Expected Sliced Transport Plans
Authors: Xinran Liu
Rocio Diaz Martin
Yikun Bai
Ashkan Shahbazi
Matthew Thorpe
Akram Aldroubi
Soheil Kolouri
Venue: ICLR 2025 Poster (2025)
Keywords: Optimal Transport
Sliced Wasserstein
Transportation Plan
Probability Metrics
Primary area: other topics in machine learning (i.e., none of the above)
TL;DR: We propose a novel method for constructing a transportation plan between two probability measures by utilizing the optimal transport plans between their slices and introduce a new transport-based metric for probability measures.

Abstract: The optimal transport (OT) problem has gained significant traction in modern machine learning for its ability to: (1) provide versatile metrics, such as Wasserstein distances and their variants, and (2) determine optimal couplings between probability measures. To reduce the computational complexity of OT solvers, methods like entropic regularization and sliced optimal transport have been proposed. The sliced OT framework improves efficiency by comparing one-dimensional projections (slices) of high-dimensional distributions. However, despite their computational efficiency, sliced-Wasserstein approaches lack a transportation plan between the input measures, limiting their use in scenarios requiring explicit coupling. In this paper, we address two key questions: Can a transportation plan be constructed between two probability measures using the sliced transport framework? If so, can this plan be used to define a metric between the measures? We propose a ‘lifting’ operation to extend one-dimensional optimal transport plans back to the original space of the measures. By computing the expectation of these lifted plans, we derive a new transportation plan, termed expected sliced transport (EST) plans. We further prove that using the EST plan to weight the sum of the individual Euclidean costs $\|x - y\|^p$ for moving from $x$ to $y$ results in a valid metric between the input discrete probability measures. Finally, we demonstrate the connection between our approach and the recently proposed min-SWGG, along with illustrative numerical examples that support our theoretical findings.

Published as a conference paper at ICLR 2025
EXPECTED SLICED TRANSPORT PLANS
Xinran Liu"), Rocío Martín Diaz Yikun Bai', Ashkan Shahbazi',
Matthew Thorpe", Akram Aldroubi*. Soheil Kolouri'
1 DDepartment of Computer Science, Vanderbilt University, Nashville, TN, 37235
Depaatmentt of Mathematics, Tufts University, Medford, MA 02155
3 DDeparteent of Statistics, University of Warwick, Coventry, CV4 7AL. UK
'Department of Mathematics, Vanderbilt University, Nashville, TN, 37235
ABSTRACT
The optimal transport (OT) problem has gained significant traction in modern ma- chine learning for its ability to: (1) provide versatile metrics, such as Wasserstein distances and their variants, and (2) determine optimal couplings between probabil- ity measures. To reduce the computational complexity of OT solvers, methods like entropic regularization and sliced optimal transport have been proposed. The sliced OT framework improves efficiency by comparing one-dimensional projections (slices) of high-dimensional distributions. However, despite their computational efficiency, sliced- Wasserstein approaches lack a transport plan between the input measures, limiting their use in scenarios requiring explicit coupling. In this paper, we address two key questions: Can a transport plan be constructed between two probability measures using the sliced transport framework? If so, can this plan be used to define a metric between the measures? We propose a 'lifting" operation extend one-dimensional optimal transport plans back the original space of the measures. By computing the expectation of these lifted plans, we derive a new transport plan, termed expected sliced transport (EST) plans. We further prove that using the EST plan to weight the sum of the individual Euclidean costs ||x- y|| for moving from to y results in a valid metric between the input discrete probability measures. Finally, we demonstrate the connection between our approach and the recently proposed min-SWGG, along with illustrative numerical examples that support our theoretical findings.
INTRODUCTION
The optimal transport (OT) problem (Villani 2009) seeks the most efficient way to transport a distribution of mass from one configuration to another, minimizing the cost associated with the transportation process. It has found diverse applications in machine learning due to its ability to provide meaningful distances, i.e., the Wasserstein distances (Peyré & Cuturi 2019). between probability distributions, with applications ranging from supervised learning (Frogner et al. 2015) to
generative modeling (Arjovsky al. 2017). Beyond merely measuring distances between probability measures, the optimal transport plan obtained from the OT problem provides correspondences between the empirical samples of the source and target distributions, which are used in various applications, including domain adaptation (Courty et al.) (2014), positive-unlabeled learning (Chapel al. 2020). texture mixing (Rabin et al., 2011), 1) color transfer (Rabin et al.) 2014), image analysis (Basu et al. 2014), and even single-cell and spatial omics (Bunne et al. 2024). to name few.
One of the primary challenges in applying the OT framework large-scale problems is its com- putational complexity. Traditional OT solvers for discrete measures typically scale cubically with the number of samples N (i.e., the support size) (Kolouri et al. 2017). Precisely, when using linear programming for solving the OT problem, the computational complexity is of order O(N3 log N)). This computational burden has spurred significant research efforts to accelerate OT computations. Various approaches have been developed to address this challenge, including the seminal work (Cuturi 2013), which introduces entropic regularization, leveraging the Sinkhorn algorithm to compute OT efficiently with quadratic time complexity; multiscale methods (Schmitzer) 2016); and projection-
"Equal contribution.

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based techniques such sliced- Wasserstein distances (Rabin et al. 2011 and robust subspace OT Paty & Cuturi 2019). Each of these methods has its own advantages and limitations.
For instance, the entropic regularized OT is solved via an iterative algorithm (i.e., the Sinkhorn algorithm) with quadratic computational complexity per iteration. However, the number iterations required for convergence typically increases the regularization parameter decreases, which can offset the computational benefits of these methods. Precisely, the complexity of Sinkhorn method is of order ON2 log(N) ( /X²),, where is the regularization parameter Dvurechensky et al. (2018); Altschuler et al. 2017). Additionally, while entropic regularization interpolates between Maximum- Mean Discrepancy (MMD) (Gretton al. 2012) and the Wasserstein distance (Feydy et al. 2019).
does not produce true metric between probability measures. Despite not providing a metric, the entropic OT provides transport plan, i.e., soft correspondences, alheit not the optimal one. On the other hand, sliced- Wasserstein distances offer linearithmic computational complexity, enabling the comparison of discrete measures with millions of samples. These distances are also topologically equivalent to the Wasserstein distance and offer statistical advantages, such as better sample complex- ity (Nadjahi et al. 2020). However, despite their computational efficiency, the sliced- Wasserstein approaches do not provide a transport plan between the input probability measures, limiting their applicability to problems that require explicit coupling between measures.
transportation plan be constructed between two probability measures using the sliced transport framework? If so, can the resulting transportation plan be used to define metric between the two probability measures? Within the sliced transport framework, the "slices" refer to the one-dimensional marginals of the source and target probability measures, for which an optimal transportation plan is computed. Crucially, this optimal transportation plan applies to the marginals (i.e., one-dimensional probability measures) rather than the original measures. To derive a transportation plan between the source and target measures, this optimal plan for the marginals must be "lifted" back to the original space.
In this paper. address two central questions: First,
For discrete measures with equal support size, N, and uniform mass, 1/N, the optimal transportation plan between marginals is represented by correspondence matrix, specifically an N x N permutation matrix. Previous works have used directly the correspondence matrix obtained for a slice as a transportation plan in the original space of measures (Rowland et al. 2019; Mahey et al., 2023). This paper provides a holistic and rigorous analysis of this problem for general discrete probability measures.
Our specific contributions in this paper are:
1. Introducing computationally efficient transport plan between discrete probability measures, the Expected Sliced Transport (EST) plan. Motivated by the first question highlighted above, construct this transport plan the average of transport plans computed via lifting scheme involving -iedimmenioonal sliced transport plans. (See Definition 2.4 below..
2. Providing a distance for discrete probability measures, the Expected Sliced Transport (EST) distance. (See Definition 2.6 and Theorem below.)
3. Offering both theoretical proof and an experimental visualization showing that the EST distance is equivalent to the Wasserstein distance (and to weak" convergence) when applied to discrete measures.
4. Illustrating the potential applicability of the proposed distance and transport plan, with a focus interpolation and classification tasks.
