Tree-Sliced Wasserstein Distance: A Geometric Perspective
Authors: Hoang V. Tran
Huyen Trang Pham
Tho Tran Huu
Minh-Khoi Nguyen-Nhat
Thanh Chu
Tam Le
Tan Minh Nguyen
Venue: ICML 2025 poster (2025)
Keywords: tree-sliced wasserstein distance
tree wasserstein distance
optimal transport
Primary area: theory->everything_else
TL;DR: In this paper, we replace one-dimensional lines in Sliced Optimal Transport with a more intricate structure, called tree systems, where OT problems on tree systems still allow closed-form solutions

Abstract: Many variants of Optimal Transport (OT) have been developed to address its heavy computation. Among them, notably, Sliced Wasserstein (SW) is widely used for application domains by projecting the OT problem onto one-dimensional lines, and leveraging the closed-form expression of the univariate OT to reduce the computational burden. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. To mitigate this issue, in this work, we propose to replace one-dimensional lines with a more intricate structure, called \emph{tree systems}. This structure is metrizable by a tree metric, which yields a closed-form expression for OT problems on tree systems. We provide an extensive theoretical analysis to formally define tree systems with their topological properties, introduce the concept of splitting maps, which operate as the projection mechanism onto these structures, then finally propose a novel variant of Radon transform for tree systems and verify its injectivity. This framework leads to an efficient metric between measures, termed Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL). By conducting a variety of experiments on gradient flows, image style transfer, and generative models, we illustrate that our proposed approach performs favorably compared to SW and its variants.

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Viet-Hoang Tran
Trang Pham
Tho Tran'
Khoi N.M Nguyen Thanh T. Chu Tam Le
Tan M. Nguyen 11
Abstract
put measures. OT has been applied in many research fields, including machine learning (Nguyen et al., 2021b; Bunne et al., 2022; Fan et al., 2022; Hua et al., 2023; Le et al., 2024a;b; Nguyen & Ho, 2024; Kessler et al., 2025; Chapel et al., 2025; Chapel & Tavenard, 2025), statistics (Mena Niles- Weed, 2019; Weed & Berthet, 2019; Liu et al., 2022: Nguyen et al., 2022; Nietert et al., 2022; Wang et al., 2022: Pham al., 2024), multimodal (Park al., 2024; Luong et al., 2024), computer vision and graphics (Rabin et al., 2011; Solomon et al., 2015; Lavenant et al.. 2018; Nguyen et al., 2021a; Saleh et al., 2022; Vu et al., 2025).
Many variants of Optimal Transport (OT) have been developed to address its heavy computation. Among them, notably, Sliced Wasserstein (SW) is widely used for application domains by project- ing the OT problem onto one-dimensional lines, and leveraging the closed-form expression of the univariate OT to reduce the computational bur- den. However, projecting measures onto low- dimensional spaces can lead to a loss of topo- logical information. To mitigate this issue, in this work, we propose to replace one-dimensional lines with a more intricate structure, called free systems. This structure is metrizable by a tree metric, which yields closed-form expression for OT problems on tree systems. We provide extensive theoretical analysis formally de- fine tree systems with their topological properties, introduce the concept of splitting maps, which operate as the projection mechanism onto these structures, then finally propose a novel variant of Radon transform for tree systems and verify its in- jectivity. This framework leads to an efficient met- ric between measures, termed Tree-Sliced Wasser- stein distance on Systems of Lines (TSW-SL). By conducting a variety of experiments on gradient flows, image style transfer, and generative models. we illustrate that our proposed approach performs favorably compared to SW and its variants.
However, OT has a supercubie computational complexity concerning the number of supports input measures (Peyré et al., 2019). To address this issue, (SW) (Rabin et al., 2011; Bonneel et al., 2015) exploits the closed-form expression of the one-dimensional OT to reduce its computational complexity. More concretely, SW projects supports of input measures onto a random line and leverage the fast computation of the OT on -inedimnnsional lines. SW is widely used in various applications, such as gradient flows (Bonet et al., 2022: Liutkus et al., 2019), clustering (Kolouri et al., 2018; Ho et al., 2017), domain adaptation (Courty et al., 2017), generative models (Desh- pande et al., 2018; Wu et al., 2019; Nguyen & Ho, 2022), thanks to its computational efficiency. Due to relying on one-dimensional projection, SW limits its capacity to cap- ture the topological structures of input measures, especially in high-dimensional domains.
Related work. Prior studies have aimed enhance the Sliced Wasserstein (SW) distance (Nguyen al., 2024a; 2020; Nguyen & Ho, 2024) or explore variants of SW (Bai et al.. 2023; Kolouri et al., 2019; Quellmalz et al., 2023). These works primarily concentrate on improving existing components within the SW framework, including the sampling process (Nguyen et al.. 2024a; 2020; Nadjahi et al., 2021), determining optimal lines for projection (Desh- pande et al., 2019), and modifying the projection mecha- nism (Kolouri et al., 2019; Bonet et al., 2023). However, few studies have focused on replacing one-dimensional lines, which play the role of integration domains, with more com- plex domains such as oneedimensional manifolds (Kolouri et al., 2019), or low-dimensional subspaces (Alvarez-Melis
1. Introduction
Optimal transport (OT) (Villani, 2008; Peyré et al., 2019) is naturally geometrical metric for comparing probability dis- tributions. Intuitively, OT lifts the ground cost metric among supports of input measures into the metric between two in-
'Equal contribution *Co-last authors 'National Univer- sity of Singapore *Movian AI "The Institute of Statisti- cal Mathematics. Correspondence to: Viet-Hoang Tran
Proceedings of the nnnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
al., 2018; Bonet et al., 2023; Paty & Cuturi, 2019; Niles- Weed & Rigollet, 2022; Lin al., 2021; Huang al., 2021; Muzellec & Cuturi, 2019). In this paper, we concentrate the latter approach, aiming to discover novel geometrical domains that meet two key criteria: (i) pushing forward of dish-iimennional measures onto these domains can be processed in a meaningful manner, and (ii) OT problems on these domains can be efficiently solved, idcally with a closed-form solution.
the product space Q S such that ( x Qt) = m(A), (( B) BBB for all measurable sets A, B. For > 1, the p- p-Wasserstein distance Wp between and (Villani, 2008) is defined as:
& (Jova d(z,y) dr de((,,) (1)
W Wo(n, inf
rEP(N.)
Sliced Wasserstein Distance. For U, P(R2), the Sliced W-Wasserssein distance (SW) (Rabin et al., 2011; Bonneel et al., 2015) between μ4, v is defined by:
Contribution. Our contributions are three-fold:
SW (W,(u, = (Ls- (, do
We introduce the concept of tree systems, which con- sist of copies of the real line equipped with additional structures, and study their topology. A key property of tree systems is that they form well-defined metric spaces, with metrics being tree metrics. This property is sufficient guarantee that OT problems on tree systems admit closed- solutions.
(2)
where 1S=-1) is the uniform distribution on Sd-1 operator R. L' (' 1 is the Radon Trans- form (Helgason, 2011):
[ f(x) 8(t (x. 0)) dx, Jgd
(t,
(3)
We define the space of integrable functions and prob- ability measures on tree system, and introduce novel transform, called Radon Trunsform on Systems of Lines. This transform naturally transforms mea- sures supported in high-dimensional space onto tree systems, and is a generalization of the original Radon transform. The injectivity of this variant holds, similar to other Radon transform variants in the literature.
and fus fu are the probability density functions of μ,, M, re- spectively. Max Sliced Wasserstein (MaxSW) distance is discussed in Appendix C.
Monte Carlo estimation for SW. The Monte Carlo method is usually employed to approximate the intractable integral in Equation (2) as follows:
We propose the Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL), and analyze its efficiency through the closed-form solution for the OT problem on tree systems, achieving a similar computational cost as the traditional SW.
L
Swp(u, =
L
1=1
(4)
where 0I...., 0r are drawn independently from ((- Using the closed-form expression of one-dimensional Wasserstein distance, when and v are discrete measures that have supports of most n supports, the computational complexity of SW,, p is O(Ln log n + Ldn) (Peyré et al., 2019).
Organization. The remainder of the paper is organized as follows. Section 2 provides necessary backgrounds of SW distance and Wasserstein distance on tree metric spaces. Section 3 provides a brief and intuitive introduction of tree systems and studies its properties, and Section 4 introduces the Radon Transform System of Lines. The novel Tree- Sliced Wasserstein distance on Systems of Lines is proposed in Section 5. Finally, Section 6 contains empirical results for TSW-SL. Formal constructions, theoretical proof's of key results, and additional matcrials are presented in Appendix.
Tree Wasserstein Distances. Given rooted tree (T, r) (T is tree graph, with one certain node called root) with non-negative edge lengths, and the ground metric dr, i.c. the length of the unique path between two nodes. For two distributions μl, v supported on nodes of T,, the Wasser- stein distance with ground cost dr, i... tree- Wasserstein (TW) (Le et al., 2019; Indyk & Thaper, 2003), admits a closed-form expression:
2. Preliminaries
In this section, we review Sliced Wasserstein (SW) distance and Wasserstein distances on metric spaces with tree metrics (TW).
W W (1, = Σ w. (I((II((T))
(5)
TET
where u. is the farther endpoint of edge from r, we is the length of e, and I(v.) is the subtree of T rooted at e,, i.e. the subtree consists of all node x that the unique path from x to r. contains Ve.
Wasserstein Distance. Let Ω be a measurable space with a metric d on Ω2 and let (4, v be two probability distributions on Q2. Let P(u,2) v) be the set of probability distributions "
2

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Adding tree structure
3
1
4
2
3
4
5
3
Figure 1: This illustration demonstrates the process of adding a tree structure to a system of lines. Left: An example of a system of lines in R², where the lines intersect, making the system connected. Right: Adding a tree structure to the connected system. In this example, only four pairs of lines are adjacent, shown by intersections, while the remaining pairs disconnected, represented by gaps. This structure is derived by taking spanning tree from graph with five nodes (representing the five lines), with edges connecting nodes where lines intersect.
3. System of Lines with Tree Structures
tree structures. Figure gives an example of a system of lines with added tree structure. pair (c. T) consists of connected system of lines C and its tree structure T of C. is called a tree system. We also denote it as L for short.
This section provides an intuitive and brief introduction of systems of lines and their additional tree structures. These structures form metric spaces, called tree systems, which serve generalization of one-dimensional lines within the framework of the Wasse-rensnn distance. We then explore the topological properties and the construction of tree systems. The ideas illustrated in Figures I, 2, and complete formal construction with theoretical proof' are presented in Appendix A.
3.2. Topological Properties of Tree Systems
A tree system C can be intuitively understood as a system of lines that are connected in certain ways. It naturally forms topological space by taking disjoint union copies of R and then taking the quotient at intersections of these copies. The disjoint union is straightforward, and the quotient follows the tree structure of L. The topological space resulting from these actions is called the topological space of a tree system and is denoted by U2g. By its construction, Si naturally carries measure induced from the standard measure each copy of R. This measure is denoted by Mec Notice that, due to the tree structure, a unique path exists between any two points of Ugg. This leads to an important result regarding the metrizability of Ωgg
3.1. System of Lines and Tree System
A line in Id can be fully described by specifying its direc- tion and a point it passes through. Specifically, a line is determined by (x,0) E Rd gá-1, a and is parameterized as 0 for E TR.
Definition 3.1 (Line and System of Lines in Rd). A line in Rd is an element (z, 8) IR gd-1, For le > 1, a system k lines Rdd is a set of i lines in Rdd.
Theorem 3.2 (Sc is metrizable by tree metric). Consider de: Ze x Pg - [0, 20) defined by:
We denote line in Rd as = (x,,0r) Here, Iı and Oz are called source and direction of l, respectively. Denote (Rd gd-1)* by T.g which is the space of systems of l lines in RRd, and an element of L_g is usually denoted by L. The ground set of system of lines L defined by:
bc(a,b) Hc (Pa.)) Va,b E Pic
(6)
where Pa.b is the unique path between and b in Ωgg Then de is a metric on Qg, which makes (Sc, dc) metric space. Moreover, de is tree metric, and the topology on Ug induced by de is identical to the topology Scc.
2: (xx, ER x L: x 2i + lx 0g for some by E RR.
For each element L. sometimes write (x. (tz, l), where ly E IR presents the parameterization of x I as = + tx 0, By a point of L, we refer to a point of the ground set Now consider a system of distinct lines in Rdd is said to be connected its points form a connected set in Rdd In this case, L. naturally has certain
The proof is presented in Theorem A. Figure 2 illustrates an example of a unique path between two points on a tree system, providing an intuitive explanation of why de is indeed a metric.
3

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
dc((aa))
de dc(Po,))
dc(Pac)
1
b
3
5
2
3
4
3
Figure 2: The same tree system L shown in Figure 1, naturally has topology derived from five copies of IR Consider three points b, c. The red zigzag line presents the unique path from a to b. Here the distance between b, i.e. da b), is the sum of four red line segments, Similar for paths between b and e; a and c. This demonstrates that the triangle inequality is satisfied for dc.
3.3. Construction of Tree Systems and Sampling Process
way. We obtain distribution over the space of all tree systems that can be sampled in this manner. The algorithm for sampling tree systems is summarized in Algorithm 1, and illustrated in Figure 3.
tree system can be built inductively by sampling lines, ensuring that each new line intersects one of the previously sampled lines. We introduce straightforward method to construct tree system: start by sampling line, and at each subsequent step, sample new line that intersects the previ- ously selected line. Specifically, the process is as follows:
Algorithm Sampling (chain-like) tree systems.
Input: The number of lines in tree systems k:
Sampling I1 ~ (([[1,1]*) and 0₃ u(
Step 1. Sampling 21 ~ Mi for an μ1 E P(RR), then 01 ~ Vn for an v1 E P(S--1). The pair (I1,01) forms the first line;
for = 2 to & do
Sample 4 ~ [[[ (, 1]) and O, ~ (S--11).
Compute x; = 2;-1 t. 0,-1.
Step At step i, sampling v4 I4-1 + l 0:-1 where 4 ~ μú for μ4 E P(R), then 0, v4 for Vi P(gd-1). The pair (x;, 0,) forms the 1th line.
end for
Return: (I1, 01), (x2,, 02), ... (2 0k).
The tree system produced by this construction has a chain- like tree structure, where the t line intersects the (i + thhh line. A general approach for sampling tree systems is provided in Appendix A.4. In practice, we simply assume all the distributions p's and v's be independent, and let:
95
8g
22
18
210
8
Mi be a distribution on a bounded subset of Rdd for instance, the uniform distribution on the d-dimensional cube [1,, 1jd, i.e. ({11111)));
Figure 3: Illustration of the sampled (chain-likke like) tree systems in Algorithm 1.
μ4 f > to be distribution bounded subset of R, for instance, the uniform distribution the interval
1], i.c. u([-1,1])
4. Radon Transform on Systems of Lines
0, to be distribution on gd-1 for instance, the uni- form distribution (Sd-1).
In this section, we introduce the notions of the space of Lebesgue integrable functions and the Radon Transform for systems of lines. Let L E Lg be system of & lines. Denote (RRd))) as the space of Lebesgue integrable functions on
Using the distributions j's and v's, we get a distribution on the space of all tree systems that can sampled by this

