Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
Authors: Khai Nguyen
Hai Nguyen
Tuan Pham
Nhat Ho
Venue: ICML 2025 poster (2025)
Keywords: Dataset distance
sliced optimal transport
data-centric machine learning
transfer learning
Primary area: general_machine_learning->everything_else
TL;DR: We introduce sliced optimal transport dataset distance, a model-agnostic, embedding- agnostic approach for dataset comparison that requires no training, is robust to variations in the number of classes, and can handle disjoint label sets.

Abstract: We introduce  sliced optimal transport dataset distance (s-OTDD), a model-agnostic, embedding-agnostic approach for dataset comparison that requires no training, is robust to variations in the number of classes, and can handle disjoint label sets. The core innovation is  Moment Transform Projection (MTP), which maps a label, represented as a distribution over features, to a real number. Using MTP, we derive a data point projection that transforms datasets into one-dimensional distributions. The s-OTDD is defined as the expected Wasserstein distance between the projected distributions, with respect to random projection parameters. Leveraging the closed form solution of one-dimensional optimal transport, s-OTDD achieves (near-)linear computational complexity in the number of data points and feature dimensions and is independent of the number of classes. With its geometrically meaningful projection, s-OTDD strongly correlates with the optimal transport dataset distance while being more efficient than existing dataset discrepancy measures. Moreover, it correlates well with the performance gap in transfer learning and classification accuracy in data augmentation.

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
Khai Nguyen
Hai Nguyen
Tuan Pham Nhat Ho
Abstract
continual learning, and fairness evaluation. Additionally, dataset distances are valuable in emerging areas such as synthetic data evaluation, 3D shape comparison, and feder- ated learning, where comparing heterogeneous data distribu- tions is fundamental. By capturing meaningful similarities and differences between datasets, these measures facilitate data-driven insights, enhance model robustness, and support novel applications across diverse fields.
We introduce sliced optimal transport dataset dis- lance (sOTTDD), a model-agnostie, embedding- agnostic approach for dataset comparison that requires no training, is robust to variations in the number of classes, and can handle disjoint label sets. The core innovation is Moment Trans- form Projection (MTP), which maps label, represented as distribution over features, to real number. Using MTP, we derive a data point projection that transforms datasets into one- dimensional distributions. The s-OTDD is de- fined as the expected Wasserstein distance be- tween the projected distributions, with respect to random projection parameters. Leveraging the closed form solution of one-dimensional op- timal transport, s-OTDD achieves (near-)linear computational complexity in the number of data points and feature dimensions and is independent of the number of classes. With its geometrically meaningful projection, s-OTDD strongly corre- lates with the optimal transport dataset distance while being more efficient than existing dataset discrepancy measures. Moreover, it correlates well with the performance gap in transfer learning and classification accuracy in data augmentation.
A common approach to comparing datasets relies prox- ies, such analyzing the learning curves of a predefined model (Leite & Brazdil, 2005; Gao & Chaudhari, 2021) or examining its optimal parameters (Achille et al., 2019; Khodak et al., 2019) on given task. Another strategy involves making strong assumptions about the similarity or co-occurrence of labels between datasets (Tran al., 2019). However, these methods often lack theoretical guar- antees, are heavily dependent on the choice of the probe model, and require training the model to completion (c.g.. to identify optimal parameters) for each dataset under com- parison. To address limitations of previous approaches, model-agnostic approaches are developed. These methods often assess task similarity based on the similarity between joint or conditional input-output distributions, occasionally incorporating the loss function as well. Principled notions of domain discrepancy (Ben-Davic et al., 2006; Mansour et al., 2009) provide a more rigorous foundation but are frequently impractical due to computational intractability or poor scalability to large datasets.
1. Introduction
More recently, approaches based on Optimal Transport (Vil- lani, 2008; Peyré & Cuturi, 2020) (OT) have demonstrated promise in modeling dataset similarities (Alvarez-Melis & Fusi, 2020; Tan et al., 2021). Authors in (Alvarez-Melis & Fusi, 2020) introduced optimal transport dataset distance (OTDD) which is based on a hierarchical OT framework drawing from (Yurochkin al., 2019). In this approach, an inner OT problem calculates the label distance between the class-conditional distributions of two supervised learning tasks. This label distance is then integrated into the trans- portation cost of an outer OT problem. yielding a dataset distance that accounts for both sample and label discrepan- cies, and can handle disjoint sets of labels from different datasets. Meanwhile, authors in (Tan et al., 2021) treat the optimal transport plan between the input distributions of datasets as a joint probability distribution and leverage con-
Dataset distances provide a powerful framework for com- paring datasets based on their underlying structures, dis- tributions, or content. These measures are essential in ap- plications where understanding the relationships between datasets drives decision-making, such as assessing data qual- ity, detecting distributional shifts, or quantifying biases. They play a critical role in machine learning workflows, enabling tasks like domain adaptation, transfer learning,
'Equal contribution 'Department of Statisties and Data Sci- ences, University of Texas at Austin, Texas, USA "Qualcomm Al Research, Qualcomm Vietnam Company Limited. Correspondence to: Khai Nguyen <khainb coanbb@texxas.du.
Proceedings of the 4nnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
ditional entropy to quantify the difference between datasets.
1. We propose Moment Transform Projection (MTP), which maps a label, a distribution over high-dimensional features, to a single scalar. MTP first projects the distribution of in- terest onto one dimension using feature projection, then computes a scaled moment to convert the projected one- dimensional distribution into a scalar. We prove that MTP is injective under certain regularity assumptions of the un- derlying dataset distributions. Using MTP, we derive data point projection that combines the label projections from multiple MTPs with feature projection to obtain one- dimensional representation of a data point. By connecting data point projection to the hierarchical hybrid projection approach (Nguyen & Ho, 2024), we show that data point projection is injective, given the injectivity of the MTP.
A key limitation of OT-based approaches is their high com- putational complexity. These methods require pairwise OT (or rntropy-regularized OT) computations across datasets and classes. This process can be prohibitively expensive, especially for applications that demand frequent dataset sim- ilarity evaluations. Solving optimal transport incurs a time complexity of (n logn) n) (Peyré & Cuturi, 2020) and space complexity of O(n²). where is the maximum num- ber of realizations in the two input distributions. Using entropic regularization (Cuturi, 2013) for approximate solu- tions reduces the complexity to On² log n2/c²) (Altschuler et al., 2017). with e denoting the accuracy level. To enhance efficiency, (L.iu et al., 2025) proposes using multidimen- sional scaling (MDS) (Cox& & Cox, 2000) to embed class labels as vectors while preserving the Wasserstein distance geometry for label distances. The authors then leverage reference distribution to form Wasserstein embeddings of datasets (Wang et al., 2013; Kolouri et al., 2021), termed Wasserstein Task Embedding (WTE). Although WTE is more efficient than OTDD, performing MDS and Wasser- stein embedding remains costly due the need for solving optimal transport. Additionally, WTE is not valid metric.
2. We propose the sliced optimal transport dataset distance (s-OTDD), the first formal dataset distance based sliced optimal transport. The s-OTDD is defined as the expected Wasserstein distance between the projected one-dimensional distributions of two input datasets, where the projections are determined by random parameters. We prove that the s- OTOD a valid distance on the space of distributions over the product of the feature space and the space of all distributions on the feature space. Moreover, we dicuss in detail the computational properties of the s-OTDD including approximation error, computational complexities, and other computational benefits.
Sliced optimal transport or sliced Wasserstein (SW) dis- tance (Bonneel et al., 2015; Rabin et al., 2012) is a well- known approach to obtain scalable optimal transport dis- tance between distributions. The SW relies on random pro- jections to exploit the closed- form of Wasserstein distance in one-dimension with On log n2) in time complexity and (n) in space complexity. The challenge in using SW dataset distance is that carrying out projection for class la- bels is non non-trivial trivial. Authors in (Bonet et al., 2024) bypass the problem by using MDS to embed class labels into vectors on a chosen manifold, then utilize SW on product of manifolds i.e., Cartan- Hadamard Wasserstenn (CHSW), as the dataset distance. Nevertheless, conducing MDS as WTE leads to extra computation which has quadratic computation in terms of the number of classes that is undesirable when having a large number of classes.
3. We show that the s-OTDD exhibits a strong correla- tion with the OTDD when comparing subsets from MNIST and CIFARIO, while being faster than existing competitors. Moreover, observe that it correlates well with the per- formance gap in transfer learning across various datasets and modalities, including NIST datasets for images, as well as AG News, DBPedia, Yelp Reviews, Amazon Reviews, and Yahoo Answers for text. Finally, we also find that the s- S-OTDD correlates well with classification accuracy in data augmentation on CIFARIO and Tiny-ImageNe
Organization. We begin by reviewing preliminaries on opti- mal transport dataset distance in Section 2. Next, we present the proposed s-OTDD and its key contributions in Section 3. Experiments demonstrating the favorable performance of the s- s-OTDD are provided in Section 4. Finally, we conclude in Section and defer proofs of key results and additional materials to the Appendices.
Our goal is to develop a model-agnostic sliced optimal transport distance for comparing datasets, which is also baheedding-agnostte requiring no additional preprocessing. Furthermore, we aim for the distance achieve (near-)lineat computational complexity with respect to the number of data points and feature dimensions, while being robust to the number of classes in the datasets. To this end, we propose a novel approach for projecting a data point-comprising a feature and a label-onto a real number. This projection in- cludes an innovative label (class) projection, which maps a label, represented as a conditional distribution over features (as in OTDD), to a scalar.
Notations. We denote P as the set of all distributions on the set X. We denote Rd as the set of real numbers in d' > 0 dimensions, N = 1, 2, 0o as the set of natural numbers, and s as the unit hypersphere in d > dimensions. For a
dataset D = (x1, 3 ..... (Zn, Bn), we define its empirical distribution as PD n E (( We denote [[x]] as the set {1, . n). For any two sequences an and bn. the
notation a, (b,,) means that a,, I Cbn, for all > 1 for a constant C. Lastly, Al denotes the factorial of A >>> 0.
Contribution. In summary, contributions are three-fold:
2