2 EXPECTED SLICED TRANSPORT
2.1 PRELIMINARIES
Given a probability measure μ E P(R") and unit vector 0 E gd Rd we define f (0, :) # a to be the 0-slice of the measure μ4, where (0,z) 0 0Th denotes the standard inner product in Rd. For any pair of probability measures with finite p-moment (p > 1) n² ,u² E Pp (R.). one can pose the following two Optimal Transport (OT) problems: On the one hand, consider the
classical OT problem, which gives rise to the p- Wasserstein metric:
(F | v1ldd(
1/8
Wpln (0(m),1)) = min rEI(u-
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60)
(8)
0.5
10
Lift
0.1
0.5
0.0
0.4
Figure 1: Our goal in this paper is to construct transportation plans between d-dimensional measures (d > 1) using their one-dimensional marginals, or slices. For two measures J (represented by green circles) and μr² (represented by blue circles), we first derive one-dimensional transport plan Ag =p² from their slices along a unit vector 0 (indicated by triangles). This one-dimensional transport plan is then lifted back to the original d-dimensional space to produce transport plan between the measures, To .²" Finally, these transport plans are aggregated over all directions E g In (a), the measures u and μ² are uniform, with their masses non-overlapping when projected in the direction of 8. In (b), n² and y² are non-uniform, resulting in overlapping masses when projected along 8. For further details, see Remark A.I in Appendix I..
where || II denotes the Euclidean norm in IRd and [(u ,²²) c P(R x R4) is the subset of all probability measures with marginals 4² and μ². On the other hand, for a given E gd-1, consider the one-dimensional OT problem:
uu-uulddouu. 1/P
W (0 02p2)=
min Ao 8gx2)
In this case, since the measures 0z 02 can be regarded as one-dimensional probabilities in P(R), there exists unique optimal transport plan, which we denote by Ag Au1-22 (see, for e.g. (Villani 2021 Thm. 2.18, Remark 2.19)], (Maggi 2023 Thm. 16.1)).
2.2
ON SLICING AND LIFTING TRANSPORT PLANS
In this section, given discrete probability measures 2², 12² E Pp(() we describe the process of slicing them according to a direction E gd-1 and lifting the optimal transport plan Ag . which solves the 1-d OT problem (2). to get a plan in I'(2², ,²²). Thus, we obtain a new measure, denoted 7 .,² in P(R R with first and second marginal μ and μ²², respectively. For clarity, first describe the process for discrete uniform measures and then extend it to any pair of discrete measures.
2.2.1 ON SLICING AND LIFTING TRANSPORT PLANS FOR UNIFORM DISCRETE MEASURES
Given E N. consider the space P(N) of uniform discrete probability measures concentrated at N particles in Rd, that is,
PRN)(R
Sz, x E RR, Vi {1, N]
N
Let n²- N Em= S2,, n Σm= S3s E (R))( where 24, My E R.d and og, denotes Dirac measure located at æ; (respectively for 8,,). Let us denote by (((-11) the uniform measure on hypersphere gd-1 C Rd. In this case, the 0-slice of μ² is represented by 0gp²² 인 Σm 80.2. and similarly for 2² Let Sy denote the symmetric group of all permutations of the elements in the set [N] := (1...., NJ.. Let Çe, E SN denote the sorted indices of the projected points {0 xz N 1/ j=1 respectively, that is,
and {0 Uj
0 c, 1(1) < 5 (2) < ... 5 50-0 Ts ((N) and 0 yr, ((1)
5 S0.9., 1(2) S ... 0 Vr, I(N (3)
The optimal matching from 1² to 12 for the problem
2 is induced by the assignment
0 , (1) + 0 Vo
V1 SisN.
(4)
We define T 2 {x
ΦN} + .VI..... ........ aNf the lifted transport map between jd² and μر² by: Tyl 2² (xx)
V1 siSS
(5)
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Rigorously, Tb " is not necessarily a function defined on (.i...., ANY but on the labels (l...., N), as two projected points 0 and 0 I, could coincide for o As result, it is more appropriate to work with lifted transport plans. Indeed, the matrix wn . E ERux given by [//N if = To (G)
ug (i.j) (1.3)= = 10 otherwise
To
(6)
encodes the weights of the optimal transport plan between 0 and 0g given by uh "²²
Ag
Σ
(1.1)0(0-2.005)))
as well as the weights of the lifted transport plan between the original measures J and μ² according the 0-slice defined by
ra To
Σ aa (i.5))(e)))))
(8)
This new measure 2/ 2 E P(R x Rd) has marginals y and x², While % is not necessarily optimal for problem D. it can interpreted as a transport plan in T(u*, n²) which is optimal when projecting j²¹ and f the direction of 0. See Figure (a) for a visualization.
" .²²
2.2.2 ON SLICING AND LIFTING TRANSPORT PLANS FOR GENERAL DISCRETE MEASURES
Consider discrete measures n² E (T.). In this section, we will use the notation je ExERe p(z) where p(z) 2 0 for all x E Rd, p(z) + 0 for at most countable many points x E R and Emer= p(x) = 1. Similarly, n²² = EVER* q(y), for a non-negative density function in Rd with finite countable support and such that EyeRs q(v) 1. Given 0 E gd-1, consider the equivalence relation defined by:
rag x if and only if 0
We denote by 5 the equivalence class of E IRd By abuse of notation, we will use interchangeably that z6 is a point in the quotient space / and also the set fa' R f. x f. x'), which is the orthogonal hyperplane to 0 that intersects æ. The intended meaning of x² will be clear from the context. Notice that, geometrically, the quotient space Rd /~10 is the line R in the direction of 0.
Now, we interpret the projected measures 0z 041² as 1-dimensional probability measures in P[IRd// /~w)) given by 0z ExTERs/NO P0z where P(50) = = p(x'), and similarly, 1² ESPERt/~O (TT") yōge, where Q(T") = Ev'gg q(v').
Remark 2.1. Notice that if P(=") = O, then p(x') = for all x' E 2", or, equivalently, pf(() o O, then P(x0) # 0 (where is any 'representative" of the class 3²) Similarly for Q.
Consider the optimal transport plan Ar .p² E T(0 0q c P(R x Rd /~a) between 0 #/ and 12 which unique for the OT problem (2) as we are considering | I-dimensional probability measures. Let us define
ut n² (I, 4)
pla)g(y)) P P(2")Q"") )
AB .p² (1(=", 7))
pf + 0 and g(v) + 0 p() 0 ((v)
which allows us to generalize the lifted transport plan given in 6 in the general discrete case: uy .n²
10 2
Σ ZERA VER4
Σ
(I, (1)
See Figure (b) for visualization.
Remark 2.2. Notice that this lifting process can be performed by starting with any transport plan Ao E T(0 u 00 )²), ) but in this article we will always consider the optimal transport plan, iee.
Ag = Ay'.a The reason why we make this choice is because this will give rise to a metric between discrete probability measures: The EST distance which will be defined in Section 2.3
Lemma 2.3. Given general discrete probability measures 4d and pi in RRd, the discrete measure
y² ua at u² defined by (9) has marginals 4² and 2², that is, tt E r(# (P)) C P(Rd Rd).
We refer the reader to the appendix for its proof.
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2.3 EXPECTED SLICED TRANSPORT (EST) FOR DISCRETE MEASURES
Leveraging on the transport plans %0 described before, this section we propose a new transport
plan 7t .² T(p', p²), which will give rise measures.
metric in the space of discrete probability
Definition 2.4 (Expected Sliced Transport plan). Les a E P(S)-1). For discrete probability measures u², u²2 in Rdd, we define the Expected Sliced Transport (EST) plan qu l²² E P(R x R2)) by
spa 2 = Beno lio
where each u². is given by 19, that is,
(10)
gpa 1² (f(z, =
10 "n'"" (H(x)8)))do(O),
V Rd..
-0-1
In other words, u²² =mTERt EVERS Un' (2, (1.1))(z.0)) where the new weights are given by
Tu'm" (I,U)= p()))(w)) Soa-. ( 5'1 (o) ifp()) + 0 and q(v) # 0
10
((E))
otherwise
consequence of Lemma 2.3) qu N²² E &T(p', .,22,, i.e. has marginals y and μ². (See Lemma A.5 in the appendix.)
Remark 2.5. The measure qu u² is well well-defined and, moreover, (as
Definition 2.6 (Expected Sliced Transport distance). Let E P((d-1) with supp(a) Sd-1, We define the Expected Sliced Transport discrepancy for discrete probability measures p .2p in Rd by
Dp(u ,12);
11-1111- vll" (((x,)))) ( ΣE DER yERR
where sp'n" x² is defined by 10 (11)
Remark 2.7. By defining the following generalization of the Sliced Wasserstein Generalized Geodesics (SWGG) dissimilarity presented in Mahey et al. (2023).
1/8
Dpln' (2:0)=
Σ #ER" yERU
| vll ((x,0))
(12)
we can rewrite 77 as
Dp(p' 11)) E (D 12:01]
Remark 2.8. Since the EST plan sp'da 1²2 a transport plan, we have that
Wp(n*, (0(m),1²) s Dp (.t.pppp ,1²).