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Rd with norm || /ll...ie.
Many variants of the Radon transform require the transform to be injective. In the case of systems of lines, the injectivity also holds for R
L' ((()) {s: 1 B: II [ [f(x)) do < 00
Theorem 4.2. R is injective for all splitting maps a E C(R², Ax-1).
Two functions f1. L' (RR)) are considered to be iden- tical if (x) (a) almost everywhere on Rdd As counterpart, Lebesgue integrable function on C is func- tion f: C - such that:
The proof of this theorem is presented in Theorem B. Denote P as the space all probability distribution on R and define probability distribution on L be a function LL' (C) such that f: - [0, 20) and ll.
|| VMI==EJ[m /1 (5(,, LEC LELUR
xo.
1. The space probability distribution on L is denoted by P(C). Then R transforms a distribution in P( to distribution in P(C). In other words, the restricted operator R2: P(RR)) - P(Z) is also well-defined.
The space of Lebesgue integrable functions L is denoted by (2). Two functions S1, f2 E (C) are considered to be identical if fi())= = f2( almost everywhere on L. The space L (2) with norm || lle is a Banach space.
5. Tree-Sliced Wasserstein Distance on Systems of Lines
Recall that has k lines. Denote the (k 1)- dimensional standard simplex as 4 ={(a))EE aj > O and Liec a 1} C RR Denote C(R", Ax-1) the space of continuous maps from E Ak-1. A map in C(RR 11- is referred to as a splitting map. Let C be sys- tem of k lines in Li. be splitting map in C(R", Ak-1), we define an operator associated to that transforms Lebesgue integrable functions on Rd to a Lebesgue inte- grable functions on L, analogous to the original Radon Transform. For f E L (R"), define:
In this section, we present a novel Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL). Consider as the space of tree systems consisting of k lines Rd that be sampled by Algorithm 1. By the remark at the end of Subsection 3.3, we have a distribution a on the space T. General cases of T, as in Appendix A.4, will be handled in similar manner. For simplicity and convenience, we occasionally use the same notation to represent both a mea- sure and its probability distribution function, provided the context makes the meaning clear.
RES: Σ-R
(x,1) (x, | ((0 a(y) (12--
( (y Ii, 0t)) dy,
5.1. Tree-Sliced Wasserstein Distance on Systems of Lines
(9)
Consider a splitting function a in C(R AR-1). Given two probability distributions μa v/ in P(Rd) with their density function fu. fv, respectively, and the Radon Transform R2 in Definition 4.1, fa and fu are transformed to R& and R2 in P(C). By Theorem 3.2, L has a tree metric de, we compute Wasserstein distance W We (R2 R2fv) (R. tion (5).
where is the 1-dimensional Dirac delta function. For I E L (RR)), we can show that R2 E (C). Moreover, we have |Rafllc & []lf In other words, the operator R2: I.d (RR) - (2) is well-defined, and linear operator. The proof for these properties is presented in Theorem B.2. We now propose a novel variant of Radon Transform for systems of lines.
system & E T. By
between R2 and R2Fu by Equa-
Definition 4.1 (Radon Transform on Systems of lines). For a C(Rd, 1)- the operator Rº:
Definition 5.1 (Tree-Sliced Wasserstein Distance on Sys- tems of Lines). The Tree-Sliced Wasserstein distance on Systems of Lines between Jr, in P(Rd) is defined by:
Rº L (RR)
II (C)
CELA
TSW-SL(H, v) S /Wac,1 (R I R2fv) (C (10)
(R2f)
is called the Radon Transform on Systems of Lines. Remark. An illustration of splitting maps and the Radon Transform Systems of Lines is presented in Figure 4. Intuitively, splitting map a indicates how the mass at a given point is distributed across all lines of a system of lines. In the case 1s = 1, there is only one splitting map which is the constant function 1, and the Radon Transform for L is identical the traditional Radon Transform.
Remark. Note that, the definition of TSW-SL depends on the space of sampled tree systems T, the distribution o on T. and the splitting function a. For simplifying the notation, omit them.
TSW-SL is a metric on P(Rd). The proof for the below theorem is provided in Appendix D.I.
Theorem 5.2. TSW-SL is metric on P(Rd).
5

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
2
0.3
0.2
0.1
0.1
0.4
f R2f
0.2
0.6
3
3
Figure 4: An illustration Radon Transform on Systems Lines. Given f E L (R*) such that f(x) = 0.6, ((y) 0.4, and is system of 3 lines. For a splitting map such that a(z) (1/6, 3/6, 2/6) and a(y) (1/4, 2/4, f is transformed to Rgf. By Equation (9), for instance, the value of Rg. at the projection of onto line (2) of C is f(x) a(())2 = 0.3.
Remark. If tree systems in T consist only one line, i.c. Wdc.1(22 (R as follows:
1, then in Definition 4.1, the splitting map a is the constant map 1, and the Radon Transform Ra now becomes identical to the original Radon Transform, as pushing for- ward measures onto lines depends only on their directions. Also, according to the sampling process described in Subsec- tion 3.3, becomes the distribution of 01. which is (S--1)) In this case, TSW-SL in Equation (10) is identical to SW in Equation (2). Furthermore, in Appendix C, we introduce Max Tree-Sliced Wasserstein Distance on Systems of Lines (MaXTSW-SL.), an analog of MaxSW (Deshpande et al., 2019).
Wee (R2.
= => W. eET
- - (13)
This expression enables an efficient and highly parallelizable implementation of TSW-SL, relies on fundamental oper- ations like matrix multiplication and sorting. Appendix B.3 provides a detailed illustration of the underlying mechanism of this expression.
Remark. Assume n > m, the time complexity for TSW- SL is O(Lkn n + Lkdn) since it primarily involves projecting onto L k lines and sorting projections on each line. This complexity is equivalent to that of SW when the number of projection directions is the same. Therefore, in our experiments, we ensure a fair comparison by evaluating the performance of TSW-SL against SW or its variants using the same number of projection directions.
5.2. Computing TSW-SL
We employ the Monte Carlo method to estimate the in- tractable integral in Equation (10) as follows:
L E
Ts I 1=1
(11)
We summarize this section with Algorithm 2 of computing TSW-SL.
where Ci....,Cy iid are referred to as project- ing tree systems. We discuss on how to compute Wee.1 (R25u, R2fxv for L E T. In applications, consider fu E P(RRa) given as follows:
Algorithm 2 Tree Sliced Wasserstein distance on Systems of Lines.
Input: 1 and v in P(R.). the number of lines in each tree system k the number of tree systems L, splitting
((x) = Σ 5(x - ai) and v(x) Σ 1=1
U1 5(x by)
map a: Rd Ak-1-
for = 1 to T. do
1=1
(12)
Sample tree system
o(9)). Project μ and onto L, to get R2, fa and R2
L: L= ((x (1) ((9) )...., (1 (1)
R2 pushes fyr fo on C. resulting in discrete measures R2fu, R2fu in P(C). In details, from definition of R2f the support of R2 is the set of all projections of support of μ onto lines of C. Moreover, the value of fu at projec- tions of a; onto 7 is equal to ar (a:)) u4. Similar for R2fu. From this detailed description of R2fus REfu, together with Equation (5), we derive a closed-form expression of
Compute Wde, 1 R2,fu).
end for
Compute TSW-SL = + 1 W RR2 R2, Sc). Return: TSW-SI.(uuu).
6

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Table 1: Average Wasserstein distance between source and target distributions of 10 runs on Swiss Roll and 25 Gaussians datasets. All methods use 100 projecting directions.
Swiss Roll
25 Gaussions
Iteration
Iteration
Methods
Time/Tten(s)
Time/len((s)
500
1000
1500
2000
2500
500
1000
1500
2000
2500
SW
5.73e-3
2.04e-3
1.23e-3
1.11e-3
1.05e-3
0.009
1.6l-1
9.52e-2
3.44c-2
2.56e-2
2.20e-2
0.006
MaxSW
2.47e-2
1.03--2
6.e-0
4.47e-3
3.45e-3
2.46
5.09c-1
2.36e-1
1.33e-1
9.70e-2
8.48e-2
2.38
SWGG
3.84e-2
1.53e-2
1.02e-2
4.49e-3
3.57e:5
0.011
3.10e-1
1.17e-1
3.38e-2
3.58e-3
2.54e-4
0.009
LCVSW
7.28c-3
1.40c-3
1.38c-3
1.38c-3
1.36c-3
0.010
3,38c-1
6.64e-2
3.06c-2
3.06c-2
3.02e-2
0.009
TSW-SL
9.41c-3
2.03e-7
9.63e-8
4.44e-8
3.65e-8
0.014
3,49c-1
9.06c-2
2.96c-2
1.20c-2
3.03e-7
0.010
MaXTSW-SL
2.75e-6
8.24e-7
5.14e-7
5.02e-7
5.00e-7
2.53
1.12c-1
8.28e-3
1.61e-6
7.32e-7
5.19e-7
2.49
6. Experimental Results
represents the change in the source distribution over time and (1SW-SL(44, v) is the gradient of TSW-SL with re- spect to j4, We initialize with μág = (O, and iteratively update μr over 2500 iterations. To compare the effective- of various distance metrics, we employ alternative dis- tances as loss functions (SW (Bonneel et al., 2015), MaxSW (Deshpande et al., 2019), SWGG (Mahey et al., 2023) and LCVSW (Nguyen & Ho, 2023)) instead of TSW-SL. Over 2500 timesteps, we evaluate the Wasserstein distance be- tween source and target distributions at iteration 500, 1000, 1500, 2000 and 2500. We use L 100 in SW variants and L 25, k 4 in TSW-SL for fair comparison. Detailed training settings are presented in Appendix E.1.
In this section, we present empirical results demonstrating the advantages of our TSW-SL distance over traditional SW distance and its variants, and how MaxTSW-SL enhances the original MaxSW (Deshpande et al., 2019) through op- timized tree construction. The splitting maps a will be selected either as a trainable constant vector or random vector, while the tree systems will be sampled such that the root is positioned near the mean of the target distribu- tion, i.e. the data mean. It is worth noting that the paper presents a simple alternative by substituting lines in SW with tree systems, focusing mainly on comparing TSW-SL with the original SW, without expecting TSW-SL to outperform more recent SW variant. Further improvements to TSW-SL could be made by incorporating advanced techniques devel- oped for SW, but we leave this for future research, choosing instead to focus on the fundamental aspects of TSW-SL.
We first utilize both the Swiss Roll (a non-linear dataset) and 25 Gaussians (a multimodal dataset) as described in (Kolouri et al., 2019). In Table 1, we present the performance and runtime of various methods on these datasets, emphasiz- ing the reduction of the Wasserstein distance over itera- tions. Notably, across both datasets, our TSW-SL method demonstrates superior performance by significantly reduc- ing the Wasserstein distance. Moreover, our MaxTSW-SL method shows a significant decrease in the Wasserstein dis- tance compared to MaxSW, highlighting its improved per- formance and effectiveness. Furthermore, we provide addi- tional results from experiments of 10, 50, 75, 100, 150, and 200-dimensiona Gaussian distributions, where target distri- bution supports were sampled from these nigh-dimensional spaces to showcase the empirical advantages of our TSW- SL in capturing topological properties. In this context, we compare the Tree Sliced Wasserstein distance on a System of Lines (TSW-SL) with Sliced Wasserstein distance (SW) to demonstrate TSW-SI's effectiveness when distribution supports lie in high-dimensional spaces. The results pre- sented in Table 2 highlight TSW-SL's superior ability to preserve the original data's topological properties compared to SW.
In order to provide a comprehensive assessment, our ex- periments focus on key tasks where SW has been widely applied in the literature: gradient flows and generative mod- els (including GANs and denoising diffusion models). We first demonstrate TSW-SL's performance gradient flow tasks using the 25 Gaussians and Swiss Roll datasets, high- lighting its ability to capture complex topological properties, with additional higher-dimensional results in the Appendix. Next, we evaluate generative models across various datasets, ranging from simple CIFAR-10 to complex STL-10, as- sessing scalability and robustness. We refer the reader to Appendix for additional experiments on color transfer and ablation studies the effect of the number of lines k on the performance of generative adversarial networks.
6.1. Gradient Flows
First of all, we conduct experiments to compare the effec- tiveness of our methods with baselines in the gradient flow task. In this task, we aim to minimize TSW-SL( v), where v is the target distribution and μ represents the source distri- bution. The optimization process is carried out iteratively as O.14 = VTSW-SL(1., v) with μ4 = AO 1), -0,44
7