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
2. Preliminaries
the unit hypersphere in d dimensions. In addition to the computational benefit, SW has been widely known for its low sample complexity (Manole et al., 2022; Nadjahi et al., 2020; Nietert et al., 2022; Boedihardjo, 2025).
In this section, we review the definition of Wasserstein dis- tance, sliced Wasserstein distance, and their computational properties, and restate the optimal transport dataset distance.
Optimal Transport dataset distance. We are given two datasets Di = (())))))) and D2 = ((rl, " where (zt, Be), (z'), E X x y, Vi E [[r]]] vj E [m]]] with X' is the space of features and y is the space of la- bels. We assume that know the support distance on the space of features X' i.e. dx X x X - R e.g., Euclidean distance. In contrast to the mild assumption of having dx, we rarely know the distance on the space of labels y e.g., we cannot tell if the label "dog" is closer to the label "cat" than to the label "bird". As a solu- tion, authors in (Alvarez-Melis & Fusi, 2020) propose to map a label into distribution over features, then use dis- tances between distributions as the proxy for the distance between labels. In particular, for the first dataset, we as- sume that (2) VI Ji),..., (In. Mr) ) ~ q(x, Y) where gXx, Y) is an unknown joint distribution. After that, we can define % (X) x(X Y = y) as the conditional distribution over features given the label y. Similarly, for the second dataset, we can obtain qy (X') for a label y'. Therefore, the distance between labels can be defined as:
Wasserstein Distance. The Wasserstein-p distance (Villani, 2008; Peyré& & Cuturi, 2020) (p 2 1) between two distribu- tions/ E P(X) and v E P())), where X' are subsets of Idd and has a ground metric dx= X' x X + R+, is defined as:
/
W v) =
inf xEll(a,)) XxX
(1)
where [1,, v) is the set of all possible transportation plan i.., all joint distributions ग (2 x²) such that T(2, X) = p(z) and xx)x')) = v(x'). When i and v are discrete i.e., " = Ei= a, and v= Σ== 8y62, with Σx= O; 2==1 3) = 1 (a; > Ovi, and 3 > Ovj), we have:
w)(m,) = min
Wi x 25),
(2)
(a) i=1 1=1
where I(a, = (7 E R2xm 1 >1 B = a). (a1,..., an), and 3 (81 (B....., 3x)) The time com- plexity and the space complexity of the Wasserstein distance is On" Oon"logg) (Peyré & Cuturi, 2020) and O(n²) in turn which are very expensive. Using entropic regularization re- duces the complexity to O(n² log n/=²) with e denoting the accuracy level. To avoid such quadratic complexity, sliced Wasserstein is proposed as an alternative solution.
dy(v.v) = ((gg 9g. (X')),
(5)
where TD is a distance between two distributions. In prac- tice, we observe (X) and (X') in empirical forms, hence, optimal transport distances naturally serve as a measure for D. Authors in (Alvarez-Melis & Fusi, 2020) suggest using the Wasserstein distance i.e., dy v2) W (X), By (X')), where we abuse the notation of the Wasserstein distance between two pdfs as the Wasserstein distance between the two distributions. Due to the expen- sive computational complexities of the Wasserstein distance, authors in (Alvarez-Melis & Fusi, 2020) propose to approx- imate two distributions by two multivariate Gaussians, then use the closed solution of Wasserstein-2 distance as the label distance. Nevertheless, the approximation can only capture the first two moments of the two distributions, hence, the resulting comparison might not be accurate. Af- ter having the label distance, the optimal transport dataset distance can be defined as:
Sliced Wasserstein distance. The sliced Wasserstein (SW) distance is motivated from the closed-form solution of the one-dimensional Wasserstein distance with ground metric dx(x, = h(x - x') with X' C R and h is a strietly
convex function:
WE(u,u) WBlun)==/ 1'' dx (FT (3), dz,
(3)
where F-1 and F are inverse CDF of μ and v respec- tively. When μ and y are discrete with at most n supports, the time complexity and the space complexity for comput- ing the closed-fone are Onglog n) (Peyré & Cuturi, 2020) and (n)) respectively. To exploit the closed-form, the SW distance relies on random projections. For p 2 1, the SW distance (Bonneel et al., 2015) of p-th order between two distributions j E P(x)) and v E P(X)) with X C IRd is defined as follow:
OTDD(DI, D2)
min PET
Myd v.). v5)),
(6)
1=
(w) = Eg- 1 (WH(ROi,, Rotv)], (4)
which is the Wasserstein distance with the ground metrie: dp (P.,, vi), (1/), 1/})) = dxx (1:,, + dy (Js, vf), which is the combination of the feature distance and the label dis- tance. As long as the feature distance and the label distance are valid metrics, the OTTD is a valid metric on the product space of X x P(2)) (Alvarez-Mer Melis & Fusi, 2020).
where RøNn and RETV are the one-dimensional push- forward distributions of μ and y through the function Rs(x) (x) = A x (derived from Radon Transform (RT) (Hel- gason, 2011)), and (S(-1) is the uniform distribution over
3

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
3. Sliced Optimal Transport Dataset Distances
exploding in value of high moments.
While the OTDD is a natural distance between two datasets, inherits the computational challenges of the Wasserstein distance. Let n denote the maximum number of data points in the two datasets, the maximum number of classes, and Mmmx the maximum number of samples per class. The time complexity of OTDD is On log n + c log maxx + d)), and the memory complexity is On" + +c²). When using a Gaussian approximation for the label distribution, the time complexity becomes (n log n + (d3 + n) As result, OTDD may not be scalable for large datasets. To address this, aim to develop sliced optimal transport version of OTDD that leverages the one-dimensional closed- form of the Wasserstein distance. This requires developing a novel approach to project a data point onto a single scalar.
Moment Transform Projection. After the feature projec- tion, we obtain an one-dimensional projection of the label distribution Fy. We then can obtain the scaled moment of FPotdy as the final desired scalar. Before giving the for- mal definition, we first state the following assumption.
Assumption (Existence of projected scaled moments), distribution l E P(RR)) with the density function Su has all projected X-th scaled moments (A A C N) given fu(x)dz < oo for
feature projection FP, if JE (FPo(=))) all 0 E gd-1 and X E A.
A
Definition 2. Given feature projection FPg : X - R and a set A C N. a distribution E P(R")) (d > 1) that satisfies Assumption 1, the Moment Transform projection MTP P((R)) + R, with 0 E sdd and À E A, is
defined as follows:
3.1. Label Projection
MTPx0(H) SMA(FPotH)
As mentioned in Section 2, a label is represented as a dis- tribution over the feature space X' Given a label y, we would like to map the label distribution qy to a scalar. To achieve our goal, there are two steps: projecting by to one- dimension through feature projection, and transforming the one-dimensional projection of (y into scalar.
(FPo(a)) (x (8)
X
When is an empirical distribution i.e., l = 1 E Sz,- the MTP projection of μ given À and a feature projection FPs is MTPx, Σ (FPo(x))) We now discuss X!
Feature Projection. Since a label is treated as a distribution over the feature space, we can project label to an one- dimensional distribution through a feature projection i.e., a mapping from FP : X + I with the projection parameter f belongs to a projection space O We can choose any feature projection methods based on the prior knowledge of the feature space X' e.g., Euclidean space (Bonneel et al., 2015), images (Nguyen & Ho, 2022), functions (Garrett et al., 2024), spherical space (Tran et al., 2024; Quellmalz et al., 2023; 2024), hyperbolic space (Bonet et al., 2023), manifolds (Bonet et al., 2024; Nguyen & Mueller, 2024), and so on. It is worth noting that a feature projection is injective if FPoAp1 = FPsipg for all 0 E θ then μl = μ2
the distributionally injectivity of the MTP.
Proposition 1. For μ, E P(IB) and injectivive fea- ture projection FP, and set A C N. MTPx, = MTPx ) for all E Sd-1 and E A implies J = V if either following conditions hold:
(1) μ and v satisfies Assumption with an infinite set A N and moment generating functions (MGFs) of FPely and FPoty exist for all ④
(2) l and satisfies Assumption I with an finite set A C N, and for all 0 E θ, the projected Hankel matrices
for any M2 P(x). The injectivity of the projection is vital to preserve the distributional metricity.
m8,0.0
me,,11
m0
m0
ao,000
Ao,a =
Scaled Moment. We now discuss how to map one- dimensional distribution into a scalar. More importantly, we want the transformation to be injective. To design such transformation, we rely on scaled moments.
mh (b.mxxx
me
ma Amax
with me is X-th moment of FPotu (similar with m are positive definite and [me < CD* XI and |me < CD* X! for some constants C and D for all X E A.
Definition 1. Given a distribution μ E P(R) with the den- sity function fa and a set A C N such that fr fu (x)dx= < oo for all À E A, the X-th scaled moment of μ is defined as:
The proof of Proposition I is given in Appendix A. and is based on the Hamburger moment problem (Reed & Simon, 1975; Chihara, 2011).
SM^(K) = / X! Fa(r)dx
(7)
3.2. Data Point Projection
The difference between the scaled moment and the conven- tional moment is that the scaled moment is scaled by the factorial function of the order A. The sealing is for avoiding
With the proposed label projection, we can propose data point projection. We recall that a data point is a pair of a
4

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
feature and a label or equivalently pair of feature and distribution over features, which is denoted as (ar, qy) E X' x P(X). For the feature domain, as discussed, we can use any feature projections based on the prior knowledge of the feature space, denoted as FPg : X + R. For the label, we can use MTP defined in the Section 3.1. Now, we need to combine the outputs of the feature projection and the label projection obtain the final scalar.
where the expectation is with respect the random projec- tion parameters (tb, 0, X, Φ) (3) ( @ a (A*) & U($) with ø E Φ, and o ) is the uniform distribution on A when A is finite or the product of vero-truncated Poisson distributions (Cohen, 1960) when A is infinite.
Proposition 2. The sliced optimal transport dataset dis- tance S-OTDD, D2) defines valid metric on P(X x P(X)) the space of distributions over feature and label- distribution pairs if the data point projection is injective.
Definition 3. Given data point (z, qy) E X' x P(X) and k 2 1, the data point projection can be defined as follows:
Proof of Proposition 2 is given in Appendix A.2. The propo- sition strengthens the geometrical benefits of the s-OTDD.
DP D..... 0.0.2.0 (1,, 4n) = ( F1)(2)
k
+
( MTPX o
(9) Numerical Approximation. The expectation in s-OTDD is intractable, hence, numerical approximation such as Monte Carlo integration must be used. In particular, we sample (v1 0,, A1, 01)....,) (VL,, (2,>L, ~ U($) & (S 8 (A*) 8 U(()) with the number of projections L > 0, then we form the following estimation:
where v0 = (v()), 2b(2)
)((+))) E st 0 Y
(A(1) .....,, ) E $ De o with O and tb are projection space of the feature projections.
Given a dataset D {(21 9ps 9p.)..... ..... (11 9gm)) }} we have the projected distribution of the corresponding em- pirical distribution through the data point projection is $.0.x. DP
OTDD (D1, D2; D);
Σ Wg(DP Pi. PP., DPt or (11)
= 1/4 E Sppt a (1)-9)))' (".. The pro- posed data point projection combines the outputs of k 2 MTPs to obtain the final projection value. It follows the principle of hicrarchical hybrid projection in (Nguyen & Ho, 2024) to retain the overall injectivity.
In practice, we do not know the number of moments. Hence, we can assume it to be infinite and use the product of zero- truncated Poisson distributions (Cohen, 1960) with some rate hyperparameters a (A Another approach is to cap the number of moments to be Xmaz 2 1 i.e., A = {1, (1,2.... Amar }, then use the uniform distribution (A**) as (A*). We now discuss the approximation error of the S-OTDD when using the Monte Caro estimation.
Corollary 1. The data point projection is injective when the feature projection and the MTP is injective i.e,, for p. E x (x)) if DP a. = DD 5.0. for all we E S, A E O X E N
The Corollary follows the fact that the data point projec- tion is composition of injective projections (Nguyen & Ho, 2024) i.., the Radon Transform projection and the MTP.
Proposition 3. For any p 2 1, and two datasets Dr and D2, we have the following approximation error:
is worth noting that we can use the same value for 0 and é when we use single feature projection project feature and to construct label projections. This approach not only reduces memory consumption for storing projection parameters but also saves computation since we need to project features only once.
OTDDD (Di,,
(D,, Dz: L)- S-OTDD; D2) S
1 VL
Var WDP A. DP
p.pe o.
(12)
where the variance is with respect to (w) 0, Φ) u()) R
u(gd-1) 8 (A*) 8 (()).
3.3. Sliced Optimal Transport Dataset Distance
The proof of Proposition 3 is given in Appendix A.3. We can see that the approximation error reduce fast when increasing the number of projections L i.e., (L221/2) noting that variance reduction can also be used (Nguyen & Ho, 2023; Nguyen et al., 2024: Lelue al., 2024). We refer the read to Algorithm in Appendix B for detailed computational algorithm.
With projections of data points, we can now introduce the sliced optimal transport dataset distance (s-OTDD).
is worth
Definition 4. Let D, and D2 be the two given datasets, PD, and Pp. be corresponding empirical distributions of D2 and Dr respectively, the sliced optimal transport dataset distance (s-OTDD) of order p> > 0 is defined as follows:
Computational Complexities. Using a feature projection with linear complexity in the number of dimensions d (e.g.. Radon transform, convolution, etc), the time complexity of S-OTDD ([ log n + dn)) and the space complexity
s-OTDD (D1, (TD,,(DD,,)) D2)
= WT (DP tp,,,, DPk (p.)))
(10)
5

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
0ead
OTBO
382
260
042
OTEO (0xbskn 80379)
WTE
O15M 120,00 prtecderm
>0100 138,00 prgedioni
Distance Correlation: CIRARLO
Dietance Correlation: CIRAR10
Didtanca Corriaaionn CIRAR10
Distance Correaation CIARR1
1,00 200
2.3
300
1.00
2014
1003
20A4
1264
2084
1800
4084
D80
0160
WTE
0:5te 126000 xi3
8-3100 (33.08 projeclant =20m!
Figure 1: The figure shows distance correlation with OTDD (Exact) of OTDD (Gaussian approximation), WTE, CHSW, and s-OTDD.
(L((+n)) where n is the number of data points and L is the number of projections. Interestingly, both time complex- ity and memory complexity do not depend on the number of classes and the maximum class size mmmaz (good for im- balanced data). In greater detail, the time complexity for applying projection for the class i is (Ln,d) where n; is the size of the class i. As a result, the total time complexity for projection O (I. E.
baselines. Furthermore, we compare the computational time of the proposed s- s-OTDD with existing dataset distance met- rics and investigate its dependency on the number of projec- tions and on the number of moments. In Section 4.2, we eval- uate the correlations of the s-OTDD with the performance gap in transfer learning on image NIST datasets (Deng, 2012) and a diverse set of text datasets (Zhang et al., 2015) including AG News, DBPedia, Yelp Reviews (with both 5-way classification and binary polarity labels). Amazon Reviews (with both 5-way classification and binary polar- ity labels), and Yahoo Answers. Also, to demonstrate the robustness and scability of proposed method, we empiri- cally validate the correlations between s-OTDD distance and transferability on large-scale dataset, which is Split Tiny-ImageNet (Le & Yang, 2015) at 224x224 resolution. Finally, we test the performance of the s-OTDD with the OTDD as the reference in distance-driven data augmenta- tion for image classification CIFARIO (Krizhevsky et al., 2009) dataset and
= ([nd). After hav- ing the projections, solving one-dimensional Wasserstein distances costs OLL log in time. For space complex- ity, (Ld(d + n)) is for storing projection parameters and projections. It is worth noting that the majority of compu- tation of S-OTDD for projecting and computing inverse CDFs of projected distributions of datasets, and such com- putation can be done independently for each dataset. After that, s-OTDD value can be obtained with linear complexity with the precomputed projected inverse CDFs. Therefore, s-OTDD is suitable for distributed and federated settings where datasets can be stored across machines.
(Le & Yang, 2015) dataset in Section 4.3. When dealing with the high dimensional datasets, i.e. CIFARIO dataset and Tiny-ImageNer dataset, we use the convolution feature projection (Nguyen & Ho, 2022) for the s-OTDD while we use the Radon Transform projection for the NIST datasets and text feature datasets. We also provide experiments compare the convolution feature projection and the Radon transform projection on the NIST datasets in Figure 12 in Appendix B. In general, the convolution- projections require fewer projections yet exhibit stronger positive correlations between distance and adaptation performance. For all experiments, we use s- --ODDD with k = 5 and (A*) be the product of k Trun- cated Poisson distributions which have the corresponding
4. Experiments
In Section 4.1, we conduct analysis for the s-OTDD in com- paring subsets of the MNIST dataset (LeCun et al., 1998) and the CIFARIO dataset (Krizhevsky et al., 2009). In par- ticular, we analyze the correlations, which Spearman's 's rank correlation denoted by p and Pearson correlation denoted by r. of the s-OTDD with OTDD. While prior work (Alvarez- Melis & Fusi, 2020) reports only Spearman's 's rank correla- tion, we also compute Pearson correlation for a more com- prehensive analysis, though Spearman's 's correlation remains our primary metric to maintain comparability with existing
6

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
rate parameters be I,..... in turn.'.
Rundima Comparktors MHIST
Runtinn Compartson: CHAR10
4.1. Empirical Analysis of Sliced Optimal Transport Dataset Distance
Distance Correlation. We treat the OTDD (Exact) as the de- sired dataset distance. After that, we compare the proposed s-OTDD with OTDD (Gaussian approximation), WTE, and CHSW in terms of correlations with the OTDD. We ran- domly split MNIST and CIFARIO to create subdataset pairs, each ranging in size from 5,000 to 10,000. For each split, we plot the results of each method and OTDD (Exact), and report their correlations. The results are fully presented in Figure 9 and Figure 10 in Appendix B. The process is repeated 10 times and is summarized in Figure/ 1. From the figure, we observe that the s- s-OTDD (10,000 projec- tions) has significant correlations with the OTDD (Exact) in both datasets. Moreover, S-OTDD shows comparable per- formance with OTDD (Gaussian approximation) and WTE while being computationally faster than them (see the next analysis).
1000
1000
10000
Pellact Nat
1000
Figure 2: The figure shows computational time of OTDD (Exact), OTDD (Gaussian approx), WTE, CHSW (1,000, 5,000, 10,000 projections), and s-ODDD (1,000, 5,000, 10,000 projections) when varying size of two datasets.
Analyris:
Ferman Corvaitenne Gel
ABC
Sdladd Kin 1:30
Butrot Nan bie
Computational Time. We compare the computational time of the proposed s- s-OTDD with existing discussed dataset distances. We randomly split the MNIST dataset and the CIFARIO dataset into two subdatasets which has the size varied from 1,000 to the full size of corresponding dataset. The total runtime includes all preprocessing steps, such as estimating the dataset's 's mean and covariance, as well as performing MDS for WTE and CHSW. We report the com- putational time in Figure 2. We observe that s-OTDD, with different numbers of projections, scales efficiently as the dataset size increases. Additionally, when moving from MNIST to CIFARIO (which has a higher feature dimen- sion), the computation time for s-OTDD does not increase significantly compared to OTDDs, WTE, and CHSW. In contrast to s-OTDD, other dataset distances struggle with relatively large datasets i.e., they are able to run for dataset sizes below 30,000 but then crash due to memory limitations (see Appendix C for the detail of the computational infras- tructure). Thank to the computational benefits, S-OTDD successfully runs on large datasets, proving that it is more scalable than other existing dataset distances.
Dallin 239 L0
crmiectin D. 20.00
Oaland 339 EXC
Didlert 201332333000 20.00
Figure 3: The figure shows Pearson correlations of s-OTDD with s-OTDD (50,000 projections) when varying number of projections from 1,000 to 50,000 in MNIST dataset and CIFARIO dataset. correlations grows very fast when increasing the number of projections, which strengthens the fast approximation rate in Proposition 3. Furthermore, to analyze how the number of projections influences the correlations with OTDD, we select a specific dataset size and compare the s- S-OTDD re- sults using fewer projections (500, 1,000, 5,000, and 10,000) against the OTDD (Exact) baseline, with results presented in Figure 9 in Appendix B. The result indicates that increasing the number of projections enhances the alignment between s- s-OTDD and OTDD (Exact). We also conduct similar projection anaylsis for CHSW in Figure 10 in Appendix B.
Number of Moments Analysis. The number of moments for the label projection is denoted by i in the paper. It is noted that the higher k the more moments information of the label distribution is gathered into the data point projec- tion. Hence we want to have k as big possible as long as it does not raises numerical issue i.c., overflow which might be due to the non-existence of the higher moments. We conduct an ablation study in which is varied in Figure 7.
Projection Analysis. In addition to investigating the ef- feet of varying the number of projections on computational time in Figure 2, we assess the dependency of s-OTDD to the number of projections. Specifically, we vary the number of projections from 1,000 to 50,000 and examine datasets of different sizes: 1,000, 5,000, 10,000, 15,000, and 20,000, computing the correlations with the configu- ration using 50,000 projections in each case. The results are reported in Figure 3. From the figure, we see that the
4.2. Transfer Learning
We apply the proposed method to transfer learning, follow- ing the OTDD framework. Transferability between source and target datasets measured using the Performance Gap (PG), defined as the accuracy difference between fine-tuning the full target dataset and using adaptation method:
All datasets and models were downloaded and evaluated at Movian Al or University of Texas Austin
PG(Ds-T) = accuracy - accuracy(Dss.)
7

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
Distance ve Adaptstion: -NIST Datasats
Distonne vs Adsptaaton: WIST Datseets
Bictnace vs Adagtations Thny tny-leegeee
1.00
dal pi
(2103
Figure 4: The figure shows correlations of OTDD (Exact) and s-OTDD (10,000 projections) with the performance gap when conducting transfer learning in *NIST datasets.
542
014 024 31A
024
Figure 6: The figure shows the Pearson and Spearman correlations between s-OTDD (500,000 projections) and the performance gain in the Split Tiny-ImageNet (224x224) experiment.
Dietance Adsptsttorc Test ClassPication
Distance Adspestion: Text Clsstfication
Split Tiny-ImageNet. We randomly divide the Tiny Im- ageNet (Le & Yang, 2015) dataset into 10 disjoint tasks, each containing 20 classes. Each image is first rescaled to 256x256 and then center-cropped to obtain a 224x224 reso- lution. We initially train ResNet-18 model on each task, serving as baseline, and subsequently freeze all layers ex- cept the final fully connected layer to fine-tune on the target task. We calculate the performance gain after transferbility by Goin(Ds-rr) = accuracy(Pss (Ds-rr) - accuracy (Dr). For distance calculations, we draw 5,000 sample pairs and compute S-OTDD with 500,000 random projections. As shown in Figure 6, s-OTDD exhibits strong correlation with the performance gain. In contrast, OTDD is infeasible at the 224x224 resolution.
Figure 5: The figure shows correlations of OTDD (Exact) and OTDD (10,000 projections) with the performance when conducting transfer learning in text datasets.
PG serves similar role to the Relative Drop metric in OTDD but offers clearer view of adaptation transferability. A smaller PG indicates better adaptation for similar datasets, while a larger PG is expected for more distinct datasets.
NIST Datasets. We use a simplified LeNet-5, freezing the convolutional layers while fine-tuning the fully connected ones. We utilize the entire dataset for s-OTDD computation, while all other methods are evaluated using random sample of 10,000 data points. Figure 4 shows a positive correlations between s- S-ODD with 10,000 projections and PG (p 0.40), which is roughly the same as OTDD (Fxact). Additionally, OTDD (Gaussian approximation), CHSW, and WTE yield correlations of p = 0.40, p = 0.16, and p = 0.37, respectively (see Figure 11 in Appendix B).
From above experiments, we observe that not only s-OTDD achieve comparable performance but also more scalable and robust to existing dataset distances in transfer learning while it offers considerably better computational speed.
4.3. Distance-Driven Data Augmentation
As discussed in (Alvarez-Melis & Fusi, 2020), data aug- mentation enhances dataset quality and diversity but lacks clear guidelines for its effective implementation. A model- agnostic dataset metric can help compare and select the most suitable augmentation strategies. Intuitively, smaller dis- tance between the augmented source dataset and the target dataset positively correlates with higher accuracy. In this large-scale experiment, we use augmented Tiny-ImageNer as the source dataset, CIFARIO as the target dataset, and Re (He et al., 2016) as the classifier. Augmentations for Tiny-ImageNet include random variations in brightnesss contrast, saturation (0.1 and hue (0-0.5). We evaluate the proposed method and OTDD (Fxact), with results pre- sented in Figure 8. For S-OTDD, we sample 50,000 data points per dataset and use 100,000 projections, while for OTDD (Exact), we sample 5,000 data points per dataset. The results are summarized in Figure 8.
Text Datasets.
We limit the target dataset to 100 examples per class, fine-tune BERT on the source domain, then adapt and evaluate it on the target domain following OTDD set- tings. For distance calculations, we sample 5,000 points for OTDD (Exact) and 10,000 points for s-OTDD (10,000 pro- jections). Sentences are embedded using BERT (base) (Li al., 2022), and distances are computed on these embed- dings. The results in Figure 5 shows that although OTDD (Exact) outweighs bit higher than s-OTDD on Spearman rank, but both score the same on Pearson (r = 0.48), Cru- cially, s-OTDD performs the computation in roughly one- tenth the time, providing a substantial vantage at the expense of modest drop in Spearman.
ad-
8

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
Olstance ve Adsptations WIST Datasets
Diszance ve Adpptstton: 'IEST Datasets
Distance ve. Adaptstion: "NIST Datasett artup
mg
301
conte swints ttjite presciin
In -ditt preseets ieperppeeeed
Oistance Adsptatiors "REST Detesets
Distance Adaptation: "RIST Datusets
Dirtance Adaptation: "ISS Datarets
Neformata Dag
Netormana (il
some
Figure 7: Experiment when varying number of moments s-OTDD from I to in *NIST Adaptation Experiment.
baabeedding-gnnostee training-free, and robust variations in class count and disjoint label sets. At the heart of s-OTDD is the Moment Transform Projec- tion (MTP), which encodes label-represented a feature distribution-into a real number. This projection enables us to represent entire datasets as one-dimensional distributions by mapping individual data points accordingly. From theoreti- cal aspects, we discuss theoretical properties of the s-OTDD including metricity properties and approximation rate. From the computational aspects, the proposed s-OTDD achieves (near- Jinearr computational complexity with respect to the number of data points and feature dimensions, while remain- ing independent of the number of classes. For experiments, we analyze the performance of s-OTDD by comparing its computational time with existing dataset distances, its cor- relations with OTDD, its dependence on the number of projections, and number of moments, using subsets of the MNIST and CIFARIO datasets. Moreover, we evaluate the correlations between S-OTDD and transfer learning perfor- mance gaps on image NIST datasets, text datasets and large scale Split Tiny-ImageNet (224x224 resolution). Finally, we test s-OTDD for data augmentation in image classifica- tion on CIFARIO and Tiny-ImageNet. Overall, s-OTDD is comparable baseline while being significantly faster, more scalable, and more robust. Future works will focus understanding the gradient flow (Alvarez-Melis & Fusi, 2021) of the s-OTDD and adapt the s-OTDD in continual learning applications (Lee et al., 2021; Ke et al., 2020; Yang & Cai, 2023; Goldfarb et al., 2024).
is model-agnostic,
Distence Adoptetiion Imageteet-- CIARRD
Distance Adaptation: tmagetiet..o CIPAR13
Figure 8: The figure shows correlations of OTDD (Exact) OTDD (Exact) and s-OTDD (100,000 projections) with the classification accuracy in data augmentation experiment.
Figure 8 shows that both OTDD (Exact) and our proposed s-OTDD method produce distances that are negatively cor- related, which is natural, with test accuracy when transfer- ring from augmented Imay- to CIFARIO. Despite processing ten times the dataset size compared to OTDD (Exact), S-OTDD is significantly faster, with processing time in about 53 x 10 seconds, whereas OTDD (Exact) takes over 74 x 103 seconds, revealing a huge improvement in efficiency. Moreover, OTDD (Exact) achieves a Spear- man correlations of 707 while s-OTDD attains a better correlations of -0774. In other words, s-OTDD provides reliable alternative distance to OTDD (Exact), achieving comparable performance while being faster.
5. Conclusion
We propose the sliced optimal transport dataset distance (s-O (s-OTDD), a versatile approach for comparing datasets that
9

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
Impact Statement
Deng, L. The mnist database of handwritten digit images for machine learning research [best of the web). IEEE signal processing magazine, 29(6):141-14,, 2012.
This paper presents work to advance the field Machine Learning. There are potential societal consequences of our work, none of which we feel must be highlighted.
Gao, Y. and Chaudhari, P. An information-geoomerii dis- tance the space of tasks. In International Conference on Machine Learning, pp. 3553-3563. PMLR, 2021.
References
Achille, A., Lam, M., Tewari, R., Ravichandran, A., Maji, S., Fowlkes, C. C., Soatto, S., and Perona, P. Tesk2vec: Task embedding for meta-learning. In Proceedings of the IEEE/CVF international conference on computer vision, pp. 6430-6439, 2019.
Garrett, R. C., Harris, T., Wang, Z., and Li, B. Validating climate models with spherical convolutional Wasserstein distance. In The Thirty-eighth Annual Conference on Neu- ral Information Processing Systems, 2024. URL https: / //openreview. net/ forum? d=mmSF fib6pI.
Altschuler. J., Niles- Weed, J., and Rigollet, P. Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration. In Advances in Neural Information Processing Systems, pp. 1964-1974, 2017.
Goldfarb, D., Evron, I., Weinberger, N., Soudry, D., and HAnd, P. The joint effect of task similarity and overpa- rameterization on catastrophic forgetting - an analytical model. In The Twelfth International Conference on Learn- ing Representations, 2024.
Alvarez-Melis, D. and Fusi, N. Geometric dataset distances via optimal transport. Advances in Neural Information Processing Systems, 33:21 21439, 2020.
He, K., Zhang, X., Ren, S., and Sun, J. Deep residual learn- ing for image recognition. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 770-778 2016.
Alvarez-Melis, D. and Fusi, N. Dataset dynamics via gradi- ent flows in probability space. In International conference machine learning, pp. 219-230. PMLR, 2021.
Helgason, S.
The Radon transform on n. Integral Geometry and Radon Transforms, pp. 1-62. Springer, 2011.
Ben-David Blitzer, J., Crammer, K., and Pereira, F. Anal- ysis of representations for domain adaptation. Advances in neural information processing systems, 19, 2006.
Ke, Z., Liu, B., and Huang. X. Continual learning of mixed sequence of similar and dissimilar tasks. Advances in neural information processing systems, 33:18493-1850 2020.
Boedihardjo, M. T. Sharp bounds for max-sliced wasserstein distances. Foundations of Computational Mathematics, pp. 1-32,2025.
Khodak, M., Balcan, M.-F. F., and Talwalkar, A. S. Adap- tive gradient-based meta-learning methods. Advances in Neural Information Processing Systems, 32, 2019.
Bonet, C., Chapel, L., Drumetz, L., and Courty, N. Hyper- bolic Sriced-Wassersee via geodesic and horospherical projections. In Topological, Algebraic and Geometric Learning Workshops 2023, pp. 334-370. PMLR, 2023.
Kolouri, S., Naderializadeh, N., Rohde, G. K., and Hoff- mann, H. Wasserstein embedding for graph learning. In International Conference on Learning Representations, 2021. URL https: wwwp//pppnnwoeeeeooeeeeee net ntt./frutf forum?
Bonet, C., Drumetz, L., and Courty, N. Sliced-Wassersie Wasserstein distances and flows on Cartan- Hadamard manifolds. arXiv preprint arXiv:2403.06560 2024.
id=AAes _W-22.
Bonneel, N., Rabin, J., Peyré, G., and Pfister, H. Sliced and Radon Wasserstein barycenters of measures. Journal of Mathematical Imaging and Vision, 1(51):22-45,, 2015.
Krizhevsky, A., Hinton, G., et al. Learning multiple layers of features from tiny images. Master's thesis, Department of Computer Science, University of Toronto, 2009.
Chihara, T. S. An introduction to orthogonal polynomials. Courier Corporation, 2011.
Le, Y. and Yang, X. Tiny imagenet visual recognition chal- lenge. CS 23/N, 7(7):3, 2015.
Cohen, A. C. Estimating the parameter in conditional poisson distribution. Biometrics, 26(2))203-211 1960.
LeCun, Y., Bottou, L., Bengio, Y., and Haffner, P. Gradient- based learning applied to document recognition. Proceed- ings of the IEEE, 86(1 12:2278-2324, 1998.
Cox, F. and Cox, M. A. Multidimensional scaling, CRC press, 2000.
Lee, S., Goldt, S., and Saxe, A. Continual learning in the teacher-student setup: Impact task similarity. In International Conference on Machine Learning, pp. 6109- 6119. PMLR, 2021.
Cuturi, M. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems, pp. 2292-2300, 2013.
10

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
Leite, R. and Brazdil, P. Predicting relative performance of classifiers from samples. In Proceedings of the 22nd international conference machine learning, pp. 497- 503, 2005.
Peyré, G. and Cuturi, M. Computational optimal transport, 2020.
Quellmalz, M., Beinert, R., and Steidl, G. Sliced optimal
transport on the sphere. Inverse Problems, 39(10) 2023.
Lelue, R., Dieuleveut, A., Portier, F., Segers, J., and Zhu- man, A. Slived-Wassersten estimation with spherical harmonics as control variates. In Proceedings of the s International Conference Machine Learning, 2024.
Quellmalz, M., Buecher, L., and Steidl, G. Parallelly sliced optimal transport on spheres and on the rotation group. arXiv preprint arXiv:2401. 16896, 2024.
Li, T., Meng, C., Yu, J., and Xu, H. Hilbert curve projec- tion distance for distribution comparison. arXiv preprint arXiv:2205. 1.005..50050 15599, 2022.
Rabin, J., Peyré, G., Delon, J., and Bernot, M. Wasser- stein barycenter and its application to texture mixing. In Scale Space and Variational Methods in Computer Vision: Third International Conference, SSVM 2011, Ein-Gedi, Israel, May 29-June 2, 2011, Revised Selected Papers pp. 435-446. Springer, 2012.
Liu, X., Bai, Y., Lu, Y., Soltoggio, A., and Kolouri, S. Wasserstein task embedding for measuring task similari- ties. Neural Networks, 181:106796, 2025.
Manole, T., Balakrishnan, S., and Wasserman, L. Minimax confidence intervals for the sliced Wasserstein distance. Electronic Journal of Statistics, 16(1):2252-2345, 2022.
Reed, M. and Simon, B. II: Fourier analysis, self- adjointness, volume 2. Elsevier, 1975.
Tan, Y., Li, Y., and Huang, S.-L. Otce: A transferability metric for cross-domain cross-task representations. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 15779-15788, 2021.
Mansour, Y., Mohri, M., and Rostamizadeh, A. Domain adaptation: Learning bounds and algorithms. arXiv preprint ar arXiv:0902.34300 2009.
Nadjahi, K., Durmus, A., Chizat, L., Kolouri, S., Shahram- pour, S., and Simsekli, U. Statistical and topological properties of sliced probability divergences. Advances in Neural Information Processing Systems, 33:20802- 20812, 2020.
Tran, A. T., Nguyen, C. V., and Hassner, T. Transferability and hardness of supervised classification tasks. In Pro- ceedings of the IEEF/CVF International Conference on Computer Vision, pp. 1395-1405, 2019.
Tran, H., Bai, Y., Kothapalli, A., Shahbazi, A., Liu, X., Martin, R. D., and Kolouri, S. Stereographic spherical sliced Wasserstein distances. International Conference on Machine Learning. 2024.
Nguyen, K. and Ho, N. Revisiting sliced Wasserstein on images: From vectorization to convolution. Advances in Neural Information Processing Systems, 2022.
Nguyen, K. and Ho, N. Sliced Wasserstein estimator with control variates. International Conference on Learning Representations, 2023.
Villani, C. Optimal transport: old and new, volume 338.
Springer Science & Business Media, 2008
Wang, W., Slepěev, D., Basu, S., Ozolek. A., and Rohde, G. K. A linear optimal transportation framework for quantifying and visualizing variations in sets of images, International journal of computer vision, 101:254-220 2699 2013.
Nguyen, K. and Ho, N. Hierarchical hybrid sliced Wasser- stein: A scalable metrie for heterogeneous joint distribu- tions. Advances in neural information processing systems, 2024.
Nguyen, K. and Mueller, Summarizing Bayesian nonpara- metric mixture posterior- - sliced optimal transport metrics for Gaussian mixtures. arXiv preprint 11.14:2411.14674, 2024.
Yang, S. and Cai, 7. Cross domain lifelong learning based on task similarity. IEEE transactions pattern analysis and machine intelligence, 45(10):11612-1166 2023.
Yurochkin, M., Claici, S., Chien, E., Mirzazadeh, F., and Solomon, M. Hierarchical optimal transport for docu- ment representation. In Advances in Neural Information Processing Systems, pp. 1601-1611, 2019.
Nguyen, K. Bariletto, N., and Ho, N. Quasi-monte carlo for 3d sliced Wasserstein. In The Twelfth International Conference on Learning Representations, 2024.
Zhang, X., Zhao, and LeCun, Character-level convolu- tional networks for text classification. Advances in neural information processing systems, 28, 2015.
Nietert, S., Sadhu, R., Goldfeld, Z., and Kato, K. Statisti- cal, robustness, and computational guarantees for sliced Wasserstein distances. Advances in Neural Information Processing Systems, 2022.

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
Supplement to "Lightspeed Geometric Dataset Distance via Sliced Optimal Transport"
first give skipped proofs in the main text in Appendix A. We then present additional materials including algorithms and additional experiments in Appendix B. Finally, we report computational devices used for experiments in Appendix C.
A. Proofs
A.1. Proof of Proposition 1
We can rewrite the MTP in Definition 2 as:
(PP()( fu(
Sar t fFPORH (t)
MTPx0(11)
Al
where fFPotA the density function of sdg. Let The o be the random variable of FPadee have:
MTPx0(1)= B[Txol
Al
BT XI
BTT
For given 0, when MTP = MTP (v) for all A, it implies
for X E A.
X!
E[Thoo X1
(1) When A is infinite i.., A = N, we have
for all A C N. Therefore, we have:
XI
A
2 =AE[TAe
XI
Since we assume that all projected scaled moment exists in Assumption I and the moment generating functions of FPoty and FP e exist for all O, using Taylor's expansion, we have:
Ele TT.
which means that moment generating function of FPoty equals moment generating function of FPotu. Therefore, conclude that FPodu FPstv for all 0 θ due to the uniqueness of the moment generating function. Due to the assumption of injectivity of the feature projection, we obtain μ v. We complete the proof.
(2) When finite i.e., A =
Amaz), we assume that projected Hankel matrices
mg,000
mg
M0,p.Amax
mg,u.0
mgge,1
m0
mp
map.0
m9,v.1
mg.
mg
Ag.r
Ag,u=
m0
me.u.
mig.u.
are positive definite for all 0 E and px < CDD and |mo.ux < CDÀN for some constants C and D for all À E A. From the Hamburger moment problem (Reed & Simon, 1975; Chihara, 201 I), we know that moments uniquely define a distribution. Therefore, we have FPotu = FPojv for all E Due to the assumption of injectivity of the feature projection, we obtain μ = v. complete the proof.
with mg,uA =
(x)do and mgg.u. = (FP((t))))
A.2. Proof of Proposition 2
From Definition& 4, we have:
(DD D2) = - Ri.....o-11s3vv//55111/10aaa
(DPp (. t.p. DPk tP.)
Since the Wasserstein distance non-negative and symmetric (Peyré & Cuturi, 2020), the symmetry and non-negativity of s-OTDD p (D,, D2) follows directly from We now prove the triangle inequality s-OTDD. Given any three datasets Dj, D2, and Dz. We want to show that:
S-OTDD,(DI,,,)) D2) S-OTDD, D3) S-OTDD, D2).
12

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
From the triangle inequality of the Wasserstein distance, have:
(OTDDD,(1),)))) D2) = (RA.AAApnr4iDDDsss0000MMMMC P.A o.
s (BA00Lo))-MMBBBsS----TTAA-- (W, b. DD D.Apo
W. DP D. o.
Using the Minkowski's inequality, we further have:
(Di, - Pin.0hoowwiiseerr0001000000000 V.A Dp
D. o. Pp.)))
(Buu..oo--0((33arrr---ooooaaaaa0 WH(DP) (W U. W.RA Dpt M.ddd #
(Pi, (D, Dz),
which completes the proof of the triangle inequality. For the identity of indiscernibles, when D1 D22 we have directly that
SPTTDD,(P,, D2) P/tp:://....PPPPAAA//////////ee (1
WD(DPP D. DPf DDt o to.t)))))))))))))))))))))))))t
(B(0AAA)) (0(((1)))((())
[0]] =0
due to the identity indiscernibles of the Wasserstein distance when Dp o. DP to addition, when for all & E 8 θ, E A, E Φ. As
,-OTDD, (D,, D2) = O, it implies that (DDPP* ". DP p. HP result, DP o. w.o.A.oHPp, DP* oo for all v 8 E θ, A, 50 E 4. From the assumption of the injectivity of
the data point projection, obtain that PD, PD, which implies D1 D2. We conlude the proof here.
A.3. Proof of Proposition
We recall the empirical approximation of the s-OTDD is:
7
SOTDD D2; = Σ
WDP
EPP., FPD,, Dpk
or HPP.).
Using Holder's incquality, we have:
LOODDD" (D,, S-OTDD D2)
E
D.))))
<
E
LD)
L
02.0 tPb.) - E DPx D...ottt ())
=
E
(DP
p 0. Br
#Pp.
#Pp, Dpt
Since
E
WDPk
Mi DPk
oi #Pb))
1
E E (DP)
"r,,,,,,, PP,, DPt
D dr #P PPp,)
L
= E
(DPp
o. DPt (.pdt
13

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
Algorithm I Computational Algorithm for s-OTDD
Input:
Two datasets D1 = {(11 yi),..., (In, Dn]} and D2 {(21, v().................... (I'm vm label sets V and 2/ parameter 1, number of moments k number of projections L, rate parameters 71,.... 1 7'k
O.
for 7 L do
Sample 02 ~ (1SS-1)
for i= 1 to n do
Compute (2;
end for
for j to m do Compute JP (x¹))
end for for j to k do
TruncatedPoisson(rr) Sample x
for a in v do
Compute MTP X (00,((90) na El= end for for a' in V12 do
FPole.)
for nl the number of samples in class a.
21
FPola!) (G)
Compute MTP xc0,0 = m.
mm
for ma is the number of samples in class a'.
M
end for end for Sample dh (
for 1 to do
DP 8. 4n.) 11))) + 1 MTP 200) ((
end for for i= to m do
DP D... 0.o.x j 9v3)) ((O)F + E (5(K) (5(K) MTP 2000 e(
end for Compute Wi = Wp
1 n
8ppt
ol - Ei= Epp a. (t()
end for
Return: +
we have:
E
B-ODD p (D,, D2; - s-DDD D2)
1
Var
Wg(DP C Or AP.. Dpk or :P'z)
T.
(Var
(DPp
(DPP Pp, DP
v.o.A. a.n. (. (Po)))
VL
due to the i.i.d sampling of the projection parameters, which completes the proof.
B. Additional Materials
Algorithms. We present the computational algorithm for s-OTDD in Algorithm 1. In the algorithm, use the same feature projection for both the label projection (MTP) and the data point projection.
Projection Analysis. As mentioned in the main text, we present the distance correlations with OTDD (Exact) of s-OTDD as a function of the number of projections in Figure and similarly for CHSW in Figure 10. For s-OTDD, we observe that increasing the number projections consistently improves the correlations for both MNIST and CIFARIO 10 datasets, indicating a clear trend. In contrast, the same does not hold true for CHSW, where the correlations do not show consistent
14

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
praue Lroce
Intles L4bcc
104
090 FAMD
094
204
2044
+0100 ortctions
0010 15,000 prgeddennt
(5,93 prepetionss
>0100 128,000 prgecional
Distance Correlation: CIRARLO
Distance Corrlation: CIFAR10
Didtance Corretation: CIRAR10
Distance Correlation: CIFRR10
Aah
2.0
2042
2864
2962
C204
1304
00000 (500 projctinn xipe
+0000 6,000 argediorgaad XLg=
0 15.200 preaciaona x2p-1
8-3100 (13.08 projeclan =20m!
Figure 9: The figure shows distance correlations with OTDD (Exact) of s- OTDD when varying number of projections.
Distance Correlation: MNISTT
Distance Correlation: WHIST
Distance Corrciation: MHIST
Distance Correlations MRIST
182
CHOW 560 orgedionn sup=!
<HOW (1383 procecann Site
Ols licce anjecting widm
CHio (1:200 procetinn xate
Distance Corrolation: EIRARLO
Distance Correlation: CIAAL10
Distance Corrdation CIARR10
Distance Correlation: CHARID
1002
2014
2264
0900
1860
2883
wig 15A
x30
Figure 10: The figure shows distance correlation with OTDD (Exact) of CHSW when varying the number of projections.
improvement as the number of projections increases. This important observation suggests that s-OTDD in its population form highly correlated with OTDD (Exact), and therefore, reducing the Monte Carlo approximation error by increasing the number of projections results in a more accurate correlations with OTDD (Exact).
Transfer Learning. We present the complete experimental results for transfer learning using the *NIST datasets in Figure 11. As highlighted in Section 4.2, the figure demonstrates clear positive correlations between s-OTDD with 10,000 projections and PG, with a Spearman's 's rank correlation coefficient of = 0.42, which is identical to that of OTDD (Exact). For the other distances, we observe that OTDD (Gaussian approximation), CHSW, and WTE yield correlations of p = 0.44, p 0.15, and = 0.43, respectively. These results reinforce the reliability of s-OTDD in capturing meaningful transfer learning behavior, comparable to OTDD (Exact) and other related metrics.
15

---

Lightspeed Geometric Dataset Distance via Sliced Optimal Transport
pist
p014
0700
07300 Klaae
vort eis piaopeaddeo
Figure 11: The figures show Pearson and Spearman correlation of OTDD (Exact), OTDD (Gaussian approximation), CHSW (10,000 projections), WTE, and S-OTDD (10.000 projections) with the performance gap when conducting transfer learning in *NIST datasets.
1MST
"ISST
Adaprttion HAST
Calf
Eat0 Livee proptond
Loma
Shlleee Adapteten HAST Bateatts
Adaphiiee ne Beleet
Dridann delegt "MIT Doteehs
SERT Setoods
Dbioree
1000
Figure 12: The figure compares two projection methods, linear projections (Radon Transform projection) and convolution projection. using the "NIST dataset experiments.
Distance vs Adaptation: "NIST Datanats
Dietance us Adeptation: "NIST Datasebs
Distance ve Adaptation: "NEST Detanate
3.3 0.0.5
Natormane Cop Pooul
Pefommnd Gan
ections
Figure 13: The figure shows Pearson and Spearman correlations of distance and Performance Gap in term of other metrics, i.c F1, Precision and Recall.
C. Computational Devices
For the runtime experiments and distance computations, we conducted tests using 8 CPU cores with 128GB of memory. For model training experiments, such as training BERT and ResNet, we used an NVIDIA A100 GPU.
16


Reviews:
[{"review_id": "M5RkPGr7bh", "reviewer": "Reviewer_1ZM5", "summary": "This paper tackles the Dataset Distance problem with a proposed sliced optimal transport dataset distance (s-OTDD) method.   \nThe core module is called Moment Transform Projection (MTP), mapping a label (represented as a distribution over features) to a real number. Then, s-OTDD is defined as the exptected Wasserstein distance between the projected distributions, and calculated by leveraging the closed-form one-dimentional optimal transport, i.e. sliced Wasserstein distance.   \nWith random projection, s-OTDD shows (near-)linear property for the Dataset Distance problem. \n\nExperiments are conducted on various benchmarks, and the main comparison method is OTDD(exact). OTDD can be regarded as the gold standard or upper bound of the s-OTDD algorithm. The results shows good correlation between OTDD and s-OTDD, while the compuation time is magnitude of faster. \n\nMoreover, several theoretical results are given: injective property, s-OTDD is a valid metric, and the approximation error.\n\n## update after rebuttal\nI appreciate the authors' response. All my questions and concerns are addressed by the rebuttal.   \nI am happy to keep my score as 4: Accept.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "First, we would like to thank the reviewer for the time and the feedback. We extend the discussion as follows:\n\n**Q1.** The line below Eq. (8), empirical distribution is missing the variable.\n\n**A1.** Thank you for pointing out the typo. We have fixed the typo in the revision.\n\n**Q2.** In Eq. (9), is the first $\\psi^{(1)}$ in the first term, the same as the $\\psi^{(k)}$ with $k = 1$ in the second term?\n\n**A2.** It is actually a typo in our Eq. (9). The summation in the second term should involve $\\psi^{(i+1)}$. We have fixed the typo in the revision. In particular,  we have the following revised definition of the data point projection:\n\n$\\mathcal{DP}^k_{\\psi,\\theta,\\lambda,\\phi}(x,q_y) = \\psi^{(1)} \\mathcal{FP}_\\theta(x) $\n\n$+\\sum_{i=1}^k \\psi^{(i+1)} \\mathcal{MTP}_{\\lambda^{(i)},\\phi}(q_y)$ \n\nwhere $\\psi=(\\psi^{(1)},\\psi^{(2)},\\ldots,\\psi^{(k+1)}) \\in \\mathbb{S}^k, \\theta \\in \\Theta, \\lambda =(\\lambda^{(1)},\\ldots,\\lambda^{(k)}) \\in \\Lambda^{k}, \\phi \\in \\Phi$.\n\nWe would like to elaborate more on Corollary 1 with the revised definition. Let $\\mathcal{FP}_\\theta(x)=t_1$ \n\nand $\\mathcal{MTP}_{\\lambda^{(i)},\\phi}(q_y)=t_i$ for $i=2,\\ldots,k$, we can rewrite the data point projection as:\n\n$\\mathcal{DP}^k_{\\psi,\\theta,\\lambda,\\phi}(x,q_y) = \\psi^{(1)} t_1+ \\sum_{i=2}^k\\psi^{(k)} t_i = \\psi^\\top t = \\mathcal{R}_\\psi(t)$,\n\nwhere $\\psi=(\\psi^{(1)},\\ldots,\\psi^{(k)})$, $t=(t_1,\\ldots,t_k)$, and $\\mathcal{R}_\\psi(t)$ is the Radon Transform of $t$ with projection parameter $\\psi$ as defined at line 162. Therefore, the data point projection is simply the Radon Transform of the stacked output of the feature projections and moment transform projections. Since the composition of injective functions is also injective, the data point projection is injective given the injectivity of the feature projection and the moment transform projections.", "claims_and_evidence": "The s-OTDD is a fast and effective approximation of the OTDD dataset distance method. This is verified by the theorecial proof (valid metric and approximatoon error), as well as experimental comparison on various benchmark dataset (correlation w.r.t. OTDD (exact)). \n\ns-OTDD is a valid distance, which is proved in Proposition 2.", "methods_and_evaluation_criteria": "Yes.   \nThe proposed method mainly leverages the one-dimensional closed-form property of the optimal transport, which also translates to the sliced-Wasserstein distance.   \nBased on this, the proposed method designed Moment Transform Projection to ensure the injective feature projection. Then it applies data point projection.   \nFinally, Sliced Optimal Transport Dataset Distance is approximated random projection over four parameters ($\\psi, \\theta, \\lambda, \\phi$) sampled from corresponding spaces. \n\nEvaluation metrics mainly include (1) distance correlation w.r.t. OTDD and other dataset distances (2) processing time w.r.t. dataset size. Overall, these metrics can capture the effectiveness and efficiency of s-OTDD.", "theoretical_claims": "Proposition 1. Existence of projected scaled moments. The proof is given in Appendix A.1 \n\nProposition 2. s-OTDD is a valid metrics. The proof is given in Appendix A.2 \n\nProposition 3. The approximation error. The proof is given in Appendix A.3 \n\nI went through all the proof and did not find theoretical issue. Because this is an emergent review, I might miss some detail due to limited review time.", "experimental_designs_or_analyses": "Overall, the experiments are sound and effective.   \n(1) Various benchmarks and tasks are conducted, e.g. image classification and text classification on MNIST, CIFAR, TinyImageNet, AG News, DBPedia, Yelp Reviews, Amazon Reviews, and Yahoo Answers.   \n(2) The important baseline and standard distance OTDD(exact), as well as a bunch of other approximation distances are compared.   \n(3) Running time comparison is given w.r.t dataset size.   \n(4) Parameter analysis are given w.r.t number of projections.", "relation_to_broader_scientific_literature": "This paper mainly provides an effective and much faster dataset distance, which improves the OTDD (Alvarez-Melis & Fusi, 2020).   \n\nThrough theoretical proof and experimental comparisons, the proposed distance s-OTDD approximates OTDD and also is much faster in speed w.r.t dataset size.", "essential_references_not_discussed": "To my knowledge, I do not know other relevant references.", "other_strengths_and_weaknesses": "The main Strengths and Weaknesses have already listed in the above sections.", "other_comments_or_suggestions": "Typos.   \nThe line below Eq. (8), empirical distribution misses the variable.", "questions_for_authors": "In Eq. (9), are the first $\\psi^{(1)}$ in the first term, the same as the $\\psi^{(k)}$ with $k=1$ in the second term?", "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": {"ethical_review_concerns": "No ethical concern,"}}, {"review_id": "cZJBKScl7g", "reviewer": "Reviewer_VKhK", "summary": "This paper is a straightforward application of sliced OT on OT-based dataset distances. The main novelty is Moment Transform Projection by which the authors could project dataset labels to scalars, enabling the use of Radon transformation and hence sliced Wasserstein distances. Authors then follow standard procedures to prove the metric properties and approximation error of the sliced OT dataset distance. Empirical results from various experiments show that the sliced OT dataset distance is both efficient and effective.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "First, we would like to thank the reviewer for the review and feedback. We would like to answer questions from the reviewer as follows:\n\n**Q3.** A weakness of the proposed sliced OTDD is the extra hyper-parameter - the number of moment $\\lambda$ as in (7). In the code, it’s set to 5. I didn’t find it discussed in the paper except in [246] where authors only briefly mentioned the principle of choosing $\\lambda$. Its impact on the projection and hence the distances is unknown as well.\n\n**A3.** Thank you for your insightful comments.  We would like to recall that the number of moments is denoted as $k$ and we did report the choice of $k=5$ at line 307-308 in the paper. For the parameter $\\lambda$, it is the order of a moment. We would like to extend the discussion on both $k$ and $\\lambda$ as follows: \n\nFor $k$, it plays the role of selecting the number of moments for the label projection. In particular, the higher value of $k$, the more moments information of the label distribution is gathered into the data point projection. For $\\lambda$, it plays the role of choosing information of distributions to extract. For example, $\\lambda=1$ leads to the information of centerness while $\\lambda=2$ leads to the information of spread. In the paper, $\\lambda_i$ for $i=1,\\ldots,k$ are random which followed the zero-truncated Poisson distribution with the rate parameter equals $i$ i.e.,  the mean  $\\mathbb{E}[\\lambda_i]= (ie^i)/(e^i-1) \\approx i$. It means that we try to capture (in high probability) the first $k$ moments of the label distribution. In the case, where we know the true number of moments of the label distribution $\\lambda^\\star$, we can set $k=\\lambda^*$ to capture all information of the label distribution. Nevertheless, checking existence of moments is expensive since we only access to samples from the dataset generating distribution and  the size of datasets is also large. Therefore, $k$ becomes a hyperparameter. However, choosing $k$ is not a hard problem since we know that we want to have as big $k$ as possible. In practice, we can start with a big $k$ and check if the value of s-OTDD exists (not overflow). If s-OTDD does not exist, you can reduce the value of $k$ via a binary search rule to have a choice of $k$. To avoid such searching algorithm, we recommend to choose $k=5$ due to the concentration of random projection. In particular, we know that one-dimensional projection becomes a Gaussian distribution in high-dimension [1,2]. Therefore, two moments are enough to capture the information of a Gaussian distribution. Since we have finite dimension, using extra 3 moments (5 in total) must be enough.\n\nTo verify the above hypothesis, we have conducted additional ablation studies in *NIST Adaptation where we vary $k\\in \\\\{1,2,3,4,6\\\\}$, resulting visually at [https://imgur.com/a/NCcqYgo](https://imgur.com/a/NCcqYgo). We see that increasing $k$ leads to better correlation to  the performance gap. Nevertheless, we found that when passing $k=3$, the correlation does not increase fast as when $k<3$. Also, we found that when setting $k$ to be too large, s-OTDD is not computable due to numerical issue i.e., overflow which might be due to the non-existence of the higher moments. We will add this discussion and the additional experiments to the revision.\n\n[1] Fast approximation of the sliced-Wasserstein distance using concentration of random projections, NeurIPS 2021, Nadjahi et al.\n\n[2] Asymptotics of Graphical Projection Pursuit, The Annals of Statistics, 1984 Diaconis et al.\n\n**Q4.**. The $\\lambda$-th scaled moment as defined in (7) is not how I usually \"scale\" a moment. What’s the rationale behind a constant scaler? Please discuss the impact of $\\lambda$.\n\n**A4.** The reason we scale $\\lambda$-th moment by $\\lambda!$ is to make sure the output of the data point projection is not biased toward high order moments. In particular, in the definition of the data point projection (Definition 3), the data point projection is a weighted average of the feature projection and multiple moment transform projections with different values of $\\lambda$. If we do not scale the moment, the value of the data point projection will be entirely based on the value of the moment transform projection with highest value of $\\lambda$. However, we know that each moment captures different information of the distribution e.g., the first moment captures the information about center, the second moment captures the information about the spread. Therefore, scaling the moment value makes the contribution of all moments \"approximately\" equal. The reason we say that they are \"approximately\" equal is that we actually imply the priority for small order moments since the factorial normalizing constant goes faster in limit than the exponential function.  We will include this discussion in the revision of the paper.\n\n**Q5.** provide more details for A.1.\n\n**A5.** Thank you for your comments.  We will expand the proofs by adding intermediate steps.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "I checked all the proofs. Proofs in A.1. jump too much between lines. Please expend them and add intermediate steps.", "experimental_designs_or_analyses": "Results from empirical experiments are convincing in verifying the claim that the proposed sliced OTDD is much more efficient than traditional OTDD, and the approximation error is acceptable.", "relation_to_broader_scientific_literature": "The paper is a sweet overlap between sliced OT and OT-based dataset distance. It's a natural application of sliced OT to another one of OT's applications. The main contribution of the paper is not to identify the overlap or the technical difficulty, but to lay the necessary theoretical foundation in a timely manner so that on one hand researchers could advance their work on top of this work and on the other hand people can use OT dataset distance in practice. \n\nSliced OT until now is a somewhat well-studied subject after Bonneel et al., 2015, Nietert et al., 2022, and others. And as the authors listed, Alvarez-Melis, D. and Fusi, N. 2020 is a milestone for OT-based dataset distance. It totally makes sense to fill gap between the two subjects.", "essential_references_not_discussed": "References are sufficient. I'd add work by Ho et al who contributed to some theoretical findings around sliced OT or Kolouri et al. as an example of the applications of sliced OT but that's not necessary.", "other_strengths_and_weaknesses": "A weakness of the proposed sliced OTDD is the extra hyper-parameter — the number of moment $\\lambda$ as in (7). In the code, it’s set to 5. I didn’t find it discussed in the paper except in 246 where authors only briefly mentioned the principle of choosing $\\lambda$. It’s impact to the projection and hence the distances is unknown as well.", "other_comments_or_suggestions": "None.", "questions_for_authors": "The $\\lambda$-th scaled moment as defined in (7) is not how I usually \"scale\" a moment. What's the rationale behind a constant scaler?\n\nPlease discuss the impact of $\\lambda$ and provide more details for A.1.", "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": "BiHMVNaqPr", "reviewer": "Reviewer_FRUy", "summary": "The authors propose a similarity measure for supervised tasks based on sliced optimal transport. The data with labels are mapped to 1D slices via a feature projection map for features and the induced moment transform projection for label distributions. Then the dataset distance is defined using the 1D optimal transport distances. The proposed distance is shown to be a proper metric. Numerically, it is effective and efficient in predicting task transferability over multiple image and text datasets.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the time and constructive feedback of the reviewer.  We would like to extend the discussion with the reviewer as follows:\n\n**Q6**. Proposition 1 gives two conditions for the MTP to be injective, and the metric properties of s-OTDD are based on the injectivity of MTP. Does the MTP used in the experiments satisfy the conditions? Please justify the numerical implementation of MTP, including the assumption of infinite number of moments, the use of $\\sigma(\\Lambda^k)$ and how the metric properties are affected.\n\n**A6**.  Checking existence of all moments of a high-dimensional distribution is impractical due to a high computational cost. Moreover, we only observe random samples from the unknown distribution which makes the estimation of moments are also random. Therefore, in experiments, we implicitly assume that the underlying distributions of datasets have infinite moments. This assumption is motivated by the fact that one-dimensional projection becomes a Gaussian distribution in high-dimension [1,2] and Gaussian distribution has infinite number of moments. We then use zero truncated Poisson distribution (as reported at line 309-310) for the value of moments since it is a distribution over infinite countable natural numbers. If the underlying distributions of datasets do have infinite moments, MTP is injective and s-OTDD is a metric.  We would like to recall that s-OTDD is still a pseudo distance without injectivity of the MTP i.e., s-OTDD satisfies the triangle inequality, symmetry, non-negativity, and one direction of identity (is 0 when two datasets are the same). Therefore, s-OTDD is still a meaningful discrepancy for datasets as demonstrated through our experiments. \n\n[1] Fast approximation of the sliced-Wasserstein distance using concentration of random projections, NeurIPS 2021, Nadjahi et al.\n\n[2] Asymptotics of Graphical Projection Pursuit, The Annals of Statistics, 1984 Diaconis et al.\n\n\n**Q7.** In Definition 3, do feature projection and MTP have to share the same $\\psi^{(1)}$?\n\n**A7.** It is actually a typo in our Eq. (9).  The summation in the second term should involve $\\psi^{(i+1)}$. We have fixed this typo in the revision. In particular,  we have the following revised definition of the data point projection\n\n$\\mathcal{DP}^k_{\\psi,\\theta,\\lambda,\\phi}(x,q_y) = \\psi^{(1)} \\mathcal{FP}_\\theta(x) $ \n\n$+ \\sum_{i=1}^k\\psi^{(i+1)} \\mathcal{MTP}_{\\lambda^{(i)},\\phi}(q_y)$,\n\nwhere $\\psi=(\\psi^{(1)},\\psi^{(2)},\\ldots,\\psi^{(k+1)}) \\in \\mathbb{S}^k, \\theta \\in \\Theta, \\lambda =(\\lambda^{(1)},\\ldots,\\lambda^{(k)}) \\in \\Lambda^{k}, \\phi \\in \\Phi$.\n\n**Q8**. In Corollary 1, if both feature projection and MTP are injective, why the data point projection, as the sum of them, is also injective? Could the authors explain why the data point projection is a composition of Radon transform projection and the MTP?\n\n**A8**. Thank you for your questions. From the revised definition in **A7.** let $\\mathcal{FP}_\\theta(x)=t_1$ \n\nand \n$\\mathcal{MTP}_{\\lambda^{(i)},\\phi}(q_y)=t_i$ for $i=2,\\ldots,k$, we can rewrite the data point projection as:\n\n$\\mathcal{DP}^k_{\\psi,\\theta,\\lambda,\\phi}(x,q_y) = \\psi^{(1)} t_1+ \\sum_{i=2}^k\\psi^{(k)} t_i = \\psi^\\top t = \\mathcal{R}_\\psi(t)$,\n\nwhere $\\psi=(\\psi^{(1)},\\ldots,\\psi^{(k)})$, $t=(t_1,\\ldots,t_k)$, and $\\mathcal{R}_\\psi(t)$ is the Radon Transform of $t$ with projection parameter $\\psi$ as defined at line 162. Therefore, the data point projection is nothing but the Radon Transform of the stacked output of the feature projections and moment transform projections. Since the composition of injective functions is also injective, the data point projection is injective given the injectivity of the feature projection and the moment transform projections.\n\n**Q9**. Can you explain more about the feature projection choices for each of the dataset?\n\n**A9.** As written at line 194, we choose the feature projection based on the prior knowledge about the feature space. For example, if we believe that the feature space is an Euclidean space,  we can use the Radon projection (linear projection). If we work with a manifold, we can use geodesic projection [3]. If we work with images, we can use the convolution projection [4]. In our experiments, for the NIST dataset and the text classification dataset,  we apply linear projection by default. For image data, we use convolutional projection. We conducted a new comparison between linear and convolutional projections, available at [https://imgur.com/a/1NVv5AC](https://imgur.com/a/1NVv5AC). The results show that convolution-based projections not only require fewer projections but also tend to exhibit a stronger positive correlation.\n\n[3] Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds, JMLR, 2025, Bonet el al.\n\n[4] Revisiting Sliced Wasserstein on Images: From Vectorization to Convolution, NeurIPS, 2022, Nguyen et al. \n\n**Q10.** Typos\n\n**A10.** We have fixed them in the revision.", "claims_and_evidence": "1. Proposition 1 gives two conditions for the MTP to be injective, and the metric properties of s-OTDD are based on the injectivity of MTP. Does the MTP used in the experiments satisfy the conditions?  Please justify the numerical implementation of MTP, including the assumption of infinite number of moments, the use of $\\sigma(\\Lambda^k)$ and how the metric properties are affected.\n\n2. In Corollary 1, if both feature projection and MTP are injective, why the data point projection, as the sum of them, is also injective? Could the authors explain why the data point projection is a composition of Radon transform projection and the MTP?", "methods_and_evaluation_criteria": "Experiments are supportive and sufficient. However, can you explain more about the feature projection choices for each of the dataset?", "theoretical_claims": "The injectivity conditions of MTP, metric properties of s-OTDD and the numerical approximation bounds are checked and correct.", "experimental_designs_or_analyses": "Experiments are valid and sound.", "relation_to_broader_scientific_literature": "A fast dataset distance can be used in tasks that involve comparisons of datasets, such as transfer learning,", "essential_references_not_discussed": "Literature was well reviewed.", "other_strengths_and_weaknesses": "The proposed method using MTP to project labels onto 1D slices is novel and interesting. The computation speed gain is significant.", "other_comments_or_suggestions": "Typos:\n1. $\\Lambda\\subset \\mathbb{N}$ instead of $\\Lambda\\in \\mathbb{N}$\n2. In Preliminaries, at line 133, $n$ should be subscript.\n3. In Algorithm 1, at line 683, $i=1'$ should be $i'=1$", "questions_for_authors": "1. In Definition 3, do feature projection and MTP have to share the same $\\psi^{(1)}$?", "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": "jAtnriPa1A", "reviewer": "Reviewer_LbD9", "summary": "The paper proposes a sliced optimal transport-based dataset distance measure that uses moment transformation projection. The method improves upon a previously proposed optimal transport based dataset distance measure and makes the dataset distance computation more efficient. The paper presents a theoretical analysis of this distance and discusses various properties of the distance measure along with the computational complexity. The paper then presents evaluation of this distance on tasks such as transfer learning and data augmentation.\n\n## Update after the rebuttal\nI thank the authors for providing detailed answers to my questions. I have updated my rating of the work (2 --> 3).", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for the time and constructive feedback. \n\n**Q11**. Please provide large-scale experiments where OTDD is too expensive, and s-OTDD excels. \n\n**A11.** In the paper, we showed that OTDD cannot be used when the dataset size reaches more than 30,000 samples on MNIST and CIFAR10 in Figure 2. In addition, we added new experiments on the large-scale Tiny-ImageNet dataset, where the OTDD baseline is not feasible to compute in **A14**.\n\n**Q12.** The datasets used in the work to compute the distance tend to be very simplistic, such as *NIST-type datasets*.\n\n**A12.** We did conduct text classification with various datasets such as AG News, DBPedia, Yelp Reviews, Amazon Reviews, and Yahoo Answers (Section 4.2). In addition, we conducted experiment on Tiny ImageNet and CIFAR-10 (32×32 resolution). We also refer the reviewer to  **A14** where we added a new experiment on Tiny-ImageNet of size 224×224.\n\n**Q13.** The reason to use Pearson Correlation between OTDD and s-OTDD for evaluation is not clear.\n\n**A13.** s-OTDD can be seen as an alternative solution for OTDD when dealing with a large scale datasets. Therefore, measuring correlation with OTDD strengthens the claim that s-OTDD can replace OTDD while being more efficient.\n\n**Q14.** A large-scale evaluation on high dimensional datasets is required to demonstrate the effectiveness of the method over OTDD.\n\n**A14.** We have added a large-scale transfer learning experiment in Tiny-ImageNet dataset (224x224 resolution) where computing OTDD is infeasible. To compute dataset distances, we sampled 5,000 examples from each sub-dataset. The result of the experiment is visible in Figure 16 at this link [https://imgur.com/a/hHvg2T2](https://imgur.com/a/hHvg2T2). In the figure, we see that s-OTDD has a relatively good correlation with the classification accuracy.\n\n**Q15.** Why is OTDD a natural choice for dataset distance? Why should a new dataset distance try to be correlated with OTDD? Why should the correlation be the Pearson correlation (wouldn't a rank correlation ...)?\n\n**A15.** OTDD is a good dataset distance since it is model-agnostic, does not involve training, can compare datasets even if their label sets are completely disjoint.  Similar to OTDD, s-OTDD is also model-agnostic (requires no modeling on data), does not need to estimate any parameters, and can handle label sets that are completely disjoint. Since s-OTDD is proposed as an alternative solution to OTDD, it needs to have a good correlation with OTDD. \n\nFrom the suggestion of the reviewer, we have added the new figures with both Pearson correlation (denoted as $r$) and Spearman's rank correlation (denoted as $\\rho$) in Figure 12-14 at this link [https://imgur.com/a/hHvg2T2](https://imgur.com/a/hHvg2T2). We observe that the values of Spearman's rank correlation and Pearson correlation are almost similar in majority of cases for s-OTDD.\n\n**Q16**. The number of random projections ... What are the implications of it for larger datasets? The time complexity for the method increases to $O(n^2 \\log n)$ and space complexity $O(n(n + d))$... how many projections should one be looking at for the best results?\n\n**A16.** In practice, $L$ should be large enough with respect to the number of dimensions $d$ not the dataset size $n$. In particular, authors in [4] empirically show that we need $L\\approx 1.22 \\sqrt{d}$. In Figure 2, we showed that the computation gap between $L=1000$ and $L=10000$ is not large. Moreover, the computations for $L$ projections are independent, hence, we can utilize parallel computing to compute s-OTDD. Therefore, having a large $L$ is not a problem. In addition, we can reduce $L$ by using more advanced sampling techniques [5].\n\n[4] Sliced Wasserstein Autoencoders, Kolouri et al.\n[5] Quasi-Monte Carlo for 3D Sliced Wasserstein, Nguyen et al.\n\n**Q17**. ... s-OTDD can work with more data than OTDD, why is it not more predictive ... ?\n\n**A17.**.  In the paper, we actually compare s-OTDD and OTDD with the same number of data samples. We want to convey that s-OTDD is much faster than OTDD while being comparable in performance.  In Figure 2 and in **A14**, we showed that OTDD cannot be used when having large or high-dimensional datasets. \n\n**Q18.** Is s-OTDD predictive of metrics other than classification accuracy?\n\n**A18.**  We can compute the correlation between s-OTDD and any function of two datasets.  We added a new experiments to show that s-OTDD also correlates well with other performance metrics such as Precision, Recall, and F1 Score in Figure 15  [https://imgur.com/a/hHvg2T2](https://imgur.com/a/hHvg2T2). \n\n**Q19.** What types of datasets/tasks (...) can s-OTDD be used to measure distances for?\n\n**A19.** s-OTDD can also be interpreted as a distance between distributions over distributions. Therefore, s-OTDD can also be adapted to compare  distributions over point clouds, distributions over histograms, distributions over 3D shapes, multi-label datasets, and so on.", "claims_and_evidence": "The theoretical claims are well supported and convincing while the empirical evaluation seems rather limited.", "methods_and_evaluation_criteria": "The evaluation methodology makes sense but are rather small scale making it unclear of whether the approach is effective practically or not.", "theoretical_claims": "The claims and proofs look good but I did not check the details of the proofs though.", "experimental_designs_or_analyses": "The applications of transfer learning and data augmentation are relevant.", "relation_to_broader_scientific_literature": "The paper builds on the paper titled \"Geometric dataset distances via OT (OTDD)\" which had initially proposed a measure to calculate dataset distances via OT. The paper proposes a sliced OT based method which improves the efficiency of the previous approach. The paper follows the same experimental protocol used in the OTDD paper and shows that their distance measure is correlated with OTDD while being more efficient.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "Strengths:\n1. The problem of measuring dataset distance is very relevant.\n2. The proposal to use OT-based method and improving the efficiency over existing OTDD work is a good contribution.\n3. Theoretical analysis of the proposed metric is sound. \n\nWeaknesses:\n1. The empirical evaluation of the method is very basic and follows the OTDD paper very closely with no new experiments or datasets to demonstrate the effectiveness of s-OTDD beyond what had already been shown. Since efficiency is the main contribution, the work should include at least a large-scale experiment on an application where OTDD is not applicable/ very expensive and s-OTDD handles it with ease.\n2. The datasets used in the work to compute the distance tend to be very simplistic such as *NIST-type datasets.\n3. The reason to use Pearson Correlation between OTDD and s-OTDD for evaluation is not clear.", "other_comments_or_suggestions": "1. A large-scale evaluation on high dimensional datasets is required to demonstrate the effectiveness of the method over OTDD. \n2. Answers to below questions will be helpful for understanding the utility of the approach beyond just being faster than OTDD.", "questions_for_authors": "1. Why is OTDD a natural choice for dataset distance? Why should a new dataset distance try to be correlated with OTDD? Why should the correlation be the Pearson correlation (wouldn't a rank correlation be a better way of measuring the effectiveness of the new metric)?\n\n2. The number of random projections that lead to the best result seem to be of the same order as the number of samples in the dataset. What are the implications of it for larger datasets? The time complexity for the method increases to $O(n^2logn)$  and space complexity $O(n(n+d))$. Can you comment about this and give a guideline of how many projections should one be looking at for the best results?\n\n3. The results of sec 4.2 and 4.3 have shown that s-OTDD achieves similar correlation to accuracy for transfer learning/data augmentation problems compared to OTDD. But since s-OTDD can work with more data that OTDD, why is it not more predictive of the transfer learning/data augmentation performance? \n\n4. Is s-OTDD predictive of metrics other than classification accuracy? \n\n5. What types of datasets/tasks (beyond single label classification datasets) can s-OTDD be used to measure distances for?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper introduces s-OTDD, a computationally efficient and model-agnostic dataset distance based on sliced optimal transport, with a novel projection scheme (Moment Transform Projection) enabling the use of 1D Wasserstein distances. The approach is theoretically sound, with clear metric properties and approximation guarantees, and is shown to scale well to larger datasets, including settings where OTDD becomes infeasible. The reviewers appreciated the method’s practicality, its clean and general formulation, and its relevance to tasks such as transfer learning and data augmentation. However, some concerns were raised about the experimental validation. In particular, the evaluation setup largely follows that of prior work on OTDD, with limited exploration of new applications or more complex empirical scenarios. While the rebuttal included an additional experiment on Tiny ImageNet, showing that s-OTDD can operate where OTDD cannot, it remained limited to relatively standard classification settings and subsampled datasets, and did not thoroughly address cases involving, for instance, significant label imbalance, multi-label tasks, or structured prediction. Additionally, while the authors clarified many technical questions, including the injectivity assumptions and hyperparameter choices (such as the number of moments), a more systematic exploration of these design decisions, as well as an expanded discussion of practical limitations (e.g., s-OTDD being a pseudo-metric when injectivity is not guaranteed), would strengthen the final version. Reviewers also noted that, although Pearson correlation with OTDD is a reasonable baseline metric, incorporating rank correlations such as Spearman’s would provide a more robust view of how well s-OTDD aligns with practical downstream performance, especially for transfer tasks, and such metrics were explored only late in the rebuttal. Despite these limitations, the paper offers a timely, well-executed contribution with both theoretical and practical value, and provides a valuable tool for scaling dataset comparison methods to modern data sizes. On balance, while some aspects of the work would benefit from further development and broader validation, I believe the strengths of the work outweigh the weaknesses, and I recommend acceptance.
Decision: Accept (poster)