In Appendix B we show that they define the same topology for the space of discrete probabiliies. Remark 2.9 (EST for discrete uniform measures and the Projected Wasserstein distance). Consider uniform measures n² E 82, 1 E Sy E PIN)(R and for 0 E gd-1, let Se, Ta E Sy be permutations that allow us to order the projected points in 3 Notice that if a = USd- ), by using the formula for each assignment given 0 and noticing that Tg o E SN. can re-write (I7) as
1 N
Dplut p3pp E0-u(3a11)
(13)
Therefore, we have that the expression for (p(..)) given by 13 coincides with the Projected Wasserstein distance proposed in (Rowland et al. 2019 Definition 3.1). Then, by applying (Rowland let al. 2019 Proposition 3.3), we have that the Expected Sliced Transport discrepancy defined in Equation (13) is metric on the space P P(N)) (Rd). We generalise this in the next theorem.
Theorem 2.10. The Expected Sliced Transport Dplcs -) defined in 77 a metric in the space of finite discrete probability measures in Rdd
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Sketch of the proof of Theorem 2.10 For the detailed proof, refer the reader to Appendix A Here, we present a brief overview of the main ideas and steps involved the proof.
The symmetry of Dp(:.) follows from our construction of the transport plan pp"."" which is based on considering family of optimal 1-d transport plans {Ag ,* leesa-1. The identity of indiscernibles follows essentially from Remark 2.8 To prove the triangle inequality we use the following strategy:
1. We leverage the fact that Pp(:-) is a metric for the space P(N)) (IR.) of uniform discrete probability measures concentrated at particles Rdd (Rowland al. (2019) to prove that Pp(:.) is also metric the set of measures in which the masses are rationals. To do so, we establish correspondence between finite discrete measures with rational weights and finite discrete measures with uniform mass (see the last paragraph of Proposition A.7
2. Given finite discrete measures μ² μ²² we approximate them, in terms of the Total Variation norm, by sequences of probability measures with rational weights ] {u J. supported on the same points as f and μ²² respectively. We then turn our attention on how the various plans constructed behave as the increases and show the following convergence results in Total Variation norm:
A me M
converges A/ .u
(a) The sequence
aEN converges % ,=
% 12
(b) The sequence
MEN
(c) The sequence gaampe
converges to 5/ ,12
MEN
As a consequence, we obtain lim Dp = Dp(u?,
Finally, given three finite discrete measures μ¹ n²² nr² proceed in point 2 by considering sequences of probability measures with rational weights {un), {uā) {nr supported on the same points as u², 2² n2² respectively, that approximate the original measures in Total Variation, obtaining
Pp(u", (2)
lim Dp(1,, (2)) S lim Dp(Hus n-c
Dp ( Dp(n' ,1² Dplue .1²)
n-000
where the equalities follows from point 2 and the middle triangle inequality follows from point
EXPERIMENTS
3.1 COMPUTATIONAL EFFICIENCY
Lx129
In practice. compute the EST plan I0 and distance [I). we sample unit vectors (0 L (or slices) uniformly from gd-1 to approximate zu' ,²² + + To,
Nue L-1124
Shrtrem 100
u², For empirical mea-
Marbo PUOOOS. V
sures y²,, n12 in RRd of size N, the computational complexity of our proposed EST method is of order O(L + 2). detailed analysis is pro- vided in Appendix C Additionally, Figure 2 presents the wall-clock time of the Sinkhorn al-
Figure 2: Wall-clock time plot of Sinkhorn algo- rithm for entropic OT with varying À and the EST method for different numbers of slices, L. as the size of the empirical measures increases.
gorithm, used compute entropic OT for various values of the regularization parameter X, and the proposed EST method, evaluated with different numbers L of slices, as the size N of the support of the discrete measures increases. The wall clock times are averaged over 10 runs.
3.2 COMPARISON OF TRANSPORT PLANS
Figure 3 iillustrates illustrasss the behavior of different transport schemes: the optimal transport plan for We(-, (e(..)), +). the transport plan obtained by solving entropically regularized transportation problem between the source and target probability measures, and the new expected sliced transport plan 7. We include comparisons with entropic regularization because it is one of the most popular approaches, as it allows for the use of Sinkhorn's 's algorithm. From the figure, we observe that while promotes mass splitting, this phenomenon is less pronounced than in the entropically regularized OT scheme. This observation will be revisited in Subsection 3.3

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Sepected Firee Tarepert
Frergg Sleed
Figure 3: Depiction of transport plans (an optimal transport plan, plan obtained from solving an entropically regularized transport problem, and the proposed expected sliced transport plan) between source (orange) and target (blue) for four different configurations of masses. The measures the left and right panels are concentrated on the same particles, respectively; however, the top row depicts measures with uniform mass, while the bottom row depicts measures with random, non- non-uniform mass. transport plans shown as gray assignments and as n heat matrices encoding the amount of mass transported (dark color no transportation, bright color more transportation), where n is the number of particles on which the source measure concentrated, and m 2n) is the number of particles which the target measure is concentrated.
3.3 TEMPERATURE APPROACH
Given je,, nr discrete probability measures, perform the expected sliced transportation scheme by using the following averaging measure o, < (Sd-1) on the sphere:
e- (2:00
da,(0) Jga-1 e (D' +²'0'' 00))))) do'
d0,
(14)
where Dp(n' .,::0) is given by 12). and 7 2 is a hyperparameter we will refer to as the temperature (note that increasing corresponds to reducing the temperature). If 0, then an U($-11). However, when * 0, a, is probability measure on $d-1 with density given by 114 which depends the source and target measures jrt and n1² We have chosen this measure because it provides a general parametric framework that interpolates between our proposed and min-SWGG (Mahey et al. (223)), as the EST distance approaches min-SWGG when + co. For the implementations, we use
scheme with the uniform measure
DD ,1:00)
a7') PE= TDJ (a""0""""""""""
(15)
where L represents the number of slices unit vectors 0 ...... 0z S Figure illustrates that as t - oo, the weights used for averaging the lifted transport plans converge to a one-hot vector, i.e., the slice minimizing (p', (p:,,=2: dominates, leading the visualization we have used source ju² and target f uniform probability measures concentrated on different number of particles. For consistency, the configurations are the same as in Figure
transport plan with fewer mass splits. For
3.4 INTERPOLATION
We use the Point Cloud MNIST 2D dataset Garcia 2023), a reimagined version of the classic MNIST dataset (LeCun 1998), where each image is represented set of weighted 2D point clouds instead
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OT pln canpln
Exoested Sieed Mon. 0.00
Expected Sloce Pan 1.0000 Source lagee sangles
Expectec Sieed Mon. T- 10.00
Expected Sliced Pen. T- 100 laget saripls
legel
sargee
OT pln
Excected Siced Ron. T 0.00
Expected Sliee Pln. T 10.000
Expected Siced Ras, 100.00
Expected Siized Ron. T 1000.00
Source "aaget carples
Sugel
bugee sanqpio
legett samplis
Saage sirqpl
Figure 4: The effect of increasing (i.e., decreasing temperature) on the expected sliced plan. The left most column shows the OT plan, and the rest of the columns show the expected sliced plan as function of increasing The right most column depicts that expected sliced plan recovers the min-SWGG Mahey al. (2023) transportation map.
Sloed
100.00
7
7
7
O
776
0
6
Jud10
Expected Sliced 10.00
7
7
6
7
7
b
6
Entrai Inter peteroeationnoooo 1=1.00
Interadation da Expected Sized Ran. 1 1.00
7
7
b
7
Entropic Interpoation.. 1=10.00
Inter beeroiidonni vs Expected Siied Ran. T=0000
6
6
7
Figure 5: Interpolation between two point clouds via ((1 ((1-t) t)x ty) #y, where y is the optimal transport plan for W2(s.)) -) (top left), the transport plan obtained from entropic OT with various regularization parameters (bottom left), and the EST for different temperatures (right).
of pixel values. Figure we illustrate the interpolation between point clouds that represent digits 7 and 6. Since the point clouds are discrete probability measures with non-uniform mass, we perform three different interpolation schemes via ((1 t)x + #7 where < < 1 for different transport plans namely:
1. 7 7", optimal transport plan for W²(+,))
2. a transport plan obtained from solving an entropically regularized transportation problem (performed for three different regularization parameters X);
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3. Y= yy: =y: the expected sliced transport plan computed using o, given by formula (14 (or 15 for implementations) for four different values of the temperature parameter T.
As the temperature increases, the transport plan exhibits less mass splitting, similar to the effect of decreasing the regularization parameter À in entropic OT. However, unlike entropic OT, where smaller regularization parameters require more iterations for convergence, the computation time for expected sliced transport remains unaffected by changes in temperature.
3.5 WEAK CONVERGENCE
Jass
Given finite discrete probability measures j and v, we consider st,, for 0 S L < 1, the Wasserstein geodesic between lr and v. particular, 14, is a curve of probability measures that interpolates μ and v, that is pi0 μ and #1 = =V. Morcover, we have that W2(He, v) = (1
Dissimilarity Based Transport Plans
10a
(1,, + 0 as t + 1, equivalently, say bh converges in the weak"-topoke -topology to v. Figure 6 illustrates that the expected sliced distance also satisfies
10
D2(pe, v) - 0 as + 1. Indeed, this experimental conclusion is justified by the following theoretical result:
Dipa v)
OT
10
Outer
Sinkhom, 1.000
Let Pn E P(22) be discrete measures with finite or countable support, where Q C. _e compact. Assume < U((d))) Then, Dp(Pa.H) + O if and only if Hn
Sinkhom, 1010.0
EST, r 0.00
10
EST, == 1.00
EST. r- 10.00
3 n..
EST, 100.00
10
10
109
We present its proof in Appendix
B
Figure 6: Discrepancies calculated from transport plans between frr and v, when - as a function of t E [0, 1].
For the experiment, μ and y are chosen to be discrete mea- sures with N particles of uniform mass, sampled from two Gaussian distributions (see Figure 6 top). For differ-
14t
ent values of time, S S 1, we compute different discrepancies, MN Σjك ||= - yソ 11?"))""
calculated for various transport plans: (1) the optimal transport plan, (2) the outer product plan H4 @ v. (3) the plan obtained from entropic OT with two different regularization parameters X, and (4) our proposed expected sliced plan computed with a, given (14) for four different temperature parameters r. As 14 converges to it is evident that both the OT and our proposed EST distance approach zero, while the entropic OT and outer product plans, as expected, do not converge to zero.
3.6 TRANSPORT- BASED EMBEDDING
Following the linear optimal transportation (LOT) framework, also referred to as the Wasserstein or transport-based based embedding framework ( (Wang et al. 2013 Kolouri et al.) 2021 Nenna & Pass] 2023 Bai et al. 2023 Martín et al., 2024), we investigate the application of proposed transport plan in point cloud classification. Let Nuu EN a₁őz, denote a reference probability measure and let Ak = Ey= 3 denote a target probability measure. Let MoodPh be a transport plan between Po and (4g, and define the barycentric projection (Ambrosio et al. 2011 Definition 5.4.2) of this plan as:
Ni. Mo
1
(o. =
i L
(16)
Tip
Note that bz(*(Howtk) represents the center of mass to which o; from the reference measure is transported according to the transport plan Ho.ue When Ho.ps the OT plan, the LOT framework of Wang et al. (2013) uses
(p(px)], == by(7"0o-1k) Zze i 1.....,, N
(17)
as an embedding & for the measure 14g. This framework, as demonstrated in Kolouri et al. (202I). inntaaiiooniivariantt embedding for sets of features and, more broadly,
can be used to define point clouds. More precisely, given a point cloud 3 = {(9) ) where Σnn By 1 and 3; represent the mass at location yj, we represent this point cloud as discrete measure pkk-

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Entropic OT, A-100
Entropic
20
-0
%96.2
896.8
992.2
Effect of Reference Size
OT
Entropic OT
EST
0=
EST. T=0.1
66.72
67.06
65.53
9
88.45
89.24
88.72
16
93.51
94.39
92.88
36
96.14
96.4
94.47
96.22
96.79
94.4
100
96.21
96.82
94.3
400
94.43
96.86
92.77
%94.3
992.3
Figure t-SNE visualization of the embeddings computed using different transport plans, along with the corresponding logistic regression accuracy for each embedding. The t-SNE plots are generated for embeddings with a reference size table shows the accuracy of the embeddings 100, and for EST, we used L 128 slices. The function of reference size N. For the table, the regularization parameter for entropic OT set to = 10, and for EST, the temperature is set to 0 with L 128 slices. Lastly, the plot on the bottom left shows the performance of EST, when N 100 and L = 128, as a function of the temperature parameter,
In this section, we use reference measure with N particles of uniform mass to embed the digits from the Point Cloud MNIST 2D dataset using various transport plans. We then perform logistic regression on the embedded digits and present the results in Figure 7 The figure shows a 2D t-SNE visualization of the embedded point clouds using: (I) the OT plan, (2) the entropic OT plan with two different regularization parameters, and (3) expected sliced plan with two temperature parameters (using N 100 for all methods). In addition, we report the test accuracy of these embeddings for different reference sizes.
Lastly, we make an interesting observation about the embedding computed using EST. As we reduce the temperature, i.e., increase r, the embedding becomes progressively less informative. We attribute this to the dependence of a, on μ4g. In other words, the embedding is computed with respect to different a, for different measures, leading inaccuracies when comparing the embedded measures. This finding also suggests that the min-SWGG framework, while meritorious, may not be well-suited for defining a transport-baas based embedding.
Additionally, we refer the reader to Appendix D for a similar classification experiment using more complex dataset: ModelNet40.
CONCLUSIONS
In this paper, we explored the feasibility of constructing transport plans between two probability measures using the computationally efficient sliced optimal transport (OT) framework. We introduced the Expected Sliced Transport (EST) framework and proved that it provides valid metric for comparing discrete probability measures while preserving the computational efficiency sliced transport and enabling explicit mass coupling. Through a diverse set of numerical experiments, we illustrated the behavior of this newly introduced transport plan. Additionally, we demonstrated how the temperature parameter in our approach offers flexible framework that connects our method the recently proposed min-Sliced Wasserstein Generalized Geodesics (min-SWGG) framework. Finally, the theoretical insights and experimental results presented here open up new avenues for developing efficient transport-ban algorithms in machine learning and beyond.
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REFERENCES
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A PROOF OF THEOREM 2.10 METRIC PROPERTY OF THE EXPECTED SLICED DISCREPANCY FOR DISCRETE PROBABILITY MEASURES
A.1 PRELIMINARIES ON EXPECTED SLICED TRANSPORTATION
Remark A.1 (Figure I. Let us elaborate on explaining Figure 7 (a) Visualization for uniform
discrete measures yi 2/4 (ds, + ōs2 + 62z) (green circles). f [60, + +8y2 + 5ys) (blue circles) in P(N) (IR)) with n = 3. Given an angle 0 (red unit vector), when sorting {0 xs xx]i=1 and {0- yj 1/133= we use permutations Çe and To given by (o(2) = 1, (1() = 2: So (3) = 3. and To (1) = 1. (3)3) = 2, To (2) = 3. The optimal transport map between 0 2 (green triangles) and 02 (blue triangles) given by the following assignment: 8 ". ", (1)
8 y1 0 yr '1y F3 ++ + y2 = ,. (3y This gives
D2
0 (2) 0 21 +
0- y/ = 0. ''7"'0 (2y 0 (3)
.²² given in (7 with represented by solid arrows the first panel. The lifted plan To na' "² defined in (8) is represented in the second panel by dashed assignments. (b) Visualization for finite discrete measures μ = 0.16I, +0.35 +0.65 (green circles), μ² 0.56y, +0.35 1..6,,=00.66 +0.26 (blue circles). When projection according a given direction 8 the locations with green masses 0.3 and 0.6 overlap, as well as the locations with blue masses 0.2 and 0.3. Thus, the mass of 0 /=² is concentrated at two green points (triangles) on the line determined by 0, each one with 0.1 and 0.9 of the total mass, and similarly O 0z is concentrated at two points (blue triangles) each one with 0.5 of the total mass.
rise to the plan A Ag'
2.3 that is, us show that each measure % .²² defined in (9) is a transport
Now, let us prove Lemma plan in [(u .²²).
Proof of Lemma 2.3 Let Rd First, if p(z) = O, then ut .n² (z, v) 0 for every y E Rd, and so
1o
a "²
[1x)
x Rd)
0 p(x) /² ({x)). Now, assume that p(x) + 0, then
Σ up (x, v) = P(1o)
plx)
(lv) Q(T")
AB
AH'#" #² (1(E" 1²)
VERd
yER <
p(x)
Σ
Ea(o) alv))
Σ
QiP No u² A6' (((F))8)))
P(Jo)
8e ER (oi:: 2=0
NEES
p(x)
Σ
Q17²)
AB
P(10)
8e ERE /ro:: ((7
p(x)
Σ
N 1h
p(x) P
P(x0) = p(x).
P(1")
8 gRER4/Nu
Thus, %0 n' ({x) R = p(x) ((x)) for every Rd. Similarly, EZERo up 1 (x.1)) v)
q(y), or equivalently, 7/ 2² (R4 x {v)) = qxx))= 12 ((v)) for every y E Rdd
Remark A.2 (Expected Sliced Transport for uniform discrete measures). Let pr ,12 E ((NN(RR) of the form p¹ = N E Ezv 122 E Then, the expected sliced transport plan between pd and 1², qp² Egra (ng H²1 defines a discrete measure on P(Rd x Rdd) supported on [(x, Vi]bisE(N] where it takes the values
Rpa s²² ((2,,0;)) = -Sg- 10 ua' "f ((I,)11))do(0)
Vi (N,EE (N].
(18)
Thus, it can be regarded as an N x N matrix whose (injy-entry is given by (8). Moreover, each N x N matrix u''r s² defined by (6) can be obtained by swapping rows from the N x N identity matrix multiplied by 1/N, there are finitely many matrices (precisely, N! matrices in total). Hence, the function 0 + ub .²" is a piece-wise constant matrix-valued function. Thus, the function 0 2/2 n²²
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(where n0 .² an in (81) is measurable function. This can be generalized for any pair of finite discrete measures in the following remarks.
Remark A.3 (Expected Sliced Transport for finite discrete measures), Consider arbitrary finite discrete measures μ >i= p0x,)0n, and μ² ΣØ= q(yi)on, i.e., discrete measures with finite support.
Fix 24, E R, then 0 + due to the fact that only for finitely many directions we obtain overlaps of the projected points 0-
plag)ally * for all but a finite number of directions. This is P(E)) (QU;))
The optimal transport plan Ag .²² is given by "matching from left to right until fulfilling
the target bins": that is, one has to order the points similarly in (3) and consider " "increasing" assignment plan. Since the order of (0-x:/1= and {A- Uj Wررy=1 changes a finite
number of times when varying 0 E ES the function O + A uu'a² takes finite mumber of possible transport plan options.
Thus, the range of 8 ugg .²² finite.
Remark A,4 (5u' .² is well-defined for finite discrete measures). First, we notice that for each the support of N2 yu', 4² is finite or countable, and so the support of % .²² is also finite or countable. Given an arbitrary point (x,, v) E Rd x Rdd, we have to justify that the function Sd-1 3 0 + u (x, v) u²² is a
(Borel)-measurable If the supports of p' and ye are finite, by Remark A.3 + ug piece-wise constant function, and so it is measurable and integrable on the sphere. For the general case, when the supports of μ and p² are countable, we refer to Remark A.14
Lemma A.5 (5m) s² is transport plan between jd and p²)) We have that 5u "² E er ,22), i.e. it has marginals μ² and y²2 This is because for each O E g 2il 2² T(p², ,2² and because is probability measure on Sd-1, Then, sp'a" 2²² is convex combination of transport plans n2p "t''a" n²² and since T(p', 1x2 a convex set, we obtain tharę!' N²² E(p' ,2²). Precisely, for every test function 6 Rdd + Je o(z)d5a' #2 (x. v)
fea [ o(z)d1h +² +'x,,""x,, v)do
JRARR
{ I ((x)dp (x)do ((z)d (x)
JRd
Similarly, Je+xR (())d5pt u² (x. = fne (v))d and so zpt u²² has marginals p² and 2²2
A.2 AN AUXILIARY RESULT
For simplicity, in this paper we consider the strictly convex cost || 1 y/ (1 < < oo). Also, in this section consider (Sdd11 and in this case denote do d0.
Proposition A.6. Let Ω C Fd be a compaet set, and let μº . P(L2). Let (HH)NEN, (PE)nEN c P(()) be sequences such that, for i = 1, 4º, - n² as + Jc, where the limit is in the weak*. topology. For each n N, consider optimal transport plans Th [" [h (2)). Then, there exists subsequence such that Yim. - "/ for some optimal transport plan I' ["
Proof. As (ii)DEN is sequence of probability measures, their mass is 1, by Banach-Alase Alaoglu Theorem, there exists a subsequence such that Yna, 7h, - 7, for some E P x ②). It easy to see that the limit y has marginals n²,, n Indeed, given any test functions has marginals yh u we have
C(2), since each %
[ ((x)dT(( ((x)d11(())) and fn (by)ddn (x. (()))(())
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and taking limit as n oo, obtain
I axx ((z)d7(x,1)
J Ma)dd' (x) and
((v)d<(z,))) ((v)di(()) (v). Axn
Now, we only have to prove the optimality of r for the OT problem between p and μ Since (xr, y) + || - y/ continuous and Yn. Y are compactly supported, by using that for each n N. Th. is optimal for the OT problem between 2n and un. we have
lim I
lim (W (e)))
| v1l
nxx
|| - ulPP 2 ""p("",p11)"
(19)
Rxx
Also, by hypothesis and (Santambrogio 2015 Theorem 5.10) we have that for any P(
lim W Wplthh v) Wp('",
and
lim W ( w²) n
So, by using the the triangle inequality for the p- Wasserstein distance get
lim Wp Wp Mi))
S lim Wplnn, (+") lim Wp + Wp(n ,12) n
2
n-000
= =0+ Wpln 122 = Wp''', n²²)
(20)
Therefore, from 20 and
19
we have that
lim W Wy m2) = Wp'n", ,²²).
In particular, in
19 we have that the following equality holds:
Jave ||x vll'd((z,)) (Wp(m',221)
As result, is optimal for the OT problem between μt' and 2
A.3 FINITE DISCRETE MEASURES WITH RATIONAL WEIGHTS
Let us denote by (RR)) the set of finite discrete probability measures in Id with rational weights, that is, μ E (R((R)) if and only if is of the form f Em q:ou; with x, Rd, qi Qivie [m] for some m N, and En q: 1. We have
(R)) C PQ(RR).
ENEN.
In the definition of an uniform discrete measure f I; 2y for some pairs of indexes j. Proposition A.7. D) defined by 11 is metric in PQ(
N Dende dc, E (B))((")) one can allow
Proof. This was essentially pointed out Remark 2.9 When restricting the space P(ny (R.). our P.(..-) and the Projected Wasserstein distance presented in (Rowland et al., 2019) coincide. Rowland et al. (2019) prove the metric property. We recall here their main argument, which is used for showing the triangle inequality. Given u² 1,,,,, E ((NN)(R) of the form n² N Σ 8z.
222 = N Σ. 5y. 2²² = 1/4 EN 6z.- Fix E 5d-1, and consider permutations Ço. Te, E E SN, so that
Ic (NN) co (11 5... 0. 5.. Ic,
yr (1) S ... 5 9.1 ) (N))
25, (1) S ... S 0 "g, (N)
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Thus, the key idea is that
{ Jen- N
N
q(c.T,E)
N
Dplu", ,1pp=
12)== =
('))7((1) (1) - (1)=0))
1d =
Σ
(x0--1()) - (
N
t=1
i=1
q(C,T,E)
11100,10000 J8d-1 N
1l,,,'()) (1) - -2,2'10
Σ
Σ 110+- - -0-1(0)11
N
i=1
SiTfEEN
Dp(u2, ,
/
Σ
|F5 (1) -o "10''1)
E
g(SiT.E)
1|ag 1(6) - 2 1(0)ll
N
S.TEES
where q E P(SN x SN x Sy) is such that q(C, T, (,) (Co, ro. Eo) are required, given that is drawn from Unif($d-1). With these alternative expressions established by the authors in (Rowland al., (019), the triangle inequality follows from the standard Minkowski inequality for weighted finite L -sspcees.
the probability that the tuple permutations
Finally, notice that we have used the fact that each jut is associated to N-indexes (I....,N) N), without asking that the points {a:} do not overlap, i.e., they could be repeated. That is, given 1² = Ei=1 Öz, E (R=) one can allow I, = Iy for some pairs of indexes i * (analogously for μ² and v²)). Thus, the proof also holds for measures in P2(RR)): Indeed, let n² ,,,,,² (R2))
0; 3 10² = - _ with r!. E E.N. First, consider the n1 as the least common multiple of (s)....,shh s/hy and, for each [m1], let i! so that
be of the form p² = >m _ 1²² = >i=
Thus, we can rewrite μ²² Σmn 6z Notice that, since μ² is a probability measure, we have n² Em= P!. Now, for each E [ng] such that i! 1, consider i! copies of the corresponding point 24 so that we can rewrite μ²¹ Eii _ (where we recall that n' = Σ - F! and the points 2, in the new expression can be repeated, i.e., they are not necessarily all different). Repeat this process to rewrite μ²² Ene my 0 nنر >>
ni
1 deg Now, consider N the least common multiple of n1, mly nig, and rewrite the measures as jr² = Em= 82,1 122 = Σ 6ge. v Σ 6z, where the points 24, V/. 2 be repeated if needed. Thus, je' 12, 11 can be regarded as measures in P(rr where Pp(t.)) behaves as metric.
A.4 THE PROOF FOR GENERAL FINITE DISCRETE MEASURES
We first introduce some notation. Consider finite discrete probability measure μ E P(RR)) of the form μ >m p'6z, with general weights p' R_ such that En p 1. For each
1). consider an increasing sequence of rational numbers (p' Jnen Q with 0 s p 5 p', such that lim, Ph, = p'. For i = m, consider the sequence (p" JEEN C Q defined by 0 5 PM : 1- -n p/ 1. Thus, lim,, P/ = 1- ---im _ P 1 m- pm Define the sequence of probability measures (Hn)neN given by M4n := Ei= Pi, E Pe( is easy to show that Mnn) converges to μ in Total Variation (i.c., uniform convergence or strong convergence): Indeed, let E > O. For each i [m=]. let N; E N such that Ip', -p' < e e/m Vn 2 N, and define N = max N Nm Now, given any set B C Rd we obtain, for n 2 N,
i (1,...,
m -1
m
(B,(B) - u(B)]
Σ {E[m]: mE
(v/n v')
(un, and a have the same support)
Σ
Iph p'1
(triangle inequality)
HE 2.EB
p²
(sum over all indexes to get independence of the set B)
(E(m]
<
This shows that lim 1| pllay 0. Moreover, this shows that in this case, i.e., when approximating a finite discrete measure y by a sequence of measures having the same support as μ. we only care about point-wise convergence.
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We will now introduce some lemmas which together with the above proposition will allow to prove the metric property of Dp(...) for finite discrete probability measures. For all them we will consider:
k² 12² two finite discrete probability measures in R4 given by n² Ei= p'os., ,²² Σ5 و
(H (pn)ner approximating sequences of probability measures n = E p2,22. "h - EØ= Unous, with rational weights [prJ, { defined in analogy to what we have already done, i.e., that, for k 1, we have that (eE) a sequence of probability measures that converges to μ² in Total Variation).
Also, for each 0, A .n² denotes the unique optimal transport plan between 0y and p 76 #²² denotes lifted transport plan between p and 2r² given as in (9) and sy " the expected sliced transport plan between y and μ² given as in (10). Similarly, for each N we consider the plans Aga th "h and sutia
"B
(Ag
converges to Ag' in Total Variation.
Lemma A.8. The sequence
nEN NEN
converges %00 u² in Total Variation.
Lemma A9. The sequence
Lemma A.10. The sequence (pth.??
converges qp "²a in Total Variation.
MEN Lemma A.11. lim Dp(uns Mi) = Dp(u', 1²²) n-000
In general, notice that since for every i E [mi]. j E [m²]] we have that lim lim P/ = p', and lim &/- pp, then we obtain that lim,- Pa ps, and lim Oh Q', where pi Liemm:]a p², Q" Eje(ma) Egद g¹, and where, for each n E N, P and Q are analogously
defined (see Subsection 2.2.2 Thus,
Ph PAQ
p'q" PQQ'
lim 2-
Vi [mi], [ma].
(21)
Proof of Lemmal A.8 The support of all the measures we are considering are finite and so. the measures have compact support. Hence, we can apply Proposition A.6 to Opp', (0)H")nEN. IEN, i i= 111, 1,2.
Specifically, given (Ag MEN E "nEN and Ag (0* 02p²) such that
(0 (4 there exists subsequence (A " un KEN
A Mon tie Ag.
(22)
As are in one dimension, the set I (0 001²² is singleton, and so we have that Ag Ag #²² the unique optimal transport plan. Since the supports of all the measures are the same, (that is, {(O 24, 0 y hee je(m))))) the weak" convergence in (22 implies the stronger convergence in Total Variation.
Now, suppose that the original sequence (Ag MEN does not converge to Ag .u? (in Total Variation).
Then, given E > 0, there exists subsequence (A, Un the )seN such that
P/ m Ag " || >
||A
(23)
But again, from Proposition A.6 using that the supports of all the measures involved are the same
set ((0 rr 0 yyy) (ele)e/ffdddd [ and the fact that I (0# 00 = (Ag (Ag'." .²² (only one optimal transport plan), we have that there exists a sub-subsequence such that
A| Fyy Pag Ag .k² |rv <..
a
contradicting
23
Since the contradiction is achieved from assuming that the whole sequence
(Agh 2 OeEN does not converge to Ag" n²² Variation.
we have that, in fact, does converge to Age in Total
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Proof of Lemma A.9 This holds by looking ( Due to the fact that the supports of j4, and p² are the same (respectively, for fe, and μ²),, we only care about the locations {(z,, he(slll(ee and then by using (21) and the convergence from Lemma the result holds true.
Proof of Lemma A.10 As pointed out before, we only care about point-wise convergence: That is, since the supports of the measures involved coincide (are the same set ((rss My ) (evemmlleeeee weak convergence, point-wiss wise convergence and convergence in Total Variation are equivalent.
Since 0 S 700 t (1(1i, Hj){) < and $ is compact, by the convergence result from Lemma A.9 and using the Dominated Convergence Theorem, we have that for each i [ma]. [ma],
lim gu (f(2) (((())))))) n
lim n-000
wh (1 4;)
gá--
wha (1),,
90 ga- n0000
lim
Iga-1 1/th u/t u² (1(2)) = zu u² (((1::11)))
Proof of Lemma 1.11
[Pp(1h P2)P Dp(u 10(m2,100 ,2))p
- max EE kE|m: [J)E
{112 - By 11550 gu' 1
(24)
where the RHS goes to + oo, due to Lemma A.I0
Theorem A.12. Dp(:s-) is metric for the space of finite discrete probability measures in Rd.
Proof.
Symmetry: The way we constructed Dp(+.) makes it so that Dp Dplue wt))
Positivity: It clear that by definition (p ,2²) 2 0.
Identity of indiscemibles:
First, if 4 = 112 =: μe, then rh 1²² = (id x id) $ p for all gd-1, Hence gat.1? .1
(id x 1 which implies Dp(u' 422 0.
Secondly, 1 ,1² are such that Dp(u! ,12) = by having
Wp(n (1)) s D, ,²) 0,
we can use the fact that (((.)) satisfies the identity of indiscernibles by being a distance. That is, Wp(m², 1²) = 0 implies -
Triangle inequality: Given pa 2², 2r² arbitrary finite discrete measures with arbitrary real weights, consider approximating sequences (u,) (H (uN))EN, (p)) in Po(Rd) as before. Notice that every subsequence of (ph)ngs (respectively of (pa) and (HC) mex) will converge to f (respectively, to n! and μ²), as every subsequence of convergent sequence convergent.
By Proposition A.7
we have that, for each Rdd
Dp(1i p4) S Dp (.) Dr (.)).
Taking the limit as + 0o, from Lemma A.11 we obtain
Dp(tt 112) Dp(u' n²3) (1)))
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Published as a conference paper at ICLR 2025
A.5 DISCRETE MEASURES WITH COUNTABLE SUPPORT
Lemma A.13. Let μt Emen p'' be discrete probability measure with countable support fm bmeN. Let a be an absolutely continuous probability measure with respect to the Lebesgue measure the sphere (we write, < Unif(Su-1)) Let
Sp : 10 S 0 Im = 0 Imm for some pair (m, m') with m # m m'}.
Then
(Sp)) = O.
Proof. First, consider distinct points Tmay Fm on the support μ4, and let
(1 Fmm') 10 E @Sat1_ 0 Im 0 Fmni
is straightforward to verify that
S(Tm, Tmm) S n span
Imme })²,
where span(f Fmm }} L is the orthogonal subspace to the line in the direction of the vector (Im ). Thus, 5(xm, Fmr) is subset of d 2-dimensional sub-sphere gd-1, and therefore
Tga (S( amm Lm/))
(25)
Since,
S
U
(( Lmm);
where M
{(Im, Fm²): m # m²}
(26)
we have that
sS
Σ
((((mm))m))) O.
(M
since M is countable (indeed, M < (supp(p) x supp(4))
Remark A.14. 12 is well-defined for discrete measures with countable support] Let & Unif(Sa-1)) Given two discrete probability measures μ² pr=)) and n²² = = (V)), with countable support, from LemmalA.13 we have that oS, 0 1, 1,2, and so o (S Therefore, similarly the case of discrete measures with finite support, given any x supp(u²), E supp(}²) we have that the map O + p(æ)o(y) from S to IR is equal to the constant functionn up to set of a-measure This implies that the function + ug' measurable. Finally, since hy we (((x,)))) for every (x, y), we have that sp 4²² is well-defined.
B EQUIVALENCE WITH WEAK* CONVERGENCE
Lemma B.1. Let Ω C. IR be a compact set, E P(l2) and consider a sequence of probability Then, for each gd-1 have
measures (pn)ner defined in Ω such that pn b that 0gpin 3 04 as n + co.
Proof. Given E gd-1 notice that (0 10-x x: x E 2 is a 1-dimensional compact set, which contains the supports of Ogp and (n)N)))N.. Thus, when dealing with the weak" -oppology continuous functions as test functions. Let y R + R be continuous test function, then
/ (((u)d8@un(e))
- 4(0 x)dpp(z)
- [ n-00c
(00 xddp(z) = ((u)d0
since the composition 0 s0 x)) is continuous function from IRd to RR.
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Lemma B.2. Let D C R be compact set, p² E P(L).. i 1,2, and consider sequences of probability measures (1e,) meN defined 52 such that plz i n² as + oc, for 1,2. Given 8 E = consider Ag .²² the unique optimal transport plan between 0zp² and 0zp². and for each N. consider Afjittin the unique optimal transport plan between 4 and 4 Then Ng A
Proof. The proof is similar that of Lemma A.8 of optimal plans one-dimension, there exists a subsequence (A Am M JEN such that
From Lemma
B.1
Proposition A.6 and uniqueness
Ag May mie
A
Now, suppose that the original sequence (A Ah JnEN does not converge to A =²² (in the weak"- topology). Thus, there exists continuous function to RR - R such that for given e > 0 there
exists a subsequence (Ag M ma, JEN with
U (()) dA
tin thi (u) - p(u) ( >E
(27)
But again, from Lemma B.1 Proposition A.6 and uniqueness of optimal plans in me-dimension, we have that there exists a sub sub-subsequence such that
Mag Wass (u) J p(u) dAg
/ p(()) dAy
"² (u),
Since the contradiction is achieved from assuming that the whole sequence (Ag..)) 22 eee does not converge to Ai ,²² have that, in fact, it does converge to A "u" in the weak weak"- toopology.
contradicting 27
Lemma B.3. Consider probability measures supported compact set S C Rdd such that μll 1² , i 22, Let 0 Sd-1, then the sequence of lifted plans (h6 thoth IEEN satisfies that there exists
Phetting in 2== for
subsequence such that 7
7" TT(N 11: Ag ))= = (r E T(r (0x0) A/ " )
where 9 0(z,)) y) = (0 x, v) for all (x, e x
Proof. Since %0 wh I(m, 2)), similar the proof of Proposition A.6 by Banach-Alaoglu Why th 3 7x, for some 7* P(Ω ?2).
Theorem, there exists subsequence Tip Big the such that 7
Again, as in the proof of Proposition A.6 it can be shown that 7- E I(p" p2). In addition, we have
(0 #7 win Mag 2. (0 07 and
(0 x 0) Mag u
Mi 22 2 Al An
Ag
(where the first one follows by similar arguments to those in Lemma B.J and the second one follows by definition of the lifted plans). By the uniqueness of the limit (for the w -coneergence, we have (0 x 8) 7 Ag Thus, 7" E T(w" TT, A1: Ag²
Theorem B.4. Let & Unif(ga-1). Consider discrete probability measures measures 4i' DE D:8 = Σデ=1 0j Ω supported on a compact set Q C Rd. Consider sequences (Wh) (pH) of discrete probability measures defined on Ω such that ph. _ 4², ,12 n²a Then o-a.s. we have that Dp 22 - Dp(H' .p²: ()) as n - cc. Moreover, Dp(pns)22) Dp(pt, ,p)) as ->
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Proof. Let define the set
S(u 12) = 1:00 EDER: T(u', u A/ Ap' "²) = 170 112
Since we are considering discrete measures, notice that
gd-1 S(psS((pppp00 (1p=),²)= S S₁ US
where S {A E gd-1 8 8m 8 Tm for some pair # m') and S₁₂ {0 gd-1 O. Ynr for some pair n + n'7.
By Lemma A.13
have (8 SS(' 122) S (Spr U.Se2) Thus,
(S 11²1 =.
and n/ w Let E S(u², ((),, and consider the lifted plans % -²
By Lemma B.3 there exists
subsequence of (10 th nne such that
t i E T( , Ag (
To
Since S(w², ,²),, we have that [(u², 12;; Ag""n? contains only one element, which 7 wx2 Hence,
Moreover, by uniqueness of weak convergence, proceeding similarly in the proof of Lemma B.2 we have that the whole sequence (1/ (i " NEN converges to " .+² in the weak"-topology,
Therefore, by definition of weak coovergeence for measures supported in compact (in our case, Ω x Ω C R x (R,d), since (x, v) > || - u||² is a continuous function, we have
lim Dp(m,, 12: a)== = lim / ||. - 111--1lpee / f (I, n+000
n /
{ || "1l"dd 11² (x.
J92
Dp(l' (u', :)p
Combining this with the fact that (S(n), xz2) and that (Dp((t, pa: meeN is bounded, that is, (p 0 < (yx || 1ll. J. by Dominated Lebesgue Theorem we obtain
lim Dplmh, 2+000
lim no+o0 Jga-:
1 do))
f lim Dp 2h: 0) do)
Jo
= -Dp("'1p2)
Corollary B.5. Let Mn P(S), where Ω C. Rdd is compact, be of the form = Exega p(1)
1h EvERs Pu (x) where p(x) and pu px( are O at all but countably many RR.. Assume < Unif(Ed-1) Then. Dp(pn, + 0 if and only if pm i μ.
Proof. If Dollin n)
then, by Remark 2.8
W Wp(Mns (1) n
O, hence pn 2 p.
n-300
n-
The converse is Corollary of Theorem B.4
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C COMPUTATIONAL EFFICIENCY
To deduce the computational complexity of the proposed method, we consider, for simplicity, two finite discrete probability measures on Rd, d > 1, concentrated at N particles, μ Σ pró., and Σ j=1 クر Consider L slices or unit vectors (0: in S We assume that no overlap occurs when projecting along different directions (as previously noted, this is almost surely the case as proven in Lemma A.13). The following is the analysis of the computational complexity of the proposed EST method:
For each one of the I. directions, {0:) we have project the locations (1, (a:) { to obtain the new projected measures concentrated at {0: xe ) and (0: yj دل respec- tively. This requires (LLNN) operations.
To compute the one-dimensional plan No for each S l S T,, we essentially need to solve a one-dimensional optimal transport problem (i.e., sorting problem), which is of order (N log(N)). Thus, when considering all the slices, this step is of order OLN (oV()))).
The lifting process that gives rise Ve does not require additional operations to be taken into account (it is performed as assignment correspondence).
The plan 7te can be represented as an N x N matrix. The (i.j)-entry is given by Et= T0, (1)1,, requiring L operations. Thus, the complexity of this step is O(LN2).
Finally, once we have qur for computing the EST-distance we require another (N2 operations.
As a conclusion, the complexity of computing the plan 7H. is O(L(Nd+ N log((N++ + N2)), or simply, O(LN2 + LNd), and that of the EST-distance is O(LL + d As it is generally the case, N is much larger than d, so the computational complexity of both the EST-plan and the EST-distance is of order
O(LL + d).2).
We recall that the complexity of using the linear programming approach for solving the classical optimal transport problem between μ and v is of order O( log(N)). Besides, for the entropic regularized version solved by using iterative algorithms like Sinkhorn's 's algorithm, the complexity is of order O log /X²), where denotes the regularization parameter (see Dvurechensky et al. (2018); Altschuler et al. (2017)).
and In Figures and 9 we report the wall-clock times of the Sinkhorn algorithm, used for computing the entropic regularized version of OT (for different values of the regularization parameter X), compared to the proposed EST distance (calculated with different numbers of slices L) between two N-sized empirical distributions. The only difference between the two figures is the scale used for the horizontal axis, which has been adjusted for visualization purposes.
Remark C.1. It is important to note that the computation of EST method can be parallelized over the number of slices L. Currently, our code does not implement parallelization, but incorporating would significantly speed up the computation of the EST plan and distance.
D ADDITIONAL EXPERIMENTAL RESULTS
To further demonstrate the efficiency of our proposed method for the classification task, we consider the widely used benchmark dataset in 3D computer vision and geometric deep learning. ModelNer40 (Wu et al., 2015). This dataset consists of objects represented as 3D point clouds, with 2048 points per object. It contains 40 different object categories. The training set comprises 9,840 samples, and the test set includes 2,468 samples.
In Table I we compute the accuracy of the INN classifier using (1) the classical optimal transport (OT) approach calculated with linear programming (LP), (2) its entropic regularized version for different regularization parameters using the Sinkhorn algorithm, and (3) our proposed EST method with different numbers of slices (7.) unit vectors in $2
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traditional classifier with Euclidean distance Published as a conference paper at ICLR 2025
Wall-Clock Time, Log Logio (Seconds)
Plan, L=128
EST Plan, L=256
EST Plan, L=512
EST Plan, L=1024
Sinkhorn, eps=0.1
-2
Sinkhor, eps=1.0
Sinkhorn, eps=10.0
Sinkhor,, eps=100.0
5000
10000
15000
20000
25000
30000
Number of Particles,
Figure Wall-clock time between Sinhkorn algorithm applied for computing entropic OT for different values of the regularization parameter À and the proposed EST method calculated with different numbers L of slices. We compare the differences in time as the size of the empirical measures increases.
EST Plan, L=128
Clol-Clock Time, Log1o Logio (Seconds)
EST Plan, L=256
EST Plan, L=512
EST Plan, L=1024
Sinkhom, eps=0.1
Sinkhom, eps=1.0
Sinkhom, eps=10.0
Sinkhom, eps=100.0
-2
14
Number of Particles (i.e., Support Size), Logz(N)
Figure 9: loga login wall-clock time plot between Sinhkorn algorithm applied for computing entropic OT for different values of the regularization parameter and the proposed EST method calculated with different numbers L of slices. We compare the differences in time as the size of N the empirical measures increases.
Of importance, used the linearized version of these three methods, as described in Subsection 3.6 Direct computation of the INN classifier purely based on OT, its entropic regularized version, and the EST distance would require 048 840 20, 152, 320 ("number testing samples number of training samples") pairwise comparisons for each of the three approaches. Instead, we compute 2, 048 + 9, 840 11, 888 ("number of testing samples + number of training samples") transport plans (with the three different methods) and then on the approximated Monge maps given by (17
We recall that the linearization technique is based on pivoting with reference measure (4o to reduce the number of pairwise comparisons. Specifically, it involves computing transport plans (using the different transportation approaches: OT, entropic regularization, EST) between Ai and the target samples (training and testing point clouds). applying the barycentric projection (16) to obtain an embedding (17) of all the training and testing samples, and using classifier in the embedding space.
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INN Classification
Sinkhorn
Sinkhorn
ESP
ESP
OT
À = 10
À
== 128
L 1024
(LP)
Accuracy
63.96%
78.93%
77.30%
79.45%
82.09%
Time per distance (Sec).
0.423
0.594
0.185
0.368
0.883
Table 1: Accuracy and time comparison of 1NN classifier implemented after the embedding.
In these experiments, for all methods, the reference measure is chosen as the uniform measure on the cube |-1, 1]", and we use a INN classifier with Euclidean distance the embedding space.
E RELATIONS BETWEEN THE AVERAGING MEASURE o, AND THE SLICING DISTRIBUTION IN ANERGY-BASED SLICED WASSERSTEIN DISTANCE
We draw an analogy between our proposed averaging measure a, defined in 14 and the slicing distribution in the energy-based Sliced Wasserstein distance Nguyen Ho (024).. Under the assumption that higher value of 1-dimensional Wasserstein distance Wp(00 pt, 0zv) will give
propose to build on the Sliced Wasserstein distance and define slicing distribution supported on S" as
better projecting direction 0, Nguyen & Ho (2024)
f(WF Qµv))
On. (0:
Jga f(W, ( O' ")
for ,4, v E (RR)), where f: [0, oc) θ (0, oo) is monotonically increasing energy function. When f is the exponential function fe (x) = e", the slicing distribution becomes
W (0
do do Ja W7 W710 1.0 do'
d0
(28)
Both our proposed o, in Equation 14 and (0: in Equation 28 employ the exponential function to reflect the optimality of slices in the resulting distances, with Wasserstein distance serving as the benchmark. However, it's important note key distinction: contrary to the Sliced Wasserstein distance as lower bound of Wasserstein distance, the Expected Sliced Transport distance is an upper bound, as for all 0 gd-1
W Wp00 04v) S Wpln, v) S Dp(u, (u,,,)) v).
Thus, the analogous energy function in a, is g(x) -- (with J 2 0) which monotonically decreasing, based on the assumption that better slicing direction 0 will result in a lower D, 0)
24


Reviews:
[{"review_id": "kqYVPHyS7G", "reviewer": "Reviewer_A36p", "summary": "They propose a novel method to give transport plans between disctrete distributions on Eucledian spaces named \"expected sliced transport plan\".\nAlthough sliced Wasserstein distances in previous studies are computationally efficient in comparison to Wasserstein distance itself, they do not propose concrete transport plans between distributions.\nTo address this problem, they consider computationally efficient transport plans between distributions via slicing and lifting discussion.\nIt is shown that the proposed plans induce a metric for probability distributions.\nIn addition, they examine the performance of plans in several numerical experiments.", "questions": "On a very minor point: it can be better to unify the representations \"transportation plan\" and \"transport plan\".", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "They succeed in proposing sliced transport plans inducing metric sfor probability measures.\nAs they claim, previous studies for sliced Wasserstein distances do not necessarily provide concrete transport plans.\nWhile their results are only for discrete measures, I believe that it should be applicable to many problems in the real world.\n\nIn addition, they propose a tempered measure $\\sigma$ on $\\mathbb{S}^{d-1}$ in Section 3.2, which makes the performance of transport better.\nThis result gives motivation to consider general distributions on $\\mathbb{S}^{d-1}$ and corresponding expected sliced transportation other than the uniform distribution.", "weaknesses": "While the motivation is computational efficiency, they do not argue it sufficiently. The discussion starting at Line 432 should be the only part referring to the computational complexities, but I believe that this is insufficient. Their claim is that the computational complexities of the proposed method does not change w.r.t. the temperature parameter $\\tau$ while that of the entropic OT changes w.r.t. the regularization parameter $\\lambda$. What the author(s) should compare here is not the complexities **within** the algorithms but those **between** the algorithms.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "kpBDimQCMa", "reviewer": "Reviewer_Uqt2", "summary": "The authors propose to use a \"lifting\" technique to recover a transportation plan from sliced transportation maps. The lifted transportation plan applies to labels of discrete measures, They proved that the plan marginalizes to the original measures. Furthermore, the authors compute the expectation of the lifted transportation plan and plug the expectation into the original transportation problem. They finally proved that this transportation cost induced by their expected lift plan defines a metric. The authors later show empirical evidence for convergence of their new plan and distance with toy data and their metric properties with a MNIST variant.", "questions": "In 144: \"As a result, it is more convenient to work with lifted transport plans.\" \nIs plan the emphasis of the sentence? Instead of maps? And the difference between, as implied in this paper, is that maps apply to the support and plans apply to labels?\n\n\nPlease read proof the paper, to fix minor errors like \"on the -(the) labels\" in line 14, \"the weights of +(the) optimal transport\", and \"the reason why we make this choice -(if) +(is)\" in line 211", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "+Paper is clear. \n\n+The lifting technique is a good fit for translating the sliced OT problem back to the OT problem on the original measures.\n\n+The authors' proposed contributions are well demonstrated in the paper, with the exception of (4) since their results are preliminary.", "weaknesses": "-One thing that I expected from the paper but didn't find is purpose of answering the two key questions.\n\nFirst question \"Can a transportation plan be constructed between two probability measures ...\". Also in line 61, \"limiting their applicability to problems that require explicit coupling between measures.\" What problems require explicit coupling between measures? Does this paper solve those problems?\n\nSecond question, \"can this plan be used to define a metric between the measures\". Maybe the interpolation experiment shows a little bit of the usage of the metric properties. Other than that, from reading this paper, I don't see the reason of studying the metric properties.\n\n-The empirical results are weak because the dataset is too simple to prove its usefulness in real applications. They're at most for proof of concept.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qY3Vm8uUFB", "reviewer": "Reviewer_fUu9", "summary": "The paper introduces expected sliced optimal transport plans with four contributions: computational efficiency, a distance for discrete probability measures, theory and experiments demonstrating equivalence with Wasserstein distances, and use case demonstrations.", "questions": "Please see above.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The topic, approach, and technical exposition were all excellent. \n- The theoretical results were interesting and clear. \n- The empirical results were interesting including the synthetic demonstrations and real world use cases.", "weaknesses": "- I found it hard to understand what the paper was aiming for from the introduction. The intent of the paper came into sharper focus for me well after the start of the technical section. Currently the introduction is relatively abstract in motivating the approach. I wonder whether the authors wouldn't be better served by more concrete motivations earlier. Often this is done by having a figure to illustrate the approach. Maybe figure 1 could be this if it were earlier? However, that figure would benefit from more explanation. \n- The introduction focuses on computational efficiency; however, I don't see any empirical results demonstrating this point. Because the datasets being used here aren't huge by machine learning standards, it seems important to be precise about what is meant by this claim. \n\nMinor confusions / questions that arise while I was reading: \n- \"Traditional models scale cubically\" What methods are being invoked here? \n- \"However, the number of iterations required for convergence typically increases as the regularization parameter decreases, which can offset the computational benefits of these methods.\" I don't understand this critique. \n- Around line 46: I see the citation to Cuturi there, but the discussion is inadequate: \"Various approaches have been 046 developed to address this challenge, including entropic regularization (Cuturi, 2013),\" \n- Why are we introducing transport maps in equation 5?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)