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Table 2: Average Wasserstein distance between source and target distributions of 10 runs on high-dimensional datasets.
Iteration 500
Iteration 1000
Iteration
1500
Iteration 2000
Iteration 2500
Time/Iter(s)
Dimension
SW
TSW-SL
SW
TSW-SL
SW
TSW-SL
SW
TSWSSL
SW
TSW-SL
SW
TSW-SL
10
4.32c-3
2.81e-3
2.94c-3
2.00c-3
2.81c-3
1.55e-3
2.23c-3
1.59c-3
2.28c-3
1.75e-3
0.010
0.015
50
50.41
39.26
45.69
21.91
42.56
11.91
38.81
4.08
35.75
1.72
0.014
0.018
75
92.39
79.71
90.79
67.99
90.07
53.92
86.58
44,91
90.31
31.61
0.015
0.018
100
130.12
117.66
128.13
103.23
128.58
93.41
129.80
80,46
128.29
75.28
0.018
0.019
150
214.09
203.30
213.71
190.62
215.05
186.77
212.90
183.52
216.32
182.63
0.020
0.022
200
302.84
289.83
301.35
283.34
303.07
276.94
302.70
279.24
301.51
279.08
0.020
0.021
Table 3: Average FID and IS score of 3 runs on CelebA and STL-10 of SN-GAN.
CelebA (64x64)
STL-10 (96x96)
CelebA (64x64)
STL- (96x96)
FID()
FID(D)
IS(()
FID(
FID(U)
IS(1)
SW (L=
9.97 1.02
69.46 : 0.21
9.08 0.06
SW 500)
9.62 0.42
53.52 0.61
10.56 4: 0.05
SI. 10, 5)
9.63 0.46
61.15 + 0.37
10,00 + 0.03
TS. (f. = 100, k 5)
8.90 0.49
51.81 + 1.02
10.74 + 0.13
TL = 17, 3)
8,98:: 0.75
65,91 + 0.64
9,75 : 0.10
TSW-SL (L 167, A 3)
8.90 0.38
52.27: 0.96
10.62= 0.18
6.2. Generative Adversarial Network
6.3. Denoising Diffusion Models
We then explore the capabilities of our proposed TSW-SL framework within the context of generative adversarial net- works (GANs). We employ the SNGAN architecture (Miy- ato et al., 2018). In detail, our approach is based on the methodology of the Sliced Wasserstein generator (Desh- pande al., 2018), with details provided in the Appendix E.3. Specifically, we conduct deep generative modeling ex- periments the non-cropped CelebA dataset (Krizhevsky, 2009) with image size 64 x 64, and on the STL-10 dataset (Wang& & Tan, 2016) with image size 96 x 96.
Finally, we concentrate on denoising diffusion models (Sohl- Dickstein et al., 2015; Ho et al., 2020), which are among the most complex generative frameworks for image generation. Diffusion models consist of a forward process that gradually adds Gaussian noise to data and a reverse process that learns to denoise the data. The forward process is defined a Markov chain of ' steps, where each step adds noise accord- ing to a predefined schedule. The reverse process, parame- terized by 6 aims to learn the denoising distribution. Tradi- tionally, these models are trained using maximum likelihood by optimizing the evidence lower bound (ELBO). However, to accelerate generation, denoising diffusion GANs (Xiao et al., 2021) introduce an implicit denoising model and em- ploy adversarial training. In our work, we build upon the framework in (Nguyen et al., 2024b) and replace the Aug- mented Generalized novel TSW-SL distance as the kernel and conducting exper- iments on the CIFAR-10 10 dataset (Krizhevsky, 2009). For a detailed description of the model architecture and training loss, we refer readers to Appendix E.4.
To demonstrate the empirical advantage of our method in enhancing generative adversarial networks, we employ two primary metrics: the Fréchet Inception Distance (FID) score (Heusel et al., 2017) and the Inception Score (IS) (Salimans et al., 2016). We omit to report the IS for the CelebA dataset as it does not effectively capture the perceptual quality of face images (Heusel et al., 2017). Table 3 presents the re- sults of SW and TSW-SL methodologies on the CelebA and STL-10 datasets, utilizing FID and IS as our metrics. We conduct experiments with two configurations of projecting directions: for 50 projecting directions, we use L 50 in
Energy distance with our
Table 4 demonstrates that our TSW-SL loss function signifi- cantly enhances FID performance compared to conventional SW, which is 22.43% over DDGAN and 2.4% over SW-DD. This improvement underscores the efficacy of our approach in generating high-quality samples with improved fidelity.
SW compared to L. = 10, h 5 and L 17, k = 3 in
TSW-SL; for 500 projecting directions, we use L = 500
in SW compared to L = 100, k 5 and L = 167, k 3 in TSW-SL. Our results reveal that TSW-SL significantly outperforms SW, demonstrating a considerable performance gap on both datasets in terms of IS and FID. We provide additional qualitative results and ablation study for our meth- ods with respect to the number of lines in Appendix E.3.
7. Conclusion
This paper proposes novel method called Tree-Sliced Wasserstein on Systems of Lines (TSW-SL). replacing the traditional one-dimensional lines in the Sliced Wasserstein (SW) framework with tree systems, providing a more geo-
8

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Table 4: Results for unconditional generation on CIFAR-10 of denoising diffusion models
8856. URL id=Aul INKKRv.....
het.eewwweeeeeeeeeeeeeeeeee forum?
Model
FID |
Time/Epoch(s)]
Bonet, C., Chapel, L., Drumetz, L., and Courty, N. Hyper- bolic Wasserstenn via geodesic and horospherical projections. In Topological, Algebraic and Geometric Learning Workshops 2023, pp. 333-370. PMLR, 2023.
DDGAN ((Xiao et al., 2021))
3.64
136
SW-DD ((Nguyen al., 2024b))
2.90
140
TSW-SL-DD (Ours)
2.83
163
Bonneel, N., Rabin, J., Peyré, G., and Pfister, H. Sliced and Radon Wasserstein barycenters of measures. Journal of Mathematical Imaging and Vision, 51:22 45, 2015.
metrically meaningful space. This key innovation enables the proposed TSW-SL to capture more detailed structural information and geometric relationships within the data compared to SW while preserving computational efficiency. We rigorously develop the theoretical basis for our approach, verifying the essential properties of the Radon Transform and empirically demonstrating the benefits of TSW-SL across range of application tasks. As this paper introduces straightforward alternative by replacing one-dimensional lines in SW with tree systems, our primary comparison is between TSW-SL and the original SW, without anticipating that TSW-SL will surpass more recent SW variants. Fu- ture research on adapting recent advance techniques within the SW framework TSW-SL remains an open area and is anticipated to lead to improved performance for Sliced Optimal Transport overall.
Bunne, C., Papaxanthos, L., Krause, A., and Cuturi, M. Proximal optimal transport modeling of population dy- namics. In International Conference on Artificial Intelli- gence and Statistics, pp. 651 55128652.. PMLR, 2022.
Chapel, L. and Tavenard, R. One for all and all for one:
Efficient computation of partial wasserstein distances on the line. In International Conference on Learning Repre- sentations, 2025.
Chapel, L., Tavenard, R., and Vaiter, S. Differentiable generalized sliced wasserstein plans. arXiv preprint arXh::2505.2200 22049, 2025.
Courty, N., Flamary, R., Habrard, A., and Rakotomamonjy.
Acknowledgements
Joint distribution optimal transportation for domain adaptation. Advances neural information processing systems, 30, 2017.
We thank the area chairs and anonymous reviewers for their comments. TL. gratefully acknowledges the support JSPS KAKENHI Grant number 23K 1243, and Mitsui Knowl- edge Industry Co., Ltd. grant.
Deshpande, I., Zhang, Z., and Schwing, A. G. Generative modeling using the sliced Wasserstein distance. In Pro- ceedings of the IEEE conference on computer vision and pattern recognition, pp. 3483-3491 2018.
Impact Statement
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Deshpande, I., Hu, Y.-T., Sun, R., Pyrros, A., Siddiqui, N., Koyejo, S., Zhao, Z., Forsyth, D., and Schwing, A. G. Max-sliced Wasserstein distance and its use for GANs. Proceedings of the IEEF/CVF conference on computer vision and pattern recognition, pp. 10648-10656, 2019.
References
Fan, J., Haasler, I. Karlsson, J., and Chen, Y. On the complexity of the optimal transport problem with graph- structured cost. In International Conference on Artificial Intelligence and Statistics, pp. 9147-9165. PMLR. 2022.
Alvarez-Melis, D., Jaakkola, T., and Jegelka, S. Structured optimal transport. In International Conference Artifi- cial Intelligence and Statistics, pp. 1771-1780. PMLR, 2018.
Goodfellow, I., Pouget-Abadic. J., Mirza, M., Xu, B., Warde-Farley. D., Ozair, S., Courville, A., and Bengio, Y. Generative adversarial networks. Communications of the ACM,
Bai, Y., Schmitzer, B., Thorpe, M., and Kolouri, S. Sliced optimal partial transport. Proceedings of the IEEE/CVF Conference Computer Vision and Pattern Recognition, pp. 1368 1-13690, 2023.
2020.
Hatcher, A. Algebraic topology. 2005.
Bonet, C., Courty, N., Septier, F., and Drumetz, L. Effi- cient gradient flows in sliced-Wi Wasserstein space. Transac- tions on Machine Learning Research, 2022. ISSN 2835-
Helgason, S. The Radon transform on R", Integral Geome- try and Radon Transforms, pp. 1-62, 2011.

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Heusel, M., Ramsauer, H., Unterthiner, T., Nessler, B., and Hochreiter, S. GANs trained by two time-scale update rule converge to a local nash equilibrium. 12 2017.
Le, T., Nguyen, T., Phung, D., and Nguyen, V. A. Sobolev transport: A scalable metric for probability measures with graph metrics. In International Conference Artificial Intelligence and Statistics, pp. 1844-9866 PMLR. 2022.
Ho, J., Jain, A., and Abbeel, P. Denoising diffusion proba- bilistic models. Advances in neural information process- ing systems, (166800-8511 2020.
Le, T., Nguyen, T., and Fukumizu, K. Generalized Sobolev transport for probability measures on a graph. In Forty- first International Conference on Machine Learning, 2024a.
Ho, N., Nguyen, X., Yurochkin, M., Bui, H. H., Huynh, V., and Phung, D. Multilevel clustering via Wasserstein means. In International conference on machine learning, pp. 1501-1509, PMLR, 2017.
Le, T., Nguyen, T., and Fukumizu, K. Optimal transport for measures with noisy tree metric. In International Con- ference on Artificial Intelligence and Statistics. PMLR, 2024b.
Hua, X., Nguyen, T., Le, T., Blanchet, J., and Nguyen, V. A. Dynamic flows on curved space generated by labeled data. In Proceedings of the Thirty-Secone International Joint Conference on Artificial Intelligence, IJCAI-23, pp. 3803-3811, 2023.
Lin, T., Zheng, Z., Chen, E.,, Cuturi, M., and Jordan, M. I. On projection robust optimal transport: Sample complex- ity and model misspecification. In International Confer- ence on Artificial Intelligence and Statistics, pp. 262-270. PMLR, 2021.
Huang, M., Ma, S., and Lai, L. A riemannian block co- ordinate descent method for computing the projection robust Wasserstein distance. In International Conference Machine Learning, pp. 4446 4455. PMLR, 2021.
Lin, Y.-W. E., Coifmman, R. R., Mishne, G., and Talmon, Tree- Wasserstein distance for high dimensional data with a latent feature hierarchy, In The Thirteenth International Conference on Learning Representations, 2025.
Indyk, P. and Thaper, N. Fast image retrieval via embed- dings. In International workshop on statistical and com- putational theories of vision, volume 2. pp. 5, 2003.
Liu, L., Pal, S., and Harchaoui, Z. Entropy regularized optimal transport independence criterion. In International Conference on Artificial Intelligence and Statistics, pp. 11247-11279. PMLR, 2022.
Kessler, S., Le, T., and Nguyen, V. SAVA: Scalable learning- agnostic data valuation. In The Thirteenth International Conference on Learning Representations, 2025.
Liutkus, A., Simsekli, U., Majewski, S., Durmus, A., and Stöter, F.-.. Sliced- Wasserstein flows: Nonparametric generative modeling via optimal transport and diffusions. International Conference on Machine Learning, pp. 4104-4113, PMLR, 2019.
Kingma, D. P. Adam: A method for stochastic optimization. arXiv preprint arXi::1412.6900 2014.
Kolouri, S., Rohde, G. K., and Hoffmann, H. Sliced wasser- stein distance for learning Gaussian mixture models. In Proceedings of the IEEE Conference Computer Vision and Pattern Recognition, pp. 3427-3436. 2018.
Luong, M., Nguyen, K., Ho, N., Haf, R., Phung, D., and Qu, L. Revisiting deep audio-text retrieval through the lens of transportation. arXiv preprint arXiv:2405. 10084, 2024.
Kolouri, S., Nadjahi, K., Simsekli, U., Badeau, R., and Rohde, G. Generalized sliced Wasserstein distances. Advances in neural information processing systems, 32, 2019.
Mahey, G., Chapel, L., Gasso, G., Bonet, C., and Courty, N. Fast optimal transport through sliced generalized Wasser- stein geodesies. In Thirty-seventth Conference on Neural Information Processing Systems, 2023. URL https: / / openreview. net/ forum? Y=nhXu
Krizhevsky, A. Learning multiple layers of features from tiny images. 2009. URL. https://ap:: semanticscholas: org/ argg/CorpusDDD 182 68744.
Mena, G. and Niles-Weed, J. Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem. In Advances Neural Information Processing Systems, pp. 4541-4551 2019.
Lavenant, H., Claici, S., Chien, E., and Solomon, J. Dynam- ical optimal transport on discrete surfaces. In SIGGRAPH Asia 2018 Technical Pupers, pp. 250. ACM, 2018.
Miyato, T., Kataoka, T., Koyama, M., and Yoshida, Y. Spec- tral normalization for generative adversarial networks, In International Conference on Learning Representations, 2018. URL https::///// htttttss:://ppeeeeee /ttp///vee/eeeeeeeeeeeeeeeeeeeee net/ forum? id=B1 QRgziTT
Le, T., Yamada, M., Fukumizu, K., and Cuturi, M. Tree- sliced variants of Wasserstein distances. Advances in neural information processing systems, 32, 2019.
10

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Munkres, J. R. Elements of algebraic topology. CRC press, 2018.
Nietert, S., Goldfeld, Z., and Cummings, R. Outlier-robust optimal transport: Duality, structure, and statistical analy- sis. In International Conference on Artificial Intelligence and Statistics, pp. 1691-11719. PMLR, 2022.
Muzellec, B. and Cuturi, M. Subspace detours: Building transport plans that are optimal subspace projections. Advances in Neural Information Processing Systems, 32, 2019.
Niles-Weed, J. and Rigollet, P. Estimation of Wasserstein distances in the spiked transport model. Bernoulli, 28(4): 2663-2688, 2022.
Nadjahi, K., Durmus, A., Jacob, P. E., Badeau, R., and Sim- sekli, U. Fast approximation of the sliced-Wasserse Wasserstein distance using concentration of random projections. Ad- vances in Neural Information Processing Systems, 34: 12411-12424, 2021.
Park, J., Lee, J., and Sohn, K. Bridging vision and lan- guage spaces with assignment prediction. arXiv preprint arXiv:2404.09932, 2024.
Paty, F.-P. and Cuturi, M. Subspace robust Wasserstein dis- tances. In International conference on machine learning, pp. 5072-5081. PMLR, 2019.
Nguyen, K. and Ho, N. Amortized projection optimiza- tion for sliced Wasserstein generative models. Advances in Neural Information Processing Systems, 35:36985- 36998, 2022.
Peyré, G., Cuturi, M., et al. Computational optimal trans- port: With applications to data science. Foundations and Trends@ in Machine Learning, 11(5-6):355-677, 2019.
Nguyen, K. and Ho, N. Sliced Wasserstein estimation with control variates. arXiv preprint arXiv:2305.00402. 2023.
Pham, T., Shimizu, S., Hino, H., and Le, T. Scalable coun- terfactual distribution estimation in multivariate causal models. In Conference on Causal Learning and Reason- ing (CLeaR), 2024.
Nguyen, K. and Ho, N. Energy-basee sliced Wasserstein distance. Advances in Neural Information Processing Systems, 36, 2024.
Quellmalz, M., Beinert, R., and Steidl, G. Sliced optimal transport on the sphere. Inverse Problems, 39(10): 2023.
Nguyen, K., Ho, N., Pham, T., and Bui, H. Distributional sliced-Wasserse Wasserstein and applications generative model- ing. arXiv preprint arXiv:2002.07367. 2020.
Rabin, J., Peyré, G., Delon, J., and Bernot, M. Wasserstein barycenter and its application to texture mixing. In In- ternational Conference on Scale Space and Variational Methods in Computer Vision, pp. 435-446, 2011.
Nguyen, K., Bariletto, N., and Ho, N. Quasi-Monte Carlo for 3D sliced Wasserstein. In The Twelfth In- ternational Conference on Learning Representations, 2024a. URL /tttpps://ppeeeevieeee forum? id-Wd47E7HEXg,
Rotman, J. An introduction to algebraic topology, volume 119. Springer Science & Business Media, 2013.
Nguyen, K., Zhang, S., Le. T., and Ho, N. Sliced Wasser- stein with random-path projecting directions. arXiv preprint arXiv:2401. 15889, 2024b.
Saleh, M., Wu, S.-C., Cosmo, I.., Navab, N., Busam, B., and Tombari, F. Bending graphs: Hierarchical shape matching using gated optimal transport. In Proceedings of the IEEF/CVF Conference Computer Vision and Pattern Recognition (CVPR), pp- 11757-11767, 2022.
Nguyen, T., Pham, Q.-H., Le, T., Pham, T., Ho. N., and Hua, B.-S. Point-set distances for learning representations of 3d point clouds. In Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV), pp- 10478-10487, 202
Salimans, T., Goodfellow, I., Zaremba, W., Cheung, V., Rad- ford, A., and Chen, X. Improved techniques for training GANs. NIPS' 16, pp. 2234-2242. Red Hook, NY, USA, 2016. Curran Associates Inc. ISBN 97815 1083881..
Nguyen, T. D., Trippe, B. L., and Broderick, T. Many pro- cessors, little time: MCMC for partitions via optimal transport couplings. In International Conference on Arti- ficial Intelligence and Statistics, pp. 3483-3514, PMLR, 2022.
Salimans, T., Zhang, H., Radford, A., and Metaxas, D. Im- proving GANs using optimal transport. arXiv preprint arXiv: arXiv:/803. 1803.0557, 2018.
Nguyen, V., Le. T., Yamada, M., and Osborne, M. A. Op- timal transport kernels for sequential and parallel neural architecture search. In International Conference on Ma- chine Learning, pp. 8084-8095. PMLR, 2021b.
Sohl-Dickstein, J., Weiss, E., Maheswaranathan, N., and Ganguli, S. Deep unsupervised learning using nonequi- librium thermodynamics. In International conference on machine learning, pp. 2256-2265. PMLR, 2015.

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Solomon, J., De Goes, F., Peyré, G., Cuturi, M., Butscher, A., Nguyen, A., Du, T., and Guibas, L. Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains, ACM Transactions on Graphics (TOG), 34(4)666, 2015.
Wang, D. and Tan, X. Unsupervised feature learning with c-svddnet. Pattern Recognition, 60:473-485, 2016.
Wang. J., Gao, R., and Xic. Y. Two-sample test with kernel projected Wasserstein distance. Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, volume 151, pp. 8022-8055. PMLR, 2022.
Tran, H., Vo, T., Huu, T., Nguyen The, A., and Nguyen, T. Monomial matrix group equivariant neural functional networks. In Advances Neural Information Process- ing Systems, volume 37. pp. 48628 48665. Curran Asso- ciates, Inc., 2024a.
Weed, J. and Berthet, Q. Estimation of smooth densities in Wasserstein distance. Proceedings of the Thirty- Second Conference on Learning Theory, volume 99, pp. 5118-3119, 2019.
Tran, H. V., Chu, T., Nguyen-Nhat M.-K., Pham, H. T., Le, T., and Nguyen, T. M. Spherical tree-sliced Wasserstein distance. In The Thirteenth International Conference on Learning Representations, 2025a. URL https:// openreview httpeviiwwneneeeeeeeeeee Corutt/oormmi
Wu, J., Huang, Z., Acharya, D., Li, W. Thoma, J., Paudel, P., and Gool, L. V. Sliced Wasserstein generative models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 3713- 3722, 2019.
9NK.
Tran, H. V., Nguyen-Nhat, M.-K., Pham, H. T., Chu, T., Le, T., and Nguyen, T. M. Distance-based tree-sliced Wasser- stein distance. In The Thirteenth International Confer- ence on Learning Representations, 2025b. URL https: / //openreview www. foet/fornee forum? id=oigttMHwee.
Xiao, Z., Kreis, K., and Vahdat, A. Tackling the generative learning trilemma with denoising diffusion GANs. arXiv preprint arXi::2112.07804 2021.
Tran, T., Tran, V., Chu, T., Pham, T., Ghaoui, E., Le, T., and Nguyen, M. Tree-sliced wasserstein distance with nonlinear projection. In Forty-second International Conference Machine Learning, 2025c. URL https: //openreview net. forum? SSSWXYV7CC...
Tran, V.-H., Pham, T., Tran, T., Le, T., and Nguyen, T. M. Tree-sliced Wasserstein distance on system of lines. arXiv preprint arXiv:2406. 13.44006117755 13725, 2024b.
Tran, V.-H.. Vo, T. N., Huu, T. T., and Nguyen. T. M. A clifford algebraic approach to e (n)-equivariant high-order graph neural networks. arXiv preprint arXiv:2410.04692. 2024c.
Tran, V.-H., Vo, N., The, A. N., Huu, T. T., Nguyen- Nhat, M.-K., Tran, T., Pham, D.-T., and Nguyen, M. Equivariant neural functional networks for transformers. arXiv preprint arXh::2410.04209, 2024d.
Villani, C. Optimal Transport: Old and New, volume 338. Springer Science & Business Media, 2008.
Vo, T. N., Tran, V.-.., Huu, T. T., The, A. N., Tran, T., Nguyen-Nhat, M.-.., Pham, D.-T., and Nguyen, T. M. Equivariant polynomial functional networks. arXiv preprint arXiv:2410.042 2024.
Vu, A.-K. N., Do, T-T.., Nguyen, V.-T. Le, T., Tran, M.-T., and Nguyen, V. Few-shot object detection via synthetic features with optimal transport. Computer Vision and Image Understanding (CVIU), pp. 104350, 2025.
12

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Notation
Rd
d-dimensional Euclidean space
|| ll
Euclidean norm
( )
standard dot product
gd-1
(d - di-dienssio hypersphere
0
unit vector
disjoint union
(X)
space of Lebesgue integrable functions on
P(X)
space probability distributions on X measures
8(-)
I-dimensional Dirac delta function uniform distribution s
u(
1
pushforward (measure)
Y(
space continuous maps from to metric in metric space
W,
p- Wasserstein distance
SW,
Sliced p- Wasserstein distance
r
(rooted) subtree
edge in graph
We
weight of edge in graph
1
line, index of line
system of lines, tree system ground set of system of lines, tree system
C
topological space of system of lines space symtems of k lines in RR structure in system of lines
L
T
number of tree systems
k
number of lines in a system of lines or a tree system
R
original Radon Transform
Ra
Radon Transform on Systems Lines
Ak-1
(I: dimimennsoo standard simplex splitting map
space of tree systems
distribution space of tree systems
13

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Supplement for "Tree-Sliced Wasserstein Distance: A Geometric Perspective"
A. Tree System
In this section, we introduce the notion of tree system, beginning with collection of unstructured lines and progressively adding tree structure to form a well-defined metric space with tree metric. It is important to note that while some statements here differ slightly from those in the paper, the underlying ideas remain the same.
A.1. System of Lines
We have definition of lines by parameterization. Observe that, line in R is completely determined by pair (z,, Rd x gd-1 via x + {- 0, E TR.
Definition A.1 (Line and System of lines in Rd A line in R is an element (x, 0) of Rd x gd-1, and the image of a line (x, 0) is defined by:
Im(r, 0) = far + te R2 C Rd.
(14)
For k >> 1, system of n lines in Hd is sequence of k lines.
Remark. A line in Ied is usually denoted, or indexed, by 1- (m), 01) E Idd x gd-1, Here, and Or are called source and direction of l, respectively. Denote (IRd x gd-1y/ by L. which is the collection of systems of k lines in R4, and an element of Lg is usually denoted by L.
Definition A.2 (Ground Set). The ground set of a system of lines C is defined by:
f= ER xL:
+ tu Ol for some to E ER
For each element (x, () E È. we sometime write (z, () as (tz: where to E R, which presents the parameterization of on 1 by source 2ạ and direction 0%, as x 24 tx A:
Remark, In other words, the ground set 2 is the disjoint union of images of lines in L:
ビ= Im().
This notation seems to be redundant, but will be helpful when define functions on C.
A.2. System of Lines with Tree Structures (Tree System)
Consider a finite system of lines C in Rdd Assume that these lines are geometrically distinct, i.e. their images are distinct. Define the graph Gc associated with L, where C is the of nodes in Gc, and two nodes are adjacent if the two corresponding lines intersect each other. Here, saying two lines in Rd intersect means their images have exactly one point in common.
Definition A.3 (Connected system of lines). L is called connected if its associated graph Gc is connected.
Remark. Intuitively, each edge of Ge represents the intersection of its endpoints. If L is connected, for every two points that each one lies on some lines in C, one can travel to the other through lines in L.
From now on, will only consider the L is connected. Recall the notion of spanning tree of a graph G, which is subgraph of G that contains all nodes of G, and also is tree.
Definition A.4 (Tree system of lines). Let L be a connected system of lines. A spanning tree T of Ge is called tree structure of C. A pair (C, T) consists of connected system of lines C and tree structure T of C is called tree system of lines.
Remark. For short, we usually call tree system lines as tree system. In a tree system (C, T), images of two lines of can intersect each other even when they are not adjacent in
Let r be an arbitrary line of C. Denote T, as the tree T rooted at and denote the (rooted) tree system as (C, Tr) if we want to specify the root.
Definition A.5 (Depth of lines in a tree system). Let (C, be a tree system. For each m > 0, a line e L is called a line of depth m the (unique) path from r in T has length m. Denote f. as the set of lines of depth m.
14

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Remark. Note that Lo = (r). Let T be the maximum length of paths in 7 start from which is called the depth of the line system. L has partition
La L1 LI...UI,,
For / L that is not the root, denote pr(1) E C as the parent of I, i.c. the (unique) node on the unique path from / r that is adjacent to I. Note that, by definition, and pr(1) intersect each other. We sometimes omit the root when the context clear.
Definition A.6 (Canonical tree system). tree system (C. T) is called a canonical tree system for all 7 E that is not the root, the intersection of and pr(I) is the source æj of
Remark. In other words, in a canonical tree system, line that differs from the root will have its source lies on its parent. For the rest of the paper, tree system (C, will be considered to be a canonical tree system.
A.3. Topological Properties of Tree Systems
We will introduce the notion of the topological space of tree system. Let (C, T) be (canonical) tree system. Consider a graph where the nodes are elements of E; (x, 1) and (x', (') are adjacent and only if one the following conditions holds:
1. = pr(('), x', and x' is the source of l'.
2. 1' = pr()), , x = x', and r is the source of I.
Let ~ be the relation on & such that (z, (x', (') if and only if (x.() and (x',t) are connected in the above graph. By design, ~ is an equivalence relation on L. The set of all equivalence classes in L. with respect to the equivalence relation as P2 C/ ~.
Remark. In other words, we identify the source of lines to the corresponding point on its parent.
We recall the notion of disjoint union topology and quotient topology in (Hatcher, 2005). For a line in Rd, the image Im(() a topological space, moreover, a metric space, that homeomorphic and isometric to R via the map + xy + t 8, The metric on (l) is (1, x') = 1/2 - tx for all x x' E Im((). For each E L, consider the injection map:
fi Im(l) [m([] E (x, 1).
x
L Lhec Im(l) now becomes a topological space with the disjoint union topology, i.e. the finest topology on E such that the map Ji is continuous for all 7 E L. Also, consider the quotient map:
π C
(x.1)
[( I)]-
Og now becomes topological space with the quotient topology, i.e. the finest topology on Q2 such that the map æ is continuous.
Definition A.7 (Topological space of a tree system). The topological space Ωr is called the topological space of a tree system (C,
Remark. In other words, Ωg is formed by gluing all images Im(() along the relation
We show that the topological space Ωg metrizable.
Definition A.8 (Paths 9g). For a and in P6 with a * b. a path from a to b in Ω2 a continuous injective map
7: [0, 1] - Pt where 70) = a and (11 = b. By convention, for a in Pcg the path from a to a Ω2 is the constant map 7: [0, G2 such that y(t) a for all & [0, 1]. For path y from a to b, the image of defined by:
1m(7) = 7(10, 1]) C ac-
(15)
Theorem A.9 (Existence and uniqueness of path in Ug), For all a and b in Ωgg there exist a path from a to b in Dg. Moreover, y is unique up to a re parammttrrizzionn, i.e. if x and of are two path from a to in Qg, there exist a homeomorphism ep: [0, 1] - [0, 1] such that ? = 7 sp.
15

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Proof. All previous results we state in this proof can be found in (Munkres, 2018; Rotman, 2013; Hatcher, 2005). For two point a, b on the real line IR, all paths from b are homotopic to each other. In other words, all paths from a to b are homotopic to the canonical path:
7a [0,1] - R - (1 (1-1)
Now consider two point b space Szg Observe that Ωg is path-connected by design and by the fact that R is path- connected. Consider curve from a to b on lig, i.c. a continuous map f: [0, - flg, and consider the set consists of sources of lines L that lie on the curve f. i.e. all the sources that belong to f((0, 1)) We choose the curve f that has the smallest set of sources. By the tree structure added L, all curves from a to have the set of sources that contains the set of sources of f. We denote the sources belong this set of f as .1,..., 8k-1, and defined:
T; inf f (s:) for all 1 <isk-
We reindex s; such that:
IT E ... xK-1
For convention, we define = a and = b. By design, for i = O.... As 1, we have s, and 8441 line on the same line in L. So by the result paths on R, there exist a path y from 8; to 8441 on 92g- Gluing Yo: " 7h-1 to get a path from 8o = a to 8k = on f2g by:
7 [0
U2
i iit1
t (i if tE
, k
K
is clear to check y is a path from a to on Ω. and the uniqueness (up to re-parameterization) of comes from homotopy of paths in R.
Remark. The image of a path from does not depend on the chosen path y by the uniqueness property, Indeed, for homeomorphism yp: [0, [0, 1], we have >(00 1]) y o ([00 1)). Denote the image of any path from to b by Pa.b-
Let be the standard Borel measure on IR, i.c. ((a, 6]) = b - for every half-open interval (a, b] in R. For / E L, denote ii as the pushforward of μ by the map t + i + t 0,, which a Borel measure on Im Denote the o-algebra of Borel sets in L and Qg as ((C) and B(Qc), respectively.
Definition A.10 (Borel measure on E and P2)). The map #z: B(22) [0, oo) that is defined by:
vB E B(L), (B Em
(B)
IEC
called the Borel measure C. Define the Borel measure on Ueg, denoted by umzz as the pushforward of ur by the map :: C - Pg.
straightforward to show that μz is well-defined, and indeed Borel measure of L. As a corollary, Ane is also Borel
measure of Pig
Remark. By abuse of notation, we sometimes simply denote both of μz and μaz as ULc
Theorem A.11 (ig is metrizable by a tree metric). Define the map da: Ue Ωe - [0, oo) by:
dc(a,b) b) Pe (Pa.s) Va, E Ωc.
(16)
Then de is a metric on Qgg, which makes (Ier, de) a metric space. Moreover. de is a tree metric, and the topology on 2g induced by de is identical to the topology of 90g-
16

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Proof. is straightforward to check that de is positive definite and symmetry. We show the triangle inequality holds for dg. Let a, b, be points of S2g. is enough to show that Pa.e is subset of Pa.o b U Pb.c. Let yo: 7 be paths L from to and from b to e, respectively. Consider the curve from a to on Ω defined by:
7: [0, 22
i i+1
J 7122 - 1) if tE
, 0,
is clear that y curve from a to We have 7((0,, 1)] is exactly the union of Pa.b and Po.e- As in the proof of Theorem A.9, the set of sources of y contains the set of sources lying on the path from So 7(0, 1]) contains Pa Pa.e-
We have the below corollary says that: If we take finite points on Ωg. together with the sources of lines, it induces a tree (as a graph) with nodes are these points; Moreover, we have a tree metric on this tree which dg.
Corollary A.12. Let y1, 92..... ym be points on Stg. Consider the graph, where t. U U E L} is the node set, and two nodes are adjacent if the (umique) path between this two nodes on Q2 does not contain any node, except them. Then this graph a rooted tree at 2,, with an induced tree metric from dg.
A.4. Construction of Tree Systems
We present a way to construct a tree system in Rd. First, we have a way to describe the structure of a rooted tree by a sequence vectors.
Definition A.13 (Tree representation). Let T be a nonnegative integer, and m..... ur be T positive integer. A sequence
s {ox To where r; is a vector of n, nonnegative numbers, is called a tree representation if zo = [1]. and for all i < T, n. is equal to the sum of all entries vector T;-1.
Example A.14. For = 5 and [n. = 21,3,4,2,3), the sequence:
8: 20 = [1]
- = [3]
-22 = 1,]
-23 = [1,0,2,0]
-24 [1,2]
-5 [0, 0, 1].
a tree representation.
For tree representation s = {=a)7=n= a tree system of type is tree system that is inductively constructed step-by-step as follows:
Step O. Sample a point x, E dRd and a direction 0, E gd-1, Define r as the line that passes through s, with direction 0,. We call r as the line of depth O.
Step On the j-th line of depth (i-1), 1). sample (æ:)) points where (z:), is the j-th entry of vector xg. For each of these points, denoted æt, sample direction 0₁ E gd-1, and define / is the line that passes through xy with direction 6,. We call the set of all lines sampled at this step as the set of lines of depth and order them by the order that they are sampled.
By this construction, we will get a system of lines L in Rd. together with a tree structure T,. The pair (c, T,) forms a tree system, which is canonical by design, and is said be of type s. Denote T, as the set of all tree systems of type s.
Remark. A tree system in of type has k = E وj= (zt), lines. Observe that constructing a tree system of type only depends on sampling k points and A directions, so by some assumptions on the probability distribution of these points and directions, we will have probability distribution on T.. Note that:
1. xy is sampled from a distribution on Rdd:
17

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
2. For all o æt is sampled from distribution on R;
3. For all I, Oz is sampled from a distribution on gd-1
We have some examples of tree presentations and distribution on T..
Example A.15 (Lines pass through origin). Consider the tree representation 8:
[1]
(17)
and the distributions on T is determined by:
1. 2, 0 E R2
2. 0, ~ U(
In this case, T is identical to the set of lines that pass through the origin 0.
Example A.16 (Coneurrent lines). Consider the tree representation
8 : [1] - [k 1],
(18)
and the distributions on T. is determined by:
1. Hr for 14, P(RR));
2. For all # r, 21 Fzj
3. For all 1, 0: US)=1
4. vr and all are pairwise independent.
In this case, T, is identical to the set of all tuples of concurrent lines.
Example A.17 (Series of lines). Consider the tree representation
[1] [1]
[1].
(19)
and the distributions on T, is determined by:
1. Fr ~ Hy for an E P(RR));
2. For all 21 in for Pr E P(R);
3. For all I, 0₂ ~ 1()
4. All ry's and all 0 is pairwise independent.
In this case, we call T, as the set of all series of l lines. This is the same as the sampling process Subsection 3.3 and Algorithm 1.
Example A.18. For an arbitrary tree representation s, the distributions on T, is determined by:
1. is sampled from the uniform distribution bounded subset of Rd for instance, P4 ~ 21(0
2. For + r, z4 will be sampled from the uniform distribution on bounded subset of R. for instance, pr 2(00 1));
3. For all 1, 0 will be sampled from the uniform distribution on gd-1, i.e 0, ~ (((d11);
4. Together with assumptions on independence between all x, 's and 0r's.
18

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
B. Radon Transform Systems of Lines
introduce the notions of the space Lebesgue integrable functions and the space probability distributions on a system of lines. Let C be system of k lines.
B.1. Space of Lebesgue integrable functions on a system of lines
Denote L (IR)) as the space of Lebesgue integrable functions on [& with norm 11:
L' (IR) {:: Rd -R Ilfll 1f(x)) dx co
(20)
As usual, functions fi, f2 E L (RR) are considered be identical if fi = f2(x) almost everywhere on F
Definition B.1 (Lehesgue integrable function on a system of lines). Lebesgue integrable function on C. is a function f: L R such that:
lifllc = Σ E11 (f(ttt)) dtz < co.
(21)
IE6
The space of Lebesgue integrable functions on is denoted by:
L'(C) = { f:C-B: Ilfll Σ [f(ta))))dt < c00 IEC
(22)
Remark. As an analog integrable functions on Rd, two functions S1, J2 E (C) are considered identical fi(x) = f2(2) almost everywhere on É. The space L (C) with norm -|- Banach space.
Recall that has & lines, we denote the (k dimeeensio standard simplex as A {(ii)ec ai 2 and Eiec a: 1} C R² Let C(Rd, 11-1) be the space continuous function from Rd to Ar-1. A map in C(Rdd 11 is called splitting map. Let L be a system of lines in IL a be map in C(IRd, Ax-1), we define an operator associated to a that transforms a Lebesgue integrable functions on Rd to a Lebesgue integrable functions on C. For f T.¹ (Ted), define:
R2s: C R
(x,1) [ f(v) a(y) 8 -(y (yy 1--a (y-x1,,0) 01) dy.
-(y))
where ò the 1-dimensional Dirac delta function.
Theorem B.2. For f E L (TRR), we have R&J E (C). Moreover, we have |R & [[ In other words, the operator:
R2: (R(()))))))))))))))(()) - (C(
(23)
well-defined, and a linear operator.
19

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Proof. Let f E (RR)) We show that || & [1 Indeed,
|RaflIc= = Σ
(ta,)) dtz
(24)
IEC JR
Σ
lEc JR JRd f(y) a(y): 6 (tz - (y -y-1x Iz, 0()) dy dtx
(25)
lEe Bd (L [f()]l alv): - 8(tx (y - 21, B1)) dtz
dy
(26)
aly): (Le 8(t (y 21, B1)) dtz dy
Σ
[f(w) -
(27)
IEC
E/ [((w) aly)
(28)
IEC
-fn |s())
Σ TEC
dly)
(29)
J (s(x)) dy
(30)
|1/ <0.
(31)
So the operator R& well-defined, and clearly a linear operator.
Definition B.3 (Radon transform system of lines), For E C(TR.,, A 1), the operator R
R L (RR)
II (C)
CELA
f
(REf)
is called the Radon transform on a system of lines.
Many variants of Radon transform require the transforms to be injective. We show that the injectivity holds in the Radon transform on system of lines.
Theorem B.4. R is injective.
Proof. Since RD is linear, we show that if R = 0, then f = 0. Let f E (²())) such that R f = 0, which means R2 = 0 for all L E LØ Fix a line index I, consider the operator:
J/L f(y) a(y): 8 (t. - (y- x1,01)) dy = 0, Vtz E RR (1), E 12d Sd-1
(32)
Note that for index 1, [(y) avy): is a function of y. Let on be fixed and O varies in Rdd By the injectivity of the usual Radon transform (Helgason, 2011), we have f(x) a(x); for all x Rdd This holds for all line index so f(z) -, E, fz) a(x); 0. So R is injective.
Remark. By the proof, we can show stronger result as follows: Let A be a subset of Lg such that for every index and 0 E sd-1, there exists A such that 0: 0, where 0 is the direction of line with index / C. Roughly speaking, the set of directions in is (Sd-1)#
Remark. Observations in (Tran et al., 2024b; 2025b;c) suggest that imposing equivariance conditions on the splitting maps may enhance the performance of the distance. Nonetheless, since the primary focus of these works is the geometric formulation of the tree-sliced distance, this aspect is not explored in detail. More broadly, equivariance and invariance are prevalent themes in machine learning, with applications across various domains, including equivariant models (Tran et al., 2024c) and equivariant metanetworks (Vo et al., 2024; Tran et al., 2024d;a;d).
20

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
B.2. Probability distributions on system of lines
Denote ((RR)) the space of all probability distribution on R2:
P(E*) {f: R [0, oo): 1 c. L' (IR")
Definition B.5 (Probability distribution on a system of lines). Let L be system of lines. A probability distribution on is function f E (C) such that f: L - [0, oo) and || = 1. The space of probability distribution on L is defined by:
P(C) {f: L - [0, oo) llflle = (L)
(33)
Corollary B.6. For f E (I(R)), we have R P(CC) In other words, the restricted of Radon Transform:
R2: P(RR)) P(C),
(34)
well-defined.
Proof. Let f E (IR)). is clear that R2f: -- 00). We show that |R2fllc- 1. Indeed,
|R2J1lc
Σ
/
R2 R2 dlx
(35)
LEC
Σ
(1 fly) a(y): 8(x -y- 21, 01)) dy)
dt
(36)
lEC
[fe dl =1
(37)
So R&f P(C), and R2 is well-defined.
B.3. An illustration of TSW-SL
In this section, we provide visual illustration of TSW-SL through four figures: Figures and 8. Each figure includes an explanatory caption; however, Figure 7 is accompanied by separate, detailed explanation below.
The explanation for Figure is as follows:
The tree consists of 9 nodes. Of these, 6 correspond to projections of the original points, while the remaining nodes-Pjj. D2. and 23 are structural nodes introduced during the construction the sampled tree. Their roles are defined as follows:
Tj is the root of the tree.
T2 is the source of line 2 and lies on line 1, representing the intersection of lines and 2.
Z3 is the source of line 3 and lies on line 2, representing the intersection of lines 2 and 3.
We compute the aggregated mass R2 (T for selected edges e,, where I(e:)) denotes subtree rooted at the endpoint of e that is farther from the root ø1. Two examples are provided below:
For edge ex:
I(es) = { 02 by, Fz, by, ash.
(T(e)) = fu + f (02) + fa(()) + fy(2a) + fngb)) + fa((a))
9
6
25
-
+
+ 0+
30
30
30
30
30'
For edge Eg:
I(es) {b2, x3, by, az);
(T(es f (b2) + fy(2z) + fy((b) + fa(a))
4
6
16
+0+
+ 30
30
30
30'
Similar computations apply to the remaining edges in the tree.
21

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Q2
b2
b
b3
f R$f
a1
3
Figure 5: This figure presents the projections of two points, and y, onto the three lines labeled 1, 2, and 3. The projections of æ are denoted by a, and those of y by b, for i 1.
R2F 1
R
2
9/30
6/30
6/30
9/30
2/30
3/30
4/30
y
6/30
y
3/30
2/30
6/30
4/30
3
3.
Figure 6: This figure presents the mass at each projection of x and y. For example, the mass at is given by Rp fa(x) a(x) 0.6 x
3
30
C. Max Tree-Sliced Wasserstein Distance on Systems of Lines.
Max Sliced Wasserstein distance. Max Sliced Wasserstein (MaxSW) distance (Deshpande al., 2019) uses only one maximal projecting direction instead of multiple projecting directions SW.
MaxSW, (axSW,(H,u)) v) = max
(,((T,,))) (E-16
(38)
MaxSW requires an optimization procedure to find the projecting direction. is a metric on space of probability distributions on Rd
We define the Max Tree-Sliced Wasserstein distance on System of Lines (MaXTSW-SL) as follows.
Definition C.1 (Max Tree-Sliced Wasserstein Distance on Systems of Lines). The Max Tree-Sliced Wasserstein Distance on Systems of Lines between probability distributions p. in P(Rd) is defined by:
MaxTSW-SL(H.1) = max Wae R(v)) R2v]
(39)
GET
MaxTSW-SL is metric on P(Rd). The proof of the below theorem is in Appendix D.2.
Theorem C.2. MaxTSW-SL distance a metric on ((Rd).
We provide an algorithm to compute the MaxTSW-SI. Algorithm 3.
22

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
2
22
e4
e3
C6
43
e3
b2
b1
e2
b3
es
Q11
e1
0.3
.1
3
Figure 7: Resulting tree after projection with nodes, including projection nodes and sampled tree nodes. The detailed explanation for this Figure provided is Section B.3
b1
b3
y
63
67
2
2
C4
65
22
02
by
33
62
68
C1
2F
Figure 8: This figure presents the outcome when x and are placed differently in space compared to Figure 5. The projection and mass computation steps remain unchanged (though aty) may differ due to the new location of y). However, the resulting tree structure changes- for example, the subtree [(ea) now contains only the point b,, while T(es) includes b2, 2a, b3, and a3-
Algorithm 3 Max Tree-Sliced Wasserstein distance on Systems of lines.
Input: Probability measures μ and v, the number of lines in system splitting function a: IRd A 1, learning rate 7,, number of iterations T.
Initialize E Rdd .2....,,. la E IR, .I...., On E gd-1
Compute L corresponded to (I1, . t4, .1...., 0z). while L not converge or reach do
21 = I1 + Va, Wd RR R2v).
for = 2 to à do
t, T. 77 W., de, RR R3v).
end for
for
1 to k: do
0: = 0: Va, W (R RBv). Normalize 0; = 0, 0:1l2.
end for end while
Compute corresponded to (I1, b2,.... .., t4, ,,.,.,
Compute W.z.1 (R2Hu, Rgv). Return: L, Wa (R3u, R2v).
23

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
D. Theoretical Proof for injectivity of TSW-SL
will leave out "alloottsurrllyycooniioon"" in the proofs, as they are straightforward to verify, and including them would unnecessarily complicate the proofs.
D.I. Proof of Theorem 5.2
Proof. Need to show that:
TSW-SL(A,)) J Wo R&v) do (C).
(40)
metric on P(RR).
Positive definiteness. For Jr, v P(RR), has TSW-SI (1) 0 and TSW-SL((,,)) 0. IF then W (RV,, 0 for all E T. Since W is metric on P(C), we have R2 RE for all L E T. Since T is a subset of Lg that satisfies the condition in the remark at the end of the proof of Theorem B.4, we conclude that μ V.
Symmetry. For U, E P(R"), we have:
TSW-SL(H,V) J W WR R2v) do (C)
(41)
[. Wac, (R REE) do
(42)
TSW-SL(V,,L) R).
(43)
So TS, = TSW-SL(U,,1))
Triangle inequality.
For M1, u2, U3 E P(RR", we have:
TSW-SI(M1, (22) TSW-SI
(44)
Wde. (R R(pe) (c) + L Wae RCHa) do (C)
(45)
W R3
Wde. (R R2ns)) do (C(
(46)
W (R (R2H1, (C(
(47)
TSW
(48)
The triangle inequality holds for TSW-SL, We conclude that TSW-SL is a metric P(R.).
D.2. Proof of Theorem C.2
Proof. Need to show that:
Max max W (R2H, R?v)
(49)
is metric on P(Rd).
Positive definiteness. For μ.P E P(R"), one has MaxTSW-SL(p..p) = and MaxTSW-SL(4) v)) > 0. If MaxTSW-SL(H, D) = O, then Wde,1 (R2/4 RBv) = 0 for all L E T. Since Wa a metric, we have R2 = Rgy for all C E T. Since T is subset of LL that satisfies the condition in the remark at the end of the proof of Theorem B.4, we conclude that μ V.
24

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Symmetry. For n. E P(R"), have:
MaxTSW-SL(H.2 = max LET W W (REH, Riv)
(50)
-mxx LET
We (Rou, REN
(51)
= MMXTSW-SL(V,,))
(52)
So MaxTSW-SL((,D)) MaxTSW-SL(x,
Triangle inequality.
For P1, U3 P(R)), we have:
MaxTSW-SL(HI, 12) + TSW-SL( p3)
(53)
max
Wd (R2 R2
+ max L'ETL
W Wdg..1 (R Rg)
(54)
max LET
Waz.1 (R RB42) + Wd ( REua)
(55)
W de, (R22H1 R()s)
(56)
LET
MuxTSW-SL(H1:Ps)
(57)
The triangle inequality holds for MaXTSW-SL. We conclude that MaxTSW-SL is a metric on P(Rd).
E. Experimental details and additional results
E.1. Gradient flows
Gradient flow is concept in differential geometry and dynamical systems that describes the evolution of point or curve under a given vector field. In the field of Sliced Wasserstein distance, this is a synthetic task that is used to evaluate the evolution of Wasserstein distance between 2 distributions (source and target distributions) while minimizing different distances (Mahey et al., 2023; Kolouri al., 2019) as a loss function.
Datasets. We use Swiss Roll, 25- Gaussanns and high-dimensional Gaussian datasets as the target distribution as in (Kolouri et al., 2019). The details of these datasets can be described as follows.
The Swiss Roll dataset is a popular synthetic dataset used machine learning, particularly for visualizing and testing dimensionality reduction techniques. It is generated using the make swiss roll function of Pytorch, which creates non-linear, three-dimensional dataset resembling swiss roll spiral shape. In the original version, it is three-dimensional dataset with each dimension representing a coordinate in the 1 axis of the points. In order to simplify it, we follow (Kolouri et al., 2019) to just consider the two-dimensional Swiss Roll dataset by reducing the second coordinates and retaining only the first and the third coordinates. We set the number of samples to equal 100.
The 25-Gaussians dataset is obtained by first create a grid of 25 points spaced evenly in a x 5 arrangement. For each grid point, we generate cluster by sampling points from a Gaussian distribution centered at that grid point, with small standard deviation. All the points from the 25 clusters are combined, shuffled randomly, and scaled form the final dataset.
The High-dimensional Gaussian datasets are generated by initializing a mean vector, M4, consisting of ones across all dimensions. Each element of this mean vector scaled by random value to introduce variability. The covariance matrix, Σ, is created as an identity matrix scaled by constant, ensuring independence among dimensions. Points are then sampled from the multivariate normal distribution using these parameters, resulting in dataset of N points in the specified high-dimensional space.
The Swiss Roll and 25- Gaussians datasets are presented Figure 9.
25

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
1.5
1.00
1.0
0.50
0.5
0.25
0.0
0.255
0.50
-1.0
-0.8
-0.6
0.0
0.2
0.4
0.6
0.8
-1.0
-0.5
0.0
1.0
1.5
Swiss Roll dataset
25- 25-Gaussians datasets
Figure 9: Swiss Roll and 25-Gaussians datasets for Gradient Flows task
Hyperparameters. For TSW-SL, we use 25, k all experiments, while 100 is set for SW and SW-variants, with 100 points generated per distribution across datasets. Following (Mahey et al., 2023), the global learning rate for all baselines is 5 10-3, For methods, we use 10-3 3 for the 25-Gaussians and Swiss Roll datasets, and x 10-2 for the high-dimensional Gaussian datasets. We also follow (Mahey et al., 2023) in setting 100 iterations for both MaxSW and MaxTSW-SL, using a learning rate of 1 x 10 for both methods.
Evaluation metrics. We use the Wasserstein distance as neutral metric to evaluate how close the model distribution ((() is to the target distribution v. Over 2500 timesteps, we evaluate the Wasserstein distance between source and target distributions iteration 500, 1000, 1500, 2000 and 2500.
We utilize the source code adapted from (Mahey et al., 2023) for this task.
E.2. Color Transfer
addition to the experimental results presented in the main text, we extend our results to show the emprical advantages of TSW-SL over SW for transferring color between images to produce results that closely match the color distributions of the target images. Given source image and target image, represent their respective color palettes matrices X and Y, each with dimensions n x 3 (where n denotes the number of pixels).
We follow Nguyen et al. (2024a) to first define the curve (t) = -n V 2(0) SWW empirical distributions over X and Y turn. Here, the curve starts from Z(0)= X and ends at Y.
where Px and Py are
then reduce number of colors in the images to 1000 using K-means clustering. After that, we iterate through curve between the empirical distribution of colors in the source image Px and the empirical distribution of colors the target image Py using the approximate Euler method. However, owing to the color palette values (RGB) lying within the set {O.... ... 255), additional rounding step is necessary during the final Euler iterations. We increase the number of iterations to 2000 and utilize a step size of as in (Nguyen et al., 2024a) for baselines and step size of 16 for our experiments. We use L = 100 in SW variants and L = 25, k = 4 in TSW-SL for fair comparison. We traverse along the curve connecting Px and Py. where Px and Py denote the empirical distribution the source and the target images respectively. More specifically, this curve (denonted as Ztt) starts from ((0) = X and ends at Y. During optimization, we minimize the loss Loss ((t), Y) to make the color distribution of the obtained image close to that of the target image Y.
26

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
We evaluate the color-transferee images obtained by various loss, including SW (Bonneel et al., 2015), MaxSW (Deshpande al., 2019), and SW variants proposed in (Nguyen al., 2024a) compare with our TSW-SI. and MaxTSW-SI approaches. For consistency, we set L 100 for the SW variants and L = 25, k 4 for TSW-SL in our comparisons. We report the Wasserstein distances at the final time step along with the corresponding transferred images from various baselines in Figure 10. TSW-SL produces images that most closely resemble the target, demonstrating a significant reduction compared to SW and its variants mentioned above with the same number lines. addition, MaxTSW-SL improves upon the original MaxSW, as highlighted by both qualitative and quantitative results.
Source
TSW SL. 24.8
SW, W2 8145.75
MaxTSW-SL W, 148.3
MaxSW, 8159.91
EOSW. W; 53.65
UCVSW, W2 = 8191.98
SOSW, W, = 46.08
DOSW, W2 = 52.9
COSW. W; 45.28
LCVSW, W2 8151.8
Target
Figure 10: Color-transferred images from various baselines with 100 projecting directions.
Evaluation metries. We present the Wasserstein distances at the final time step alongside the corresponding transferred images to evaluate the performance of different methods.
We utilize the source code adapted from (Nguyen et al., 2021a) for this task.
E.3. Generative adversarial network
Architectures. We denote as our data distribution. Subsequently, we formulate the model distribution Vi. as a resultant probability measure generated by applying a neural network Go to transform a unit multivariate Gaussian (e) into the data space. Additionally, we employ another neural network Tş to map from the data space to a singular scalar value. More specifically, T3, represents the subset neural network of T's that maps from the data space feature space, specifically the output of the last ResNet block, while T3, maps from the feature space (the image of T&s single scalar. Formally, T3 T3 To We utilize the subsequent neural network architectures for G and T3:
CIFAR10:
G 122 (~~ e: c: ()), - - 4x4x 256 (Dense, Linear) ResBlock up 256 ResBlock up 256 - ResBlock up 256 BN, ReLU, x conv, Tanh.
-Te, x E [-1, 32xx32x3 - ResBlock down 128 - ResBlock down 128 - ResBlock down 128 - ResBlock 128 - ResBlock 128.
-Tp; x E RR 121288888_ - ReLU Global sum pooling (128) 1 Spectral normalization ).
-To(x) T32 (To, (z)).
CelebA:
GGg:2 122 (~ e: ./ 1)) - 4 x 256 (Dense, Linear) - ResBlock up 256 - ResBlock up 256 - ResBlock
up 256 ResBlock up 256 BN, ReLU,, - x conv, Tanh.
-To, : E 1-1, 1| 32x32x3 ResBlock down 128 - ResBlock down 128 - ResBlock down 128 - ResBlock down
128 - ResBlock 128 - ResBlock 128.
-Tp; x E 128xx888
ReLU
Global sum pooling( (12)) - 1 Spectral normalization! ).
-Tp(x) T3 (Ty, (z)).
27

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
SW L = 50
TSW-SL (50 lines)
SW L 500
TSW-SL (500 lines)
Figure 11: Randomly generated images of different methods CIFARIO and CelebA SN-GAN
STI,-10:
-G R 128 (~ (0, 1)) 3 256 (Dense, Linear) ResBlock up 256 ResBlock up 256 ResBlock up 256 - ResBlock up 256 ResBlock up 256 BN, ReLU, - 3 x 3 conv, Tanh
-Tg, x E 1- 1|__]]=xxx x2x32x3 ResBlock down 128 ResBlock down 128 ResBlock down 128 ResBlock down ResBlock 128 ResBlock 128.
128 ResBlock down 128
-Tp² x E 128x888
ReLU
Global sum pooling(128) - 1( Spectral normalization) )
-Ty(x) To (T)(())
We use the following bi bi-optimization problem to train our neural networks:
min (Exo [min (0, -1 Ts(x))) + Egne [min (0, -1 1- (G)()))), 8r.82
min E x x zm [ (o.
where the function T . (Ts, (x), (7A, (x)) which is the concatenation vector of Ts, (x) and Te, (Ta, (x)) is an estimator of the sliced Wasserstein distance.
Training setup. our experiments, we configured the number of training iterations to 100000 for CIFAR1O, STL-10 and 50000 for CelebA. The generator Go is updated every 5 iteration, while the feature function Tş undergoes an update each iteration. Across all datasets, we maintain a consistent mini-batch size of 128. We leverage the Adam optimizer (Kingma, 2014) with parameters (31, 322) (0, 0.9) for both Ge and T with the learning rate 0.0002. Furthermore, random samples generated from the generator compute the FID and Inception scores. For FID score evaluation, the statisties of datasets are computed using all training samples.
50000
Results. For qualitative analysis, Figure 11 displays selection of randomly generated images produced by the trained models. is is evident that utilizing our TSW-SL as the generator loss significantly enhances the quality of the generated images. Additionally, increasing the number of projections further improves the visual quality of images across all estimators. This improvement in visual quality is corroborated by the FID and IS scores presented Table 3.
We utilize the source code adapted from (Miyato et al., 2018) for this task.
28

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
Table 5: Performance of different methods on STL-I0 dataset on SN-GAN architecture
No. of lines
Methods
Total line
No. of trees
FID
IS
per tree
SW
50
69.46
9.08
Orthogonal-SW
50
63.61
9.63
TSW-SL (ours)
51
3
17
65.93
9.75
TSW-SL (ours)
52
4
13
62.91
9.95
TSW-SL (ours)
50
5
10
61.15
10.00
Additional results. To fully show the empirical advantage of our methods, conducted additional experiments Adversarial Neural Networks on the CIFAR-10 dataset and STL--10 dataset. First of all, Table 6 presents the average FID and IS scores for different methods on the CIFAR- 10 dataset. For 50 projecting directions, our TSW-SL method with 10 trees and lines each (L = 10, = 5) achieves the best performance, outperforming the standard SW method. Similarly, for 500 projecting directions, TSW-SI (7. 100, k = 5) shows superior results compared SW. This demonstrates the consistent effectiveness of our approach across different numbers of projecting directions. Additionally, Table illustrates the performance of generative models the STL-10 (96 96) dataset with different numbers of trees and lines compared utilize the SN-GAN architecture (Miyato et al., 2018) 1) for STL- 10. For SW and Orthogonal-S -St,, we conduct experiments using 50 projecting directions. Our TSW-SL method is tested with three distinct configurations: 10 trees with 5 lines each, 13 trees with lines each, and 17 trees with 3 lines each. All hyperparameters remain consistent with those used in our main paper. To evaluate the models, we generate 50000 random images.
with SW and orthogonal-So In our experiments,
Table 6: Average FID and IS score of 3 runs CIFAR- 10 of SN-GAN
50 projecting directions
500 projecting directions
FID(()
IS(1)
FID(L)
IS(1)
SW (L = 50) TSW-SL (L 10, k 5)
16.80 + 0.45
7.97 + 0.05
SW (L = 500)
14.23 + 0.84
8.25 + 0.05
15.44 + 0.42
8.14 + 0.05
TSW-SL (L 100, 5)
13.27 0.23
8.30 + 0.01
TSW-SL (I. 17, k 15.9 + 0.35
8.10 + 0.04
TSW-SL (I. 167, k 3)
14.18 + 0.38
8.28 + 0.07
E.4. Denoising diffusion models
In this section, provide details about denoising diffusion models, class of generative models that have shown remarkable success in producing high-quality samples. We first describe the forward and reverse processes that form the foundation of these models. Then, we introduce the concept of denoising diffusion GANs, which aims to accelerate the generation process. Finally, we explain how our proposed TSW-SL distance can be integrated into this framework.
The process in diffusion models is typically divided into two main parts: the forward process and the reverse process.
The forward process defined as:
((1T))))) Il (=))))))1) q(11)1=1))
N(xa; v1 8,11-1.
where the variance schedule 3i,..., . Br can be constant or learned hyperparameters. The reverse process is defined as:
T
po( p(xT) II po (Ze- 110)),
po(x.- (2)) = 1), Eo(xr, 1)),
29

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
where po( 1) and Ex are functions that provide the mean and covariance for the Gaussian and are defined using MLPs.
The model is trained by maximizing the variational lower bound on the negative log-likelihod:
Pe (o(Too)) q ((1111/2))
B Po (3r)] & Fag log
I,,
While traditional models have successfully generated high-quality images without the need for adversarial training. However, their sampling process involves simulating a Markov chain for multiple steps, which can be consennnn.ing. To accelerate the generation process by reducing the number of steps T, denoising diffusion GANs (Xiao et al., 2021) propose utilizing an implicit denoising model:
Po (1 = po (22,,
with (O(, T)-
Subsequently, adversarial training is employed (Xiao et al., 2021) to optimize the model parameters
min
where Dado refers to either the GAN objective the Gensen-Shannon divergence (Goodfellow al., 2020). We follow the proposed Augmented Generalized Mini-batch Energy distances of (Nguyen et al., 2024b) leverage our TSW-SI distance for Dadu-
More specifically, as described by Nguyen et al. (2024b), the adversarial loss is replaced by the augmented generalized Mini-batch Energy (AGME) distance. For two distributions μ and v, with a mini-batch size n >> 1, the AGME distance using a Sliced Wasserstein (SW) kernel is defined as:
AGME (H, (H, p: = GME (A, 87)
where ū fu and py with the mapping (a) (Ir, (xx) for nonlinear function g Rd - R. The GME is the generalized Mini-batch Energy distance (Salimans al., 2018), given by:
-V) GMB (v. - -x) E Pt-)].
where X, L.d. Sm Y
Lid. ~ ymm and
m Px = m
6ay x= (II,...,Za.) F.m)
In the equation above, D denotes a distance function that can calculate the distance between two probability measures. To evaluate how well TSW-SL compares to other SW variants in capturing topological information, particularly when the supports lie in high-dimensional spaces, we replace D with both TSW and SW variants. We then train the generative model assess which distance metric better quantifies the divergence between two probability distributions. lower FID score indicates effective distance measure.
Experimental setup. For our experiments, we adopted the architecture and hyperparameters from Nguyen et al. (2024b), training our models for 1800 epochs. For TSW, we employed the following hyperparameters: L 2500, k = 4. For the vanilla SW and its variants, we adhered to the approach outlined in Nguyen et al. (2024b), using = 10000. This consistent setup allowed us to effectively compare the performance of our proposed methods against existing approaches while maintaining experimental integrity.
30

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
FID plot. Figure 12 illustrates the FID scores of SW-DD and TSW-SL-DD across epochs. Due to the wide range of FID values, from over 400 in the initial epoch to less than 3.0 in the final epochs, present the results logarithmic scale for improved visualization. The plot shows that TSW-SL-DD achieves a greater reduction in FID scores compared to SW-DD during the final 300 epochs.
SW-DD TSW-SL-DD
Log(FiD)
3-
2-
1
0
250
500
750
1000
1250
1500
1750
Epochs
Figure 12: FID score over epochs between SW-DD and TSW-SL-DD
E,5. Computational infrastructure
The experiments on gradient flow, color transfer and generative models using generative adversarial networks are conducted on single NVIDIA A100 GPU. Training generative adversarial networks CIFARIO requires 14 hours, while CelebA training takes 22 hours. Regarding gradient flows, each dataset's experiments take approximately 3.5 hours. For color transfer, the runtime is 15 minutes.
The denoising diffusion experiments were conducted parallelly on 2 NVIDIA A100 GPUs and each run takes us 81.5 hours.
F. Further Discussions
The proposed approach, tre-sliced-Wasssesseen on tree-structured systems of lines (TSW-SL), lays foundation for further research for TSW (2025b) propose novel splitting map for TSW-SL which takes into account the relative position between supports and lines in the tree systems. (Tran et al., 2025c) derive the non-linear Randon transform for the TSW-SL. Additionally, Tran et al. (2025a) extend the TSW-SL for probability measures supported on a sphere.
systems, especially, for its applications with dynamic-support measures. For examples, Tran et al.
Notice that the QuadTree (i.e., partition-based) and clustering- based tree metric sampling for tree- Wasserstein (Le et al., 2019; Indyk& & Thaper, 2003; Lin et al., 2025) may not be applicable for applications with dynamic-support measures, e.g.. diffusion models and generative models, since the constructed/sampler trees are bounded, i.c. from the root leaves, limited finite number of nodes. Additionally, these approaches may not include the Radon transforms and guarantee its injectivity which play the key role for applications with dynamic-support measures.
For p-order Wasserstein with > 1. The tree- wre-Wassersttin (TW) is the 1-order Wasserstein for probability measures on a tree. For more general settings, e.g., with > 1, or for probability measures on graph, one may consider a scalable variant- Sobolev transport (Le et al., 2022) which also yields closed- expression for fast computation, and Sariant-Sobolev generalizes the TW (i.e., for p = 1, Sobolev transport is equal to TW).
31

---

Tree-Sliced Wasserstein Distance: A Geometric Perspective
G. Broader Impact
The novel Tree-Sliced Wasserstein distance on a System of Lines (TSW-SL) introduced this paper holds significant potential for societal advancement, By refining optimal transport methodologies, TSW-SL enhances their accuracy and versatility across diverse practical domains. This approach, which synthesizes elements from both Sliced Wasserstein (SW) and Tree-Sliced Wasserstein (TSW), offers enhanced resilience and adaptability, partieularly in dynamic scenarios. The resulting improvements in gradient flows, color manipulation, and generative modeling yield more potent computational tools. These advancements promise to catalyze progress across multiple sectors. In healthcare, for instance, refined image processing could elevate the precision of medical diagnostics. The creative industries stand to benefit from more sophisticated generative models, potentially revolutionizing artistic expression. Moreover, TSW-SL's proficiency in handling dynamic environments opens new avenues for real-time analytics and decision-makee in fields ranging from finance environmental monitoring, By expanding the applicability of advanced computational techniques wider array of real-world challenges, TSW-SL contributes technological innovation and, consequently, to the enhancement of societal welfare.
32


Reviews:
[{"review_id": "pRLtLHTs0H", "reviewer": "Reviewer_gLiQ", "summary": "The paper proposes a novel variant of Sliced Wasserstein (SW) distance, termed Tree-Sliced Wasserstein Distance on Systems of Lines (TSW-SL). The key innovation is replacing one-dimensional projection lines in SW with tree systems, which allow for better preservation of topological structures while maintaining computational efficiency. The authors provide theoretical analysis proving the injectivity of their proposed Radon Transform on Systems of Lines, discuss metric properties, and derive a closed-form solution for OT problems on tree systems. Empirical results demonstrate that TSW-SL improves upon SW in tasks such as gradient flows, generative models, and denoising diffusion models.", "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": "Based on the two sections discussed below in the review, it appears the Reviewer may have fundamentally misunderstood our framework. Let us clarify this step by step:\n\n**Claims and Evidence.** The term $\\alpha(a_i)_l$ represents the mass allocated to the projection of point $a_i$ onto line $l$, not the location of $a_i$ on line $l$. This misunderstanding seems to be the root of the confusion noted in the review. The second paragraph now does not align logically. Furthermore, our method TSW-SL, similar to SW, involves sorting the projections of data points on each line, as detailed in lines 284-291 of our paper. We have meticulously discussed these concepts in Section 4 where the Radon Transform is defined, and provided explicit formulations in Section 5.2.\n\n**Methods and Evaluation Criteria**.\n1. We define the value $w_e$ as the distance between two consecutive points on each line within the tree systems.\n2. The terms \"root\" and \"subtree\" are used in their typical graph-theoretical sense, while \"the root is positioned near the mean of the target distribution\" pertains to the sampling method used for the root in our experiments.\n3. The same misunderstanding from Claims and Evidence persists: $\\alpha$ defines mass distribution across $k$ projections, not their locations.\n\n---\nWe highly encourage the Reviewer to revisit the paper, as the current review suggests that several key points may have been overlooked. We sincerely appreciate the time and effort put into the review and are especially grateful for its constructive aspects. The foundation of our work is significant, as it serves as the backbone for several other studies—some of which, focusing on special cases of our framework, have already been well-received and published (See [4], [5]).\n\n---\nAll the tables are provided in: https://sites.google.com/view/tree-sliced-wasserstein-distan.\n\nFor **Q3 and Q4**, please refer to the above discussion, as they stem from a misunderstanding.\n\n---\n**Answer Q1.** To the best of our knowledge, our TSW-SL is currently the only tree-sliced framework that can be effectively applied to large-scale generative tasks involving the transportation of a training measure to a target measure in Euclidean space. Other works on Tree-Sliced Wasserstein, such as [1], [2], [3] are mainly designed for classification or clustering tasks and are not applicable to generative settings. This limitation stems from their reliance on clustering-based or nearest-neighbor search frameworks for computing slices—a strategy that is theoretically unsuitable (as the clustering must be recomputed each time the training measure is updated, **rendering previous clustering results irrelevant**) and empirically inefficient (since clustering is significantly more computationally expensive than linear projection methods).\n\nWe did not include these methods as baselines because applying them in our generative setting is *infeasible*.\n\nWhile approaches in [3] do not apply to our setting, it provide a closed-form OT solution in tree metrics (see Prop. 1, [3])—crucial for deriving Eq. (13). However, our focus differs fundamentally from [3].\n\n**Answer Q2.** We conducted experiments on the gradient flow task in response to the two scenarios mentioned by the Reviewer.\n1. Increasing $k$, reducing $L$: Table 1.\n2. Fixing $L$, increasing $k$: Table 2.\n\nOverall, the results show that our method is robust across different $k$ and $L$ values. For a fair comparison, we set $k$ and $L$ so that $N \\times k$ matched the total projection directions in SW and its variants.\n\nWe also explored fixing $L$ while increasing $k$. As shown in Table 3, this improves performance in GAN tasks. Notably, TSW with total 40 directions outperformed SW with 50, underscoring our method's effectiveness.\n\n**Answer Q5.** Intuitively, given $N$ points in $\\mathbb{R}^d$ and a tree system consisting of $k$ lines, the projection results in a total of $kN+k$ points on the tree structure. Specifically, each of the $N$ points gives $k$ projections, contributing $kN$ points, while the additional $k$ points come from the $(k-1)$ intersections among the lines and the root of the tree. These points together form a tree in the graph-theoretic sense. Here, $w_e$ denotes the length of edge $e$ in this tree, and the notion of a subtree follows standard definitions in graph theory. A detailed summary of how a set of points on tree systems forms a tree metric space is presented in Corollary A.12, Appendix A.\n\n---\nWe thank the Reviewer for the constructive feedback, as well as for pointing out typos and missing references, which we will address. If the Reviewer finds our clarifications satisfactory, we kindly ask you to consider raising the score. We would be happy to address any further concerns during the next stage of the discussion.\n\n---\n*References.*\n\n[4] Hoang Tran et al., Distance-Based Tree-Sliced Wasserstein Distance. ICLR 2025\n\n[5] Hoang Tran et al., Spherical Tree-Sliced Wasserstein Distance. ICLR 2025", "claims_and_evidence": "The claim that TSW-SL provides a better geometric perspective than SW by capturing more structural information seems insufficient. The role of a system of lines in TSW-SL is analogous to the random projections in SW. In SW, each data point $ a_i $ projects onto $ \\theta_i $ via the dot product $ a_i^T \\theta_i $, enabling 1D sorting of points within a distribution. However, TSW-SL lacks this sorting property because each data point $ a_i $ is projected onto a line $ l $ based on $ \\alpha(a_i)_l $, where $ \\alpha $ is a predefined hyperparameter, thereby losing the spatial information of $ a_i $.  \n\nFor instance, if $ \\alpha $ is a constant vector for all data points $ \\{a_i\\} $, the total mass on each line $ l $ remains exactly $ \\alpha_l $ for any probability distribution, making the TSW-SL distance between any two distributions zero. This suggests that the geometric and topological properties of TSW-SL depend primarily on $ \\alpha $, which is not a compelling justification for its ability to capture meaningful structural information.", "methods_and_evaluation_criteria": "The definition and formalization of tree systems are clear and well-structured. The method introduces a novel tree system as an alternative to random projections, differing from classical tree structures such as QuadTree and ClusterTree (although this distinction is not explicitly mentioned in the main text). However, the method appears incomplete, particularly regarding Eq. (13):\n\n1. **Setting the value of $ w_e $**: How should $ w_e $ be determined? For instance, should it be based on the Euclidean distance between $ x_i $ and $ x_{i+1} $? This is not specified.  \n\n2. **Defining the root and subtree $ \\Gamma(v_e) $**: The statement that \"the root is positioned near the mean of the target distribution\" is vague. How exactly should the root be set, and what constitutes the subtree $ \\Gamma(v_e) $?  \n\n3. **Choosing $ \\alpha $**: This is the most critical aspect. Given a distribution $ \\mu $ with points $ \\{a_i\\} $ and a distribution $ \\nu $ with points $ \\{b_j\\} $, where $ \\{a_i\\} $ and $ \\{b_j\\} $ may differ, how can a universal $ \\alpha $ be generated for all these points?  \n\n4. **Training $ \\alpha $**: It is mentioned that $ \\alpha $ can be \"a trainable constant vector,\" but how should it be trained? If it is set as a constant vector for all points, then the total mass on each line $ l $ is exactly $ \\alpha_l $, assuming $ \\sum_i u_i = 1 $ for probability distributions.", "theoretical_claims": "The theoretical derivations appear solid, with proofs provided in the supplementary material. The injectivity of the Radon Transform on Systems of Lines is well-supported. However, offering additional intuition behind certain proofs—such as why tree metrics naturally yield closed-form solutions—would enhance clarity.", "experimental_designs_or_analyses": "1. The experimental setup is reasonable, but the comparisons primarily focus on SW and a few of its variants (MaxSW, SWGG, LCVSW). It would be beneficial to evaluate the method against other tree-Wasserstein distance approaches, such as QuadTree [1], FlowTree [2], and ClusterTree [3], along with their sliced versions.\n - [1] Piotr I., Nitin T. *Fast Image Retrieval via Embeddings*, 2003.  \n - [2] Backurs, A., Dong, Y., Indyk, P., Razenshteyn, I., & Wagner, T. *Scalable Nearest Neighbor Search for Optimal Transport.* ICML, 2020.  \n - [3] Le, T., Yamada, M., Fukumizu, K., & Cuturi, M. *Tree-Sliced Variants of Wasserstein Distances.* NeurIPS, 2019.  \n\n2. The paper lacks a hyperparameter study on the impact of different tree configurations on performance, particularly for the hyperparameters $ k $, $ L $, and $ \\alpha $. Although Appendix E.3 provides an ablation study on the number of lines, it only considers values in {$3,4,5$}. A more convincing analysis would explore a broader range, such as $ k = ${$5, 10, 20, 50$}.  \n\n - For instance, if $ k $ increases, can $ L $ be reduced while maintaining the same accuracy?  \n\n - With a fixed $ L $, does increasing $ k $ improve accuracy?  \n\nThese relationships remain unclear and would benefit from further investigation.", "relation_to_broader_scientific_literature": "The paper effectively situates itself within the broader optimal transport and machine learning literature. It builds upon foundational works on Sliced Wasserstein distances and tree-based OT metrics but could benefit from a more direct comparison with recent tree-Wasserstein approximations.", "essential_references_not_discussed": "The paper primarily cites literature on SW variants and tree-based metrics but does not discuss more recent developments in tree-Wasserstein methods (e.g., QuadTree [1], FlowTree [2], and ClusterTree [3]).\n\n[1] Piotr Indyk, Nitin Thaper. Fast image retrieval via embeddings, 2003.\n\n[2] Backurs, A., Dong, Y., Indyk, P., Razenshteyn, I., & Wagner, T. Scalable nearest neighbor search for optimal transport. ICML, 2020.\n\n[3] Le, T., Yamada, M., Fukumizu, K., & Cuturi, M. Tree-sliced variants of Wasserstein distances. NeurIPS, 2019.", "other_strengths_and_weaknesses": "Strengths:\n\n1. Well-motivated theoretical contributions.\n2. Empirical validation includes a diverse set of experiments.\n3. Computational efficiency is maintained compared to SW.\n\nWeaknesses:\n1. The experimental validation primarily focuses on SW and lacks comparisons with broader tree-Wasserstein distance approaches.\n2. The connection between tree structures and improved topological preservation needs further clarification. It remains unclear how the tree system captures topological information (see the discussion in the Claims and Evidence section).\n3. The study lacks a detailed ablation or hyperparameter analysis on the sensitivity of performance to tree parameters ($k$, $L$, and $\\alpha$).", "other_comments_or_suggestions": "A small suggestion:\n\nLine 288: The proof for the below theorem is provided in Appendix D.1. → The proof for the theorem below is provided in Appendix D.1.", "questions_for_authors": "1. Have you considered alternative tree-Wasserstein distances beyond tree systems, such as QuadTree, FlowTree, and ClusterTree, including their sliced versions? If so, how do they compare?  \n\n2. How sensitive is the performance of TSW-SL to the choice of hyperparameters, including $ k $, $ L $, and $ \\alpha $? A more detailed ablation study on different tree configurations would be valuable:  \n   - If $ k $ increases, can $ L $ be reduced while maintaining the same accuracy?  \n   - For a fixed $ L $, does increasing $ k $ improve accuracy?  \n\n3. How should $ \\alpha $ be set for any data points $ a_i $ or $ b_j $? How should $ \\alpha $ be trained? If it is set as a constant vector for all points, then the total mass on each line $ l $ is exactly $ \\alpha_l $, assuming $ \\sum_i u_i = 1 $ for probability distributions.  \n\n4. In Line 299, why is TSW-SL identical to SW when $ k = 1 $? In SW, all data points are projected onto a 1D line via the dot product $ a_i^T \\theta_i $, resulting in a 1D sorting of points. However, in TSW-SL with $ k = 1 $, all masses $ \\{u_i\\} $ or $ \\{v_i\\} $ are projected onto the same line without a sorting relationship. Then, using Eq. (13), the system consists of a single line/edge, and the value of Eq. (13) simplifies to $w_e \\cdot \\left( \\sum_i u_i - \\sum_i v_i \\right)$. If $ \\mu $ and $ \\nu $ are probability distributions, then this equals zero. Is this interpretation correct?  \n\n5. How is $ w_e $ calculated, and what constitutes the subtree $ \\Gamma(v_e) $ in Eq. (13)?  \n\nThere may be a misunderstanding of Eq. (13), but to be honest, the key steps in calculating the final distance appear to be missing—specifically, the choice of $ \\alpha $ and the value of $ w_e $.  \n\nI will increase my score if the above concerns are addressed.", "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": "jYiipDh01Y", "reviewer": "Reviewer_8fkW", "summary": "The authors study the sliced-Wasserstein distance and propose replace projecting measures onto one-dimensional lines with a more complex structure, which they call a tree system. They propose a novel variant of Radon transforms for tree systems which leads to an efficient metric which they call Tree-Sliced Wasserstein on Systems of Lines (TSW-SL). The time complexity to compute TSW-SL is $O(Lkn\\log n + Lkdn)$, where $L$ is the number of tree systems sampled, $k$ is the number of lines, and $n$ is the number of projections on each line. Finally, the authors conduct experiments showing the effectiveness of TSW-SL on (1) the gradient flow task where the TSW-SL is better able to minimize Wasserstein distance between a target and source distribution than standard SW distance, (2) GANs where the TSW-SL is used in the adversarial loss term and (3) de-noising diffusion models.\n\n### Update after rebuttal\n\nThanks to the authors for their response. I maintain my score.", "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": "**Answer Q1+Q2.**  It appears that the Reviewer may have misunderstood certain key aspects of our paper, as several important points seem to have been overlooked.\n\nRespectfully, we do not claim that TSW-SL is a simplified version of the sliced Wasserstein (SW) distance. On the contrary, TSW-SL is a more general framework than SW. In fact, SW can be seen as a special case of TSW-SL when the underlying tree system consists of only a single line.\n\nThis generalization stems from the novel splitting mechanism, denoted by $\\alpha$, which is not present in traditional sliced approaches. Instead of projecting the entire mass of a point $x$ onto a single line and computing the 1D Wasserstein distance for each slice, our method allows $\\alpha$ to split the mass of $x$ across multiple projections—each corresponding to a line in the tree system. The Wasserstein distance is then computed over this richer tree structure. This is the key reason that TSW-SL serves as a non-trivial generalization of SW.\n\nTo the best of our knowledge, such a mechanism does not exist in any prior sliced Wasserstein variant, as those methods operate strictly with single-line projections in each slice.\n\nFinally, there is no direct connection between GSW and TSW-SL due to a fundamental difference in their formulations: TSW-SL is defined over tree systems, whereas GSW operates within the line setting.\n\n**Answer Q3.** To the best of our knowledge, our TSW-SL is currently the only tree-sliced framework that can be effectively applied to large-scale generative tasks involving the transportation of a training measure to a target measure in Euclidean space. Other works on Tree-Sliced Wasserstein (TSW), such as [1], [2], [3], are mainly designed for classification, regression, or clustering tasks and are not applicable to generative settings. This limitation stems from their reliance on clustering-based or nearest-neighbor search frameworks for computing slices—a strategy that is theoretically unsuitable (as the clustering must be recomputed each time the training measure is updated, **rendering previous clustering results irrelevant**) and empirically inefficient (since clustering is significantly more computationally expensive than linear projection methods).\n\nWe did not include these methods as baselines because applying them in our generative setting is *infeasible*.\n\n**Experiments with GSW.** All the tables are provided in: https://sites.google.com/view/tree-sliced-wasserstein-distan.\n\n*Gradient Flow.* We included GSW as a baseline to evaluate alongside our methods on the gradient flow task using the 25 Gaussians dataset. As shown in Table 4, TSW-SL consistently outperforms both maxGSW and GSW (circular). While GSW (homogeneous-polynomial) converges faster, TSW-SL slightly surpasses it in performance during the final training epochs.\n\n*Generative modeling.* We further conducted experiments comparing GSWAE and TSWAE on the generative modeling task using MNIST dataset, following the setup from [KNSBR '19]. As reported in Table 5, TSW-SL achieves superior performance over GSW in minimizing the distance between reconstructed samples and the prior, and between the reconstructed samples and the target distributions.\n\n\n\n*Denoising diffusion.* We were unable to include GSW in the diffusion task due to time constraints. However, we would like to highlight the promising potential of our tree-sliced approach. For this task, we adopt the same experimental setup as the recent variant of SW in [5]. The best performance reported in [5] is $2.70$, compared to $2.90$ from the vanilla SW method. Our approach, TSW-SL, achieves a score of $2.83$.\n\nNote that, our method serves as a foundational replacement for the SW framework, introducing a tree-based structure rather than focusing on optimizing specific components like the sampling method, as done in [5]. Therefore, we do not anticipate a significant performance boost. Nonetheless, because our work establishes a solid foundation, a recent follow-up study (see [4]) that builds upon one instance of our tree-based framework (they use concurent-lines tree structures) has achieved a performance of $2.53$ on the same diffusion task. This result highlights the potential and promising direction of future research in tree-sliced approaches.\n\n---\nWe thank the Reviewer for the constructive feedback. If the Reviewer finds our clarifications satisfactory, we kindly ask you to consider raising the score. We would be happy to address any further concerns during the next stage of the discussion.\n\n---\n*References.*\n\n[1] Indyk & Thaper. Fast image retrieval via embeddings, 2003.\n\n[2] Backurs et al. Scalable nearest neighbor search for optimal transport. ICML, 2020.\n\n[3] Le et al. Tree-sliced variants of Wasserstein distances. NeurIPS, 2019.\n\n[4] Tran et al. Distance-Based Tree-Sliced Wasserstein Distance. ICLR, 2025.\n\n[5] Nguyen et al. Sliced Wasserstein with Random-Path Projecting Directions. ICML, 2024.", "claims_and_evidence": "The authors claim that given a system of lines $\\mathcal{L}$, one can produce a chain-like tree structure. This tree structure induces a metric on the space induced by the system of lines $\\mathcal{L}$, $\\Gamma_\\mathcal{L}$ -- the space of the disjoint union of copies of $\\mathbb{R}$ with their intersections quotiented out. One can then take several systems of lines, $\\mathcal{L}_1, \\dots, \\mathcal{L}_k$, and use their Radon transform on systems of lines to compute their tree-Wasserstein distance on systems of lines. Additionally, this TSW-SL is a proper metric between probability measures. All claims are well supported and I did not see anything problematic.", "methods_and_evaluation_criteria": "The benchmark datasets and tasks that the authors use are standard for evaluation of SW distances. There are couple of datasets that perhaps the authors could add to their flow minimization experiments, perhaps MNIST so we can see the performance of TSW-SL on a more realistic dataset. Another interesting realistic benchmark/task that the authors could consider is alignment of multi-modal RNA seq datasets, especially because their high-dimensional Gaussian dataset samples distributions over $\\mathbb{R}^{200}$ and RNAseq data tends to be inherently much higher-dimensional.", "theoretical_claims": "All proofs are in the supplement and I did not check them.", "experimental_designs_or_analyses": "I looked through all experiments. I think the gradient flows experiments could be expanded as currently, these experiments are only done with synthetic datasets (Gaussians and Swiss Roll). I think in [KNSBR '19], they do similar experiments and include MNIST to show the performance of GSW on a more realistic dataset. It would be nice to see the performance of TSW-SL under similar conditions. Additionally, I think that there are several variants of sliced-Wasserstein distance (e.g. GSW and correspondingly, max-GSW) and it would be nice to see how TSW-SL compares to GSW.\n\n\"Generalized Sliced Wasserstein Distances\" [KNSBR '19]", "relation_to_broader_scientific_literature": "I am not very familiar with the sliced-Wasserstein distance literature. However, to the best of my knowledge, this falls along the same line of work as [WSBOR '13], [KTOR '16], and [KNSBR '19]. While [KNSBR '19] already considered replacing the linear projection in standard sliced-Wasserstein distance with non-linear projections, this paper explicitly projects measures onto tree structures. Once the measures are projected onto tree structures, the computation of OT on trees is also well known from [IT '03] along with a large body of follow-up work on using tree structure to approximate OT.\n\n\"A Linear Optimal Transportation Framework for Quantifying and Visualizing Variations in Sets of Images\" [WSBOR '13]\n\"A continuous linear optimal transport approach for pattern analysis in image datasets\" [KTOR '16]\n\"Generalized sliced-Wasserstