW and EST as predictors. We also consider two extensions: RG-se combines EBSW, PWD, and EST, and RG-seo combines all six variants. For each instance, we have constrained version and an unconstrained version discussed.
We evaluate our methods in several parts, each with a distinct goal. First, in Section 4.1 we test practical use via k-NN on ShapeNetV2, reporting accuracy under different metries. Second, in Section 4.2 we benchmark RG variants against Wormhole across MNIST point clouds, ShapeNett2, MERFISH Cell Niches Zhang et al. (202T). and seRNA-seq atlas Persad al. (2023), reporting M//MSE/MEE in low-data regimes. Third, in Section 4.3 we combine our framework with Wormhole to introduce RG-Wormhole, a hybrid that matches Wormhole's performance while requiring far less training time. We compare training time under varying batch sizes and epochs, as well as embedding, reconstruction, barycenter, and interpolation quality, In Appendix [B.I] we run MoG simulations to verify that our methods approximate the true Wasserstein from low to high dimensions. In Appendix B.3] we visualize metric-induced geometry with UMAP Mcinnes et al.) (2018). In Appendix B.6 we compare RG with classical methods in the many-pain setting. To illustrate robustness, we further evaluate the method under inter-class and intra-class settings in Appendix B.7 Finally, we investigate whether there are consistent patterns in the optimal RG weights across datasets; see Appendix B.S. Throughout, N denotes the number of training-set set sizes, and My the number of samples drawn from the training set, yielding M Mo(Ma pairs used to estimate RG coefficients.
2
4.1 POINT CLOUD CLASSIFICATION
We evaluate unconstrained RG variants over classification task over 10-class ShapeNetV2 with 500 training samples (5=500) and estimate RG weights from 10 samples (MM=10) drawn from the training set. The details of the experimental setting and full results are provided in Appendix B.2
Results. Table reports NN-N accuracy on ShapeNetV2 under different metrics. As expected, WD achieves the best accuracy, with 84.2% at k=5. Among single sliced-based metrics, SW and EBSW, are the strongest, though they cap at about 72.5% top-1. Our RG methods close much of the gap to Wasserstein. Both RG-s and RG-e consistently achieve around 82.5% top-1 accuracy with high
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Table Approximation quality of Wormhole and RG variants across four datasets under training set size 100 samples. Each cell reports H², MSE, and MA) with respect to the exact Wasserstein distance.
MNIST
S2
MERFISH
scRNA seq
Methods
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
Wormhole
1.28
4.3
10
0.55
7.0
7.8
(cunatr.)
0.94
10
2.3xx
1um1
1.8
3.0
19
3.7
RGre (coner)
0.66
2.3
1.7
10
1.0 10
18x
10
3.0 10
100
1-3
10
1.0
RG-0 (consir))
0.77
14
10
20
0.06
6.2
10-3
188
10
0.75
4.8
10-3
6.0 10-3
0.99
6.1
10
0.0
10-3
RG-a (cmmir))
0.8
9.5
10
10-1
0.92
14x
10
8.3
10
091
1.5
10
3,0%
10
1.00
2.4
10
3.0
10-3
RG 300 (constr)
0.85
9.0
20
0.1
1.7
1.0
10
0.2
18x 10
3.0% 10
1.00
2.2
2.0
(Inconor:)
(unconsez)
7.0
(unconatr.)
0.77
3.
6.7
2.8
6.9
se (unconsrr)
883 4.0
10
1.5 10
099
0.9
10
7.8
098
29
10
1.0 10
100
3.0
10
4.0
1073
RG-aco (cncoralt))
A.93
4.0 10
1 10
0.8
9.8
30
7.8 10
097
8.8
x 10
2.0
0.99
0.6
7.0
airplane
lamp
chair
data
RG-seo Wormhale
0.103
Figure ModelNet40: HG-Wormhole variant in reconstruction experiment.
correlation Wasserstein (R2 0.9). The multi-metric extensions further improve stability: RG-se and RG-seo reach up to 83.5% accuracy with R² as high as 0.93, essentially matching Wasserstein.
4.2 COMPARISONS OF RG VARIANTS Vs. WORMIIOLE IN LOW-DATA REGIMES
We compare our RG framework with Wormhole within the same training sizes, matching the pre- processing of (Haviv et al.) 2024) across four datasets spanning dimensionality: MNIST pixel point clouds (2D), ShapeNetV2 point clouds (3D), MERFISH Niche Cells (254D), and scRNA-seq (2,500D). We train N E {10, 50, 100, 200) random pairs and evaluate RD MMSE/MA against exact WD. For fairness, the number training pairs for Wormhole equals the number used to estimate the linear coefficients for RG variants, i.e., Mo=N Full results appear in Figures X 13 with settings in Appendix B.4
Table summarizes the Mo= 100 case, and other Mn follow the same pattern.
Results. Across all four datasets, RG variants consistently outperform Wormhole small training sizes. Wormhole is weaker primarily because it is data hungry and its performance improves as add samples, yet under comparable budgets it still trails methods. By contrast, RG variants are already accurate with few pairs, with unconstrained variants are slightly stronger, whereas constrained variants converge faster and are preferable the very smallest sizes. RG-se and RG-seo are the strongest when given sufficient samples though the latter can lag at the tiniest sizes before its weights settle but becomes top pop-eerfonnng quickly and still requires far fewer pairs than Wormhole.
4.3 RG-WORMHOLE: ACCELERATING WORMHOLE WITH REGRESSION OF WASSERSTEIN
The previous comparison reveals clear trade-off. RG framework is lightweight and data-efficient, but it does not produce Euclidean embeddings and therefore cannot support interpolation experiments. Wormhole, in contrast, learns Euclidean embeddings that enable interpolation and reconstruction. but is computationally heavy because training requires many Wasserstein evaluations (pairwise distances within each mini-bated and reconstruction losses), which slows and raises training cost.

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RG-seo Wormhole
Lamp
Lamp
Figure 3: ModelNet40: RG- GG-Woomhhle variant in interpolation experiment.
RG- 2G-Wormhole. To combine the strengths of both, we introduce PG-Wormhole. We first calibrate a RG surroaate on small set of exact Wasserstein pairs from the same data domain and freeze its weights. We then keep the Wormhole architecture, optimizer, and schedule unchanged, and simply replace every use of the Wasserstein distance with the calibrated surrogate in both the encoder (pairwise distances in the batch) and the decoder (reconstruction loss). No other component is modified. This substitution makes each training step much faster while preserving the performance.
We run five experiments of both models to empirically show that RG- Worrhoe much faster than Wormhole while keeping similar effectiveness. First, we measure training time by training Wormhole and RG- RG-Wormhole under the same optimizer and schedule, sweeping batch sizes 4-20 and reporting wall-clock time for training-set sizes N (10, 50, 100, 200). Second, assess encoders via R /MEE//M between learned pairwise distances and exact Wasserstein. Third, we evaluate decoders via the Wasserstein loss between each input shape and its reconstruction. Fourth, we examine barycenters by decoding each class's 's mean embedding and visualizing results. Finally, we study interpolation by decoding linear paths between two embeddings and visualizing the trajectories. Across all experiments, hyperparameters match Wormhole; the only change in RG- "G--ormhhoee is replacing every use of the Wasserstein distance in the encoder and decoder losses with the calibrated unconstrained RG variants. For RG-Worrhoold we estimate the RG coefficient using 10 random training samples 10)=10) before plugging into Wormhole. We provide some results in Figure though the details of experimental settings and full results can be found in Appendix B.5
Results. Replacing every Wasserstein call Wormhole with a calibrated RG variants preserves per- formance while cutting compute. First, in the training-time comparison (Figure 14 in Appendix B.5 RG-Wormhole is far faster than Wormhole across all batch sizes and training budgets, with a very large gap. As batch size increases, Wormhole's 's time grows almost exponentially. while RG- Wormhole rises only slightly. close to linear or even flat. Next, we verify that the trained models have similar quality. For the encoder, Figures 15 and 16 in Appendix [B. show pairwise distances that align with the ground-tre -tunh Wasserstein and embeddings that match Wormhole, with essentially identical R², MSE, and MAE. For the decoder, Figures a Appendix evaluuaeee reconstructions against the original point clouds using the Wasserstein distance, and both RG- Wormholle and Wormhole pro- duce very small and nearly identical distances. Finally we test whether NGG-Wrrhhole preserves the geometry needed for downstream use. The decoded class barycenters from RG- Wormhole are clean and class consistent and they match those from Wormhole, we refer to Figure in Appendix [B.5] We also interpolate by moving linearly in the embedding space and decoding along the path, and the trajectories from RG- Wormhule are smooth and semantically meaningful with no visible artifacts, we refer to 20 in Appendix B.5 Overall RG- GG--oomhole matches Wormhole while training much faster, which makes a practical choice when compute is limited.
CONCLUSIONS
We introduced a regression framework mapping Wasserstein to sliced Wasserstein distances under meta-distribution of random distribution pairs. Two simple linear models enable lightweight estimation, leading to the RG framework for few-shot Wasserstein approximation. We derived constrained and unconstrained forms and validated them through Mixture of Gaussian simulations, point cloud classification, and metric-space visualizations, where the surrogate closely matched the exact distance. Compared to Wormhole MNIST, ShapeNetV2, MERFISH, and scRNA-seg, method achieved better performance in low-data regimes. Replacing Wasserstein calls in Wormhole with our method yielded IG--orrhooe, preserving accuracy while greatly reducing training time.
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Supplement to "Fast Estimation of Wasserstein Distances via Regression on Sliced Wasserstein Distances"
A DETAILS
A.1 DETAILS OF REMARK 3
We derive the gradient:
E
w
T s(h) (1,20 - w
VwB
(w ((1)) (H,D) - W 1)))
(w ( (M, - wo(,
VWB
T S (1,, 5) (1,, v)
Wi ( (M, 2) wWw
(13)
= E Vow 1/40) (1, S (1.2)
2E V (1,, (WW
wWp(H1,)
(14)
= 2 5 (1, DS R (1,, U)T 21F
S (n,
(15)
Setting the gradient to O, we obtain
WLSE (V,))) (u,
E ( w), (),
(16)
which completes the proof.
A.2 DETAILS REMARK
From the definition, we recall the model:
K
K
w)(,,v) = Σ (,, + Σ (1- - wn) (M,
(H, v) +e.
(17)
k=1
k=1
With K 1, we rewrite the model as follows:
Wp(u,v) ( WSL.
SU +
(18)
which is equivalent
W Wp(,vv) ((((,,) SUp(M, v)) +c.
(19)
Since equation 19 is again an unconstrained linear model, we can obtain the least- squares estimate by following Appendix A.I
E (L, W wp(,
WOLSE
(20)
B((SU,(U,V) -
which concludes the proof.
B EXPERIMENTS
B. GAUSSIAN SIMULATION
We study how fower-upper bound pair approximates the Wasserstein distance as dimension grows. We simulate 3-component Gaussian mixtures for d=1... 100 (10 seeds), with 200 points per component, For each pair we compute the exact Wasserstein and six sliced-based metries. Focusing on RG-o, RG-s, and RG-e, we fit a constrained weight w E [0, 1] and report the estimated weight w and R² versus the exact Wasserstein.
Results. We refer to Figure 4 for the result. The fits are strong for all three methods and all dimensions: R² always above 0.8 and quickly rises to z 0.9-1.0. We also see a clear pattern the weights: dimension grows, the weight on the lower bound goes down, so the upper-bound metric gets more weight and eventually dominates. In short, high dimensions favor the upper bound, while lower dimensions rely more the lower bound.
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Oytional ase desension: Ravo
Oytimaal and disenddent
Optinal dwrdinnniio
netedar
734
Figure 4: Optimal w" and R2 in each dimension.
Table 3: k-NN accuracy on point-cloud classification ShapeNetV2 dataset.
Methods
R²
k=1
k=3
k=
k=10
k=15
WD
83.6% + 0.0%
83.5% + 0.0%
84.2% 1 0.0%
82.9%
+ 0.0%
79.2%
+ 0.0%
SWD
72.4%
0.0%
71.4%
0.0%
70.4%
+ 0.0%
69.0%
0.0%
66.7%
0.0%
PWD
42.6%
0.0%
42.9% 0.0%
40.4%
+ 0.0%
39.3%
1 0.0%
39.0%
0.0%
EBSW
72.5%
0.0%
69.2%
0.0%
60.4%
1 0.0%
0.0% 67.9%
65.3%
0.0%
EST
Max-SW
60.3% 1. 0.0%
54.6% 1 0.0%
57.7% 1 0.0%
57.6% 1 0.0%%
56.8%
0.0% 10.8%
0.0% 37.6%
1: 0.0%
32.9%
RGss
0.868
0.02
82.1% 0.1%
81.7%
0.1%
80.8% 1: 0.1%
79.4%
1 0.2%
75.5%
0.2%,
RG-e
0.926
0.04
0.1% 82.5%
82.2%
0.1%
80.9%
1: 0.2%
79.6%
0.3%
75.7%
0.3%
RG-o
0.774 = 0.38
65. 1%
0.3%
67.7% + 0.3%
67.6%
+ 0.5%
66.7%
+ 0.5%
66.0%
0.5%
RG-se
0.935
82.5%
RG seu
0.937 0.01
82.8% 0.4%
83.3% 1 0.5%
83.5% 1 0.7%
82.3% 1 0.7%
77.9%
0.75
B.2 POINT CLOUD CLASSIFICATION
Experimental settings. We construct 10-class subset, centralize, normalize each shape so that all coordinates lie in [-1, 1]3, and uniformly subsample 2,048 points per shape. For each class select 50 training examples and 100 test examples. We then compute pairwise distance matrices between train and test sets under different metrics, and evaluate classification accuracy using k-nearest neighbor classifier with k E (1,3, 5, 10, 15). Besides the six individual sliced-based metrics, we include all RG variants in unconstrained version. We 10 samples drawn from the training set estimate the linear coefficient of RG variants.
B.3 METRIC SPACE VISUALIZATION
Experimental settings. We visualize the geometry each metric induces on ShapeNetV2. From 10 categories, randomly sample 500 shapes per class, normalize each shape that all coordinates lie in 1]3 and keep 2,048 points per shape. For every method, we compute the pairwise distance matrix, then feed to UMAP to obtain 2D embeddings. We 10 samples drawn from the training to estimate the linear coefficient of RG variants.
The result is visual in Figures 5 Across methods, the true Wasserstein produces well- separated class clusters with clear margins. The RG variants produce embeddings that are visually very close to the Wasserstein embeddings, preserving both local compactness and the global arrangement of classes. By contrast, single sliced baselines are weaker. SWD and EBSW keep some structure but blur boundaries, while Max-SW and Min-SWGG show more mixing and noise.
Results.
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ShapelletV2:
Embeddings
Embeddings
ShapeNetV2: Embeddings
ShapeNietV2: Embeddings
ShapeleetV2: Embeddings
ShapetietV2: Embeddings of EBSW
ShapeNatV2: Embeddings of EST
ShapeNittV2: Emboddings of RG
ShapeNetV2: Embeddings of RG-se
ShapelletV2: Embeddings of RG-seo
ShapeNetVV: Embeddings of Wasserstein
Figure Embeddings of methods ShapeNetV2 datasct.
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Table 4: Wormhole training hyperparameters.
Component
Setting
Batch size
Optimizer / LR
Adam, 1r 10
LR schedule
ExponentialL.B final factor = 0.1 over all epochs 2,000 epochs (20,000 steps)
Epochs
Transformer depth Attention heads
num_layers 3
num heads =
Embedding dim
emb dim 128
MIP hidden dim
mlp_dim = 512
Attention dropout Decoder coeff.
0.1
coeff_des = 0.1
B.4 COMPARISON OF RG VARIANTS VS. WORMHOLE IN LOW-DATA REGIMES
Experimental Settings. We compare our proposed RG framework against Wormhole, a state-of-the- art Wasserstein approximation method. To ensure fairness, follow the exact preprocessing protocol of Haviv et al. (024). We consider four datasets spanning wide range of dimensionalities: (i) MNIST point clouds, obtained by thresholding x 28 grayscale images and treating the active pixels as 2D point coordinates; (ii) ShapeNetV2 point clouds, where each CAD model is uniformly sampled into 2,048 points in 3D and normalized; (iii) MERFISH Cell Niches, where each cell represented by the 50 μm neighborhood of its cene-expression profile embedded in 254-dimensional space; and (iv) scRNA-seq atlas data, where cells are aggregated into MetaCells that form -.m00-dimeniional cene-expression point clouds. We vary the number of training pairs N E {10, 50, 100, 200} by drawing pairs uniformly, and evaluate on 10,000 independently sampled test pairs. For each dataset and training size, report R MSE, and MAE with respect to the exact Wasserstein.
The original Wormhole codebase is built on JAX and TensorFlow, which are not compatible with our environment. Accordingly, we reimplemented Wormhole in PyTorch.
Data Preprocessing. We follow the same preprocessing pipeline as Haviv et al. 2024).
MNIST Point Clouds. We turn MNIST 28x28 images into 2D point clouds by thresholding pixel values at 0.5 and keeping the coordinates of the active pixels.
ShapeNetV2 Point Clouds. We use ShapeNetCore..v v2 with 15k points per shape. Each shape is normalized to fit inside a unit cube with coordinates in [-1, 1]s We then split each shape into 10k training points and 5k test points, and randomly sample 2,048 points from each point cloud.
MERFISH Cell Niches. We scale each gene's expression to [-1, 1] and divide by vd, where di is the number of genes. For each cell, spatial positions to find its 11 nearest neighbors within a 50 pm radius, keeping only cells with enough neighbors with its cell-type label.
scRNA-seq. We select 2,500 highly variable genes, normalize counts (library-size 10a and log(1+z) (xx)),)),, and scale each gene to [-11 1] divided by Vd (d-2500), We then cluster cells with K-means. For each cluster seed, we consider it as cloud, labeled by the seed's annotation.
Wormhole training hyperparameters. We follow the Transformer autoencoder setup of Wormhole with the configuration in Table 4
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PAIST PoRT Cless Worethode Masberstels
PAIST PORT Creer Warrenoe Massorstats
NAIST PoRT Clea Werrinde Wassersdde
PINST PORT CrasEt Morehole
30 12:11
PUTTI Choul 8D siaeernd Moseense
MUSTPPU Clicl 8D sharerd
Figure 6: MNIST Point Cloud: Wormhole and RG variants (comstraineeuuncosstraeee))) across training set sizes of 10, 50, 100 and 200.
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PAIST PoRT Cleas: Worehhole Masberstess
PAIST PORT Coeer Warrenoe Massorstats
NAIST PoRT Clea Werrinde Wassersbde
PINST POLT Classt Morehole
38:110"
MEST Aold Doath Messenees
MSST Nold Wassente
MEIT Nowd Ovach Mitt Jomorett Weyonte
MSIT Nor Crech Miro Rereeste
WHIET Peld Crach
Figure 7: MNIST Point Cloud: Wormhole and RG variants (comstraineeuuncossttaeeee)) across training set sizes of 10, 50, 100 and 200.
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Spapelerri:: Morritiie Messersiin
Stagemerr2: Mornhole Misserstin
Stapeherr2: Wornhhole Wissorsten
Shopeletr2: Warshole Wasterstein
113
30 38:110"
Plyyrbit BD anteerti
Habie Mic.
Figure 8: ShapeNetV2 Point Cloud: Wormhole and RG variants (constraine/uuncosstrineee) across training set sizes of 10, 50, 100, and 200.
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Spapelerr2:: Morritile Messersiin
Stagemerr2: Momahole Missrrstin
Stepeterr2: Wornhhole Wissorsten
Shopelerr2:: Warshole Wasterstein
10:10"
13:120 112
30 33:12"
30 38-112"
Pogustest Miro Emmormtt Recoontlle
Esyulaathh Min IomommAc Raccassol
BhacsborS Mire Lomoomtti Ricosson
Figure 9: ShapeNetV2 Point Cloud: Wormhole and RG variants (constraincduucconstrinnne)) across training set sizes of 10, 50, 100, and 200.
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MARFISH LeB niches: Monsholl Wasserrtin
MARIISH CAR Niches: Monehole messersten
MINIIIH Call Miches: Morreiole Misserstee
PILIITSH Call Miches monehole nioieeoil missorrite
30
MIROEe Set Mitres Bibe Wossenside
PERRES botthiiee Mhr Eoseete Rosessie
Figure 10: MERFISH Cell Niches: Wormhole and RG variants (constraineduunconstraneee)) across training set sizes of 10, 50, 100, and 200.
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MARFISH CeB niches: Monsholl Wassersttn
MARIISH CAR Niches: Momehole messersten
MINIIIH Call Miches: Morreiile Misserstte
PILNITSH Call Miches: monehole missbrrite
MENABB Call
HISrID Det
Figure 11: MERFISH Cell Niches: Wormhole and RG variants (constraineduuncoosttaanee)) across training set sizes of 10, 50, 100, and 200.
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200 oeg Arte: Momehole an Massrrstn
SCAMA-BE DEE ARMEL Monehtol Missorsten
SCAMA ARMe th nohole Mastorstiin
SCAMA sne ARMIL rehoode sorstein
-0
0uthh barotol Mesteriet
00 Fith Someto Weseesbee
00030 iitr boretal Werestee
197000 M0 Icemed Ressoman
0.5
Figure 12: scRNA-seq: Wormhole and RG variants (constaiinc/uncoosttaanee)) across training set sizes of 10. 50, 100, and 200.
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100 oeg Artea: Momeholl an Massrrttn
SCAMA-BB DEE ATME Monehtol Missorsten
SCAMA one Afte to nohole Mastorstiin
SCAMA sne Aftec rehoode Mai
12
- Resemstee
A08M5rogg Comoeeeta to Rosomatee
Figure 13: scRNA-seq: Wormhole and RG variants (ounstaiinc/unnoosttaaneee)) across training set sizes of 10. 50, 100, and 200.
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Figure 15: V22: RG- KG-Wormhole (constrained model) vs. Wormhole.
Figure 16: ShapeNetV2: RG-Wommhole (unconstrained model) Wormhole
B.5 RG-WORMIIOLE: ACCELERATING WORMITOLE WITH REGRESSION OF WASSERSTEIN
Experimental Settings. We run five experiments to show that RG- RG-Wormhole is much faster than Wormhole with similar effectiveness. First, we measure training time by training both models under the same optimizer and schedule, sweeping batch sizes from 4 to 20 and reporting wall-clock time for training sets of 10, 50, 100, and 200 pairs. Second, we assess encoders by computing R /MSE/AA between pairwise distances in the learned embedding space and exact Wasserstein. Third, we evaluate decoders by reporting the Wasserstein loss between each input and its reconstruction. Fourth, we examine barycenters by decoding the mean embedding of each class and visualizing results. Fifth, we study interpolation by decoding linear paths between two embeddings and illustrating trajectories. Across all experiments, hyperparameters match Wormhole; the only change in RG- Wormhole is replacing Wasserstein in encoder and decoder losses with the calibrated unconstrained RG. We use 10 samples from the training set to estimate RG coefficients. Except for embedding experiment which uses ShapeNetV2 dataset, other experiments use ModelNet40 dataset, same as Haviv et al.) 2024).
Training time comparison of Wormhole and RG-Wormhole methods on point cloud datasets with varying number of training samples.
Figure 14:
B.6 COMPARISON BETWEEN RG AND CLASSICAL APPROXIMATION METHODS.
Experimental Settings. We conducted direct runtime and accuracy comparisons between our methods and two widely used classical OT accelerations, Sinkhorn and Linear OT, under both single- pair computation and large-scale pairwise computations. For Sinkhorn and Linear OT, no training phase is required. For example, in RG-s, we estimate the optimal regression weight using 5 randomly selected samples (M = 10 training pairs). The training cost (2.891 seconds (s)) is not included in the inference-tne time table below, as we focus on inference time only. For Sinkhorn, we conduct a grid search over the entropic regularization hyperparameter in (0.01, 0.05, 0. ) and the number of
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airplane
cup
lamp
vase
thair
data
Reconstructon
0.058 airplane
0.039
044
0.047
0 053
cup
lamp
data
reconstruction
W 0.115 airplane
W 128
103
0.094
W 0.222 chair
data
reconstruction
= 0.061 airplane
0.1 1.121
084 lamp
0.090 vase
0. 090 chair
data
Wormhole
reconstruction
Figure 17: ModelNet40: RG- Wormhole vs Wormhole reconstruction experiment.
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airplane
cup
lamp
vase
chair
data
RG- Wormhole
reconstruction
0.105 airplane
0.133
0.079
0.156
098
cup
lamp
data
reconstruction
W 0.071
W 0.138
088
0.152
W 0.303
Figure 18: ModelNet40: RG- CGG-Wor reconstruction experiment.
iterations (500, 1000), and report the best-performing configuration for each dataset. We do not exceed 1,000 iterations because the runtime already surpasses that of exact Wasserstein.
For inference runtime comparison, we use point clouds from ShapeNetV2, where each object consists of 2,048 points in 3D, and measure inference time the number of pairwise computations in seconds increases (see Table 5). These time measurements are relative and may not be precise due system conditions.
For performance comparison, we evaluate each method's approximation quality relative to exact Wasserstein using R²2 in Table Each value is computed over 5,000 test pairs across four datasets: MNIST point clouds, ShapeNetV2, MERFISH cell niches, and scRNA-seq.
Results. From the results above, several observations are clear. First, Sinkhorn performs the worst among the compared methods. Even with 1,000 iterations, it fails to converge on MNIST point clouds and ShapeNetV2 (negative R²), while also being slower than both Linear OT and RG-s. However, this may be due to suboptimal choice of the entropic regularization parameter. In theory. the time complexity of Sinkhorn is O(NN 2 /c) for (N M)² pairs. Therefore, Sinkhorn is computationally expensive in this setting. Second, Linear OT provides reasonable accuracy and moderate speed, but its approximation quality is consistently below that RG-s, and its runtime grows much faster as the number of distributions increases. As discussed, the time complexity of Linear OT is O(NN M)n²/ /c) Third, the RG methods provide more accurate estimates than both Sinkhorn and Linear OT. In addition, RG-s, RG-e, RG-o, and RG-seo all faster than Linear OT. RG-seo achieves a speed comparable to Linear OT while offering better approximation quality, whereas optimization-bant based SW variants require substantially more computation time.
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Wermhole
airplane
bed
chair
lamp
tollet
Wormhole
airplane bed
chair
cup
lamp
tollet
Wormhole
bed
cup
tailet
Wormhole
airplane
bed
chair
cup
lamp
toilet
Wormhole
cup
RG-soo Wormhale
airplane
bed
chair
lamp
tallet
Figure 19: ModelNet40: RG- Wormhole barycenter experiment.
Table Comparison inference runtime among RG and classical approximation methods.
Samples (1 pair)
10 Samples 50 Samples
100 Samples
200 Samples
Method
(45 pairs)
(1,225 pairs)
(4,950 pairs)
(19,900 pairs)
Wasserstein
0.604s
13.351s
354.823s
1,427.276s
5,030.966s
Sinkhorn (iters=500)
0,441s
10.444s
286.203s
1,154.826
4,628.393s
Sinkhorn (iters=1,000)
0.482s
21,012s
572.441s
2,308.735:
9,336.621s
Linear OT
0.634s
3.141s
15,473s
31.121s
92.134s
RG-s
0.002s
0.090s
2.043s
8.15
21.408s
RG-e
0.0005s
0.023s
1.737s
7.060s
18.429s
RG-o
0.013s
0.136s
6.634s
15.310s
54.900s
RG-se
0,002s
0.113s
3,780s
15.170s
39.950s
RG-seo
0.013s
0.147s
7.017s
28.840s
94.200s
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Wormhalc
Lamp
Lamp
RG-o Wormhole
Lamp A
Lamp
RG-s Wormhale
Lamp
Lamp
RG-e Wormhale
Lamp
RG-se Wormhole
Lamp
RG-seo Wormhole
Lamp A
Lamp
Figure 20: ModelNet40: RG-Wormhole barycenter experiment.
Table 6: Comparison in performance among RG and classical approximation methods.
MNIST Point Cloud (10,000 test pairs)
ShapeNetV2 (10,000 test pairs)
MERFISH Cell Niches (10,000 test pairs)
scRNA-seq (10,000 test pairs)
Method
Sinkhorn
-1.970
-1.549
0.764
0.964
Linear OT
0,697
0,918
0.842
0.984
RG-s
0.925
0,942
0,964
0,989
RG-e
0.910
0.919
0.961
0.991
RG-o
0.767
0.753
0,809
0.923
RG-se
0.933
0.948
0.984
0.996
RG-seo
0.934
0.948
0.969
0.992
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Figure 21: ModelNer40: Intra-class Experiment.
Figure 22: ModelNer40: Inter-class Experiment.
B.7 EXPERIMENT INTER-CLASS AND INTRA-CLASS FOR RG.
Experimental Settings. We conducted experiment to test how well RG generalizes across intra- class and inter-class pairs. trained RG only on intra-class pairs and tested on inter-class pairs, and vice versa, using five scenarios for each setting ShapeNetV2.
Concretely, for the intra-class - inter-class setting, (A, - (A, B), we ran five experiments. In each experiment, we sampled 10 intra-class training pairs from (A, A), estimated the regression weight, and then constructed a test set by sampling 50 objects from class and 50 objects from class B. This yields 100 distributions in total and therefore 4,950 inter-class test pairs B).
For the inter-class + intra-class setting, (A, B) + (A, A), we again ran five experiments. In each experiment, we sampled five objects from class A and five objects from class B to form 10 inter-class training pairs (A, B), estimated the regression weight, and then built a test set by sampling 100 objects from class A, which yields 4,950 intra-class test pairs (A, A).
Although this is intentionally suboptimal (the regression weight is learned from a restricted subset of pairs), RG still achieves reasonably high lt2 (e.g., 0.902, 0,846, 0.932, 0,805), compared to about 0.95 when training on the full pair distribution. This shows that RG generalizes well out-of-sample data, with performance degradation that is present but moderate.
Results. We present the results in Figures 21
B.8 EXPERIMENT OF OPTIMAL WEIGHTS OF RG.
Results. We visualized the estimated coefficients of the RG methods for all applications in Figures 23 The coefficient patterns vary substantially across datasets, indicating that there is fixed set of coefficients that universally applies to all cases. This further underscores the need for the proposed regression framework, which captures the dataset-specific uncertainty in the association between Wasserstein and sliced Wasserstein distances (i.e., the meta-distribution) on the full pair distribution.
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MNIST Point Cloud: Optimal weights of RG (constr.)
1.0
0.0
Weight
02-
RGe
RG-o
RGs
RG
RG-sed
Figure 23: MNIST Point Cloud: Optimal weight of RG variants (constrained) across different training samples.
MNIST Point Cloud: Optimal weights (unconstr.)
16
14.
Mualty
10
08.
0.4
02
RG-
RGe
RG4
RG
8G-s00
Figure 24: MNIST Point Cloud: Optimal weight of RG variants (unconstrained) across different training samples.
ShapeNetV2: Optimal weights (constr.)
Songaree
Meslght
08
RG
RGo
RGs
RG-se
RG-500
Figure 25: ShapeNetV2: Optimal weight of RG variants (constrained) across different training samples.
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ShapeNetV2: Optimal weights (unconstr.)
1.25
0.75
0.8
RG-
RGJ3
RGs
RG se
RG-soe
Figure 26: ShapeNetV2: Optimal weight of RG variants (unconstrained) across different training samples.
MERFISH Cell Niches: Optimal weights of RG (constr.)
1.0
Weight
0.8
02-
R5se
R5-0
RGa
RG
NG-
Figure 27: MERFISH Cell Niches: Optimal weight of RG variants (constrained) across different training samples.
MERFISH Cell Niches: Optimal weights of RG (unconstr.)
Midee
RGa
RGo
RGs
RG-49
RG-180
Figure 28: MERFISH Cell Niches: Optimal weight variants (unconstrained) across different training samples.
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ScRNA-seq Atlas: Optimal weights of (constr.)
0.
RGe
RGo
RG3
RG
NF
Figure 29: scRNA-seq Atlas: Optimal weight of RG variants (constrained) across different training samples.
ScRNA-Seq Atlas: Optimal weights of RG (unconstr.)
Grygar
Weightt
-00
RG-
RGo
RG.s
RG-08
RG-Mo
Figure 30: scRNA-sco seq Atlas: Optimal weight of RG variants (unconstrained) across different training samples.
34


Reviews:
[{"review_id": "H6GH0lyDBb", "reviewer": "Reviewer_mmm4", "summary": "This paper introduces a novel regression-based framework for fast estimation of Wasserstein distances between probability distributions. Instead of directly solving the optimal transport (OT) problem or training deep neural networks (e.g., Wasserstein Wormhole, DWE), the authors propose to regress the true Wasserstein distance on a collection of Sliced Wasserstein (SW) and lifted SW distances, which act as lower and upper bounds respectively. \n\nTwo linear regression models are introduced:\nUnconstrained model, admitting a closed-form least-squares solution;\nConstrained model, which enforces the upper/lower bound structure with half as many parameters.\n\nOnce trained on a small set of distribution pairs, the model can predict Wasserstein distances for new pairs at the cost of computing SW distances, significantly reducing computational overhead.\n\nThe authors validate the approach on Gaussian mixtures, ShapeNetV2 point clouds, MNIST point clouds, MERFISH cell niches, and scRNA-seq datasets, showing that:\n\nThe regression estimator achieves high accuracy even with a few training samples.\n\nIt outperforms Wasserstein Wormhole in low-data regimes. Substituting it into Wormhole yields RG-Wormhole, which preserves accuracy while substantially reducing training time.", "questions": "### **Questions**\n\n**Q1. Notation confusion around $P(\\mu,\\nu)$ in Eq. (7)**  \nThe notation of $P(\\mu,\\nu)$ is confusing. From Eq. (7), it appears that $P(\\mu,\\nu)$ represents the *meta-distribution* used to define the expectation in Eq. (8). In this case, $\\mu,\\nu$ should be dummy random variables drawn from that meta-distribution. However, as currently written, $P(\\mu,\\nu)$ seems to be defined as a function of $\\mu,\\nu$ themselves, which are undefined at this point. Could the authors please clarify this notation and its relation to the sampling process in practice?\n\n**Q2. Sampling and training details**  \nHow are the training pairs $(\\mu_i, \\nu_i)$ sampled for regression fitting?  \n- Are they generated synthetically (e.g., Gaussian mixtures) or drawn from actual datasets (e.g., ShapeNet, MNIST)?  \n- What is the typical number of samples $M_0$ used for estimating regression coefficients, and how sensitive are results to $M_0$?  \nProviding more details would help assess reproducibility and stability.\n\n**Q3. Choice of sliced directions ($L$) and stability**  \nThe paper fixes $L$ as the number of slicing directions for each SW computation. How sensitive are the regression results to the choice of $L$?  \nIn high dimensions, random projections can yield high-variance estimates of SW distances—does the regression mitigate or amplify this variance?", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper presents a conceptually simple yet impactful framework for approximating Wasserstein distances through regression on sliced Wasserstein (SW) and lifted SW distances.\nThis formulation is the first to explicitly treat Wasserstein estimation as a supervised learning problem over a meta-distribution of random distribution pairs, bridging the gap between classical OT theory and data-driven surrogate modeling.", "weaknesses": "### **Weaknesses**\n\n**1. Limited contribution and unclear computational advantage**  \nThe main technical novelty of this paper lies in Equations (8) and (11), which propose to approximate the Wasserstein distance using a linear regression model on various sliced Wasserstein (SW) and lifted SW distances. While this formulation is conceptually interesting, I have several reservations about its contribution and computational benefits.\n\n**1.1 Computational complexity**  \nThe claimed time complexity is $O(MKL n(\\log n + d))$, where $n$ is the average number of support points per distribution, $M$ is the number of sampled distribution pairs, $K$ is the number of SW variants used, and $L$ is the number of slicing directions.  \nHowever, this is not necessarily more efficient than established OT approximations. For instance, Sinkhorn OT has a complexity of $O(n^2 / \\varepsilon)$ and can exploit GPU parallelism. It is therefore unclear in which practical regime the proposed regression estimator is computationally faster or more scalable. A more quantitative runtime comparison would be needed.\n\n**1.2 Lack of transport plan**  \nThe proposed approach only estimates the **transportation cost**, not the **transport plan**. Hence, it cannot support downstream applications that depend on the coupling (e.g., barycenters, alignment, gradient flows). This limitation makes it less general than classical OT accelerations such as Sinkhorn or low-rank OT methods. If the goal is to compute the Wasserstein distance for a single pair of distributions, this regression-based method would likely not outperform existing solvers.\n\n**1.3 Missing comparison to linear OT**  \nIf the intended use case is computing pairwise Wasserstein distances among many distributions, it would be natural to compare against **Linear Optimal Transport** (see [Bunne et al., 2020, “Linear Optimal Transport”](https://arxiv.org/abs/2008.09165)), which precomputes a reference map and achieves $\\mathcal{O}(C(K,2) n + K n^3)$ complexity for $K$ distributions, where O(n^3) can be improved to o(n^2/\\epsilon) if we use Sinkhorn to construct the embedding. Such a baseline would help clarify the advantages (if any) of the proposed regression-based approach for large-scale pairwise computation.\n\n1.4 \nI suggest including **direct runtime and accuracy comparisons** against classical OT accelerations such as **Sinkhorn** and **Linear OT** in two regimes:  \n(1) a single pair computation, and  \n(2) pairwise computations among many distributions.\n\n---\n\n**2. Minor technical and presentation issues**\n\n**2.1 Typographical error**  \nEquation (13) in Appendix A.1 appears to have a notational error: the term $\\omega^{\\top} S_p^{(k)\\top}$ is not correctly written, as both $\\omega$ and $S_p^{(k)}$ are vectors.\n\n**2.2 Unclear expression in Figure 1**  \nFigure 1 may be misleading. $W_p(\\mu,\\nu)$ is a scalar value, not a vector; similarly, each $S_p^{(k)}(\\mu,\\nu)$ is also a scalar. Thus, describing $W_p$ as an “L2 projection” onto the “span” of the $S_p^{(k)}$ values is conceptually vague, since the term *span* is not well-defined in this scalar regression context. The authors may consider rephrasing this illustration or providing an intuitive geometric explanation.\n\n**2.3 Limited theoretical analysis**  \nWhile the paper provides clean closed-form regression formulas, it lacks theoretical characterization of the estimator’s bias, variance, or generalization error. Since the model relies on a finite sample of distribution pairs, it would strengthen the paper to analyze how the regression error propagates when predicting Wasserstein distances for unseen distributions.\n\n---\n\n**Overall comment:**  \nThe idea of regressing Wasserstein distances on sliced variants is creative and potentially useful for few-shot estimation, but the **computational advantage, theoretical justification, and empirical comparisons** to existing OT approximations remain insufficient. The paper would benefit significantly from a clearer positioning against established methods (Sinkhorn, linear OT) and from a more rigorous analysis of its approximation properties.", "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": "gNU6N1ZmjN", "reviewer": "Reviewer_AafT", "summary": "The paper's central hypothesis is that for a given meta-distribution $\\mathbb{P}(\\mu, \\nu)$, the true $W_p$ can be accurately and efficiently modeled as a simple linear function of various sliced Wasserstein distances. The core technical insight is to use as predictors a vector of SW based metrics that includes both established lower bounds (e.g., SW, Max-SW, EBSW) and upper bounds (e.g., PW, Min-SWGG, EST).\n\nThe authors claim the weights $\\omega$ for this regression $W_p \\approx \\omega^\\top S_p(\\mu, \\nu)$ are a stable characteristic of $\\mathbb{P}$ and can be learned in a few-shot setting by computing a small number of ground-truth $W_p$ pairs. This yields the RG surrogate which is claimed to be far more accurate than any single SW baseline and outperform deep learning estimators (like Wasserstein Wormhole) in low-data regimes.", "questions": "- The RG-seo model requires computing 6 SW-based predictors, some of which (Max-SW, Min-SWGG) require their own iterative optimization. The paper's time complexity analysis and RG-Wormhole training plot (Fig 14) do not provide a direct inference-time comparison. What is the wall-clock time (for a single pair) comparing RG-seo (at inference) to a standard, converged entropic-regularized $W_p$ (Sinkhorn) approximation?\n- How sensitive is the learned regressor to the choice and number of slicing directions (L) and to the specific mix of lower vs upper SW variants?\n- What if we train an RG model only on intra-class pairs (e.g., (chair, chair)) and test its $R^2$ on inter-class pairs (e.g., (chair, airplane))? vice versa?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- This is a nicely presented paper. The authors' central (and successful) bet is that for a given problem domain $\\mathbb{P}$, $W_p$ doesn't just lie somewhere in this interval, but at a consistent relative position that a simple linear model can capture. Thus moves beyond using SW as a fast proxy and instead models the approximation error by bracketing the true value.\n\n- The paper's primary claim of data-efficiency is validated with solid empirical evidence. Table 2 for example demonstrates that with only 100 training pairs, the SOTA deep baseline (Wormhole) is completely unusable (e.g., $R^2$ of 0.65 on ShapeNetV2, -3.6 on MERFISH, 0.04 on scRNA-seq). In stark contrast, the proposed unconstrained RG variants achieve $R^2$ scores of 0.93-0.99 across all datasets, even in the 2,500-dimensional scRNA-seq domain. This high correlation translates directly to downstream utility. For example in the k-NN classification task (Table 1), the RG-seo surrogate (83.5\\% accuracy @ k=5) performs almost identically to the true, expensive $W_p$ (84.2\\% accuracy @ k=5), while the best single SW baseline is not competitive.\n\n- By replacing $W_p$ inside the Wormhole training loop the authors achieve training speedup. Figure 14 shows almost flat scaling of compute with batch size for RG-Wormhole compared to Wormhole’s quadratic growth, showing that the surrogate can be dropped into OT-based architectures without much performance loss.", "weaknesses": "- The entire method relies on a bootstrap phase where $M$ ground-truth $W_p$ pairs are computed to fit the regression weights $\\omega$. The paper frames this as a minor setup cost because $M_0=10$ (yielding $M = \\frac{10 \\times 9}{2} = 45$ pairs) works for their experiments. However, the paper seems to provide no analysis on how to determine the sufficient $M$ for a new problem.\n- OLS is only optimal if the error term $\\epsilon$ has constant variance. However figures 6-13 suggests otherwise. Would a simple weighted least-squares (WLS) model yield more accurate surrogate?\n- Def 3 adds inductive bias and has half of the parameters, which is often helpful when having limited observed samples. In Table 2 (the $M_0=100$ low-data setting), the unconstrained model is superior in nearly every case (e.g., RG-seo on ShapeNetV2: 0.95 $R^2$ unconstrained vs. 0.92 constrained). This holds even for data size ($M_0=10$) in the appendix (e.g., Fig 8, ShapeNetV2: RG-seo 0.93 unconstrained vs. 0.90 constrained). The empirical justification for the constrained model seems week here.", "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": "SH2HmqICkl", "reviewer": "Reviewer_txMr", "summary": "In many modern machine learning problems, the datasets are collections of distributions, i.e. a meta-distribution over distribution. In those cases, the Wasserstein distance (WD) is an alluring metric for analysis, but constitutes and significant bottleneck in applications that require many pairwise comparisons. This paper proposes a method to estimate the WD by learning a mapping from computationally cheap Sliced Wasserstein (SW) variants to the true WD. The learning process relies on a small number of \"ground-truth\" WD calculations to fit the estimator, which can then be applied to new pairs.\n\nThe proposed method regresses the true WD onto a set of its sliced variants. The predictors include SW-based distances that are known lower bounds (e.g., standard SW, Max-SW) and \"lifted\" SW distances that serve as upper bounds (e.g., Projected Wasserstein, Min-SWGG). The paper details two linear models for this regression: an unconstrained model with a closed-form least-squares solution, and a constrained model. The constrained model pairs lower and upper bounds, reducing the number of parameters and adding an inductive bias, which is intended for \"few-shot\" learning scenarios with limited ground-truth samples.\n\nThis regression estimator (dubbed \"RG\") is evaluated in two applications. First, as a standalone estimator, it is applied to a k-NN classification task, where its accuracy is reported to be close to that of the exact WD. The paper also shows it achieves higher accuracy than the Wasserstein Wormhole model in low-data regimes. Second, the authors propose \"RG-Wormhole,\" a hybrid that replaces the expensive WD calculations in the Wormhole training loop with the fast RG estimate. This substitution is shown to reduce training time substantially while reportedly maintaining comparable performance on tasks like point cloud reconstruction and interpolation", "questions": "The paper highlights the \"explainable\" nature of linear regression as a reason for its use. Given this, have the authors analyzed the learned coefficients ($\\omega$)? Is there a discernible pattern in the learned weights, such as a consistent difference in those assigned to lower bounds versus upper bounds? Similarly, in the constrained model, is there a consistent pattern in the weights for paired \"lifted\" (upper) and \"unlifted\" (lower) SW distances across different datasets or dimensions (e.g., does the model learn to trust one more than the other)? The Gaussian simulation in the appendix (Fig 4) hints at this, but a broader analysis on the real datasets would be valuable.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The method is elegant, principled, and provides a practical way to \"bootstrap\" a collection of cheap estimates to learn a high-quality one. The approach is clearly effective, as demonstrated by extensive comparisons and benchmarks across multiple datasets and tasks (k-NN, embedding, reconstruction). Furthermore, the data efficiency is a major plus as the ability to learn a good regression model from very few \"ground-truth\" Wasserstein distance calculations makes this highly applicable. Finally, the application within the \"RG-Wormhole\" model is a strong contribution, as it directly addresses the primary computational bottleneck of a current method.", "weaknesses": "The presentation of \"Propositions\" 1 and 2 (Section 3.2, Page 6) is a notable weakness. These appear to be standard, well-known closed-form solutions for unconstrained and constrained linear regression, respectively. Presenting these basic results as \"propositions\" with proofs in the appendix feels unnecessary and could be interpreted as an attempt to add theoretical weight where none is needed. An algorithmic and empirical paper like this does not require such justifications for using standard linear regression.", "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": "fX5Ac9mTD9", "reviewer": "Reviewer_A3f7", "summary": "This paper proposed a method to estimate the Wasserstein distance efficiently. Specifically, the proposed method utilizes the fact that the sliced Wasserstein distance is the lower bound of the Wasserstein distance, and the lifted sliced Wasserstein distance is the upper bound of the Wasserstein distance. Then, the proposed method is the weighted average of the sliced Wasserstein distance and the lifted Wasserstein distance and provides a better estimation of the Wasserstein distance by learning these weights.", "questions": "See the weakness section.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* This paper is well-written and easy to follow.\n* The proposed method is simple, and it is intuitively correct that the proposed method is a better estimation of the Wasserstein distance than the sliced Wasserstein distance and the lifted Wasserstein distance.", "weaknesses": "* The proposed method is incremental and trivial. It is just a linear combination of the sliced Wasserstein distance and the lifted sliced Wasserstein distance.\n* The experimental results are not comprehensive. The main motivation of the proposed method is to efficiently estimate the Wasserstein distance. However, this paper did not evaluate and discuss this efficiency, e.g., the time required to compute, the time required for training.\n* There are a lot of papers that proposed the efficient approximation of the Wasserstein distance, but this paper did not compare the proposed method with them in the experiments. Specifically, it is at least necessary to compare the proposed method with the sliced Wasserstein distance, the lifted sliced Wasserstein distance, the max-sliced Wasserstein distance, and other types of approximation of the Wasserstein distance.", "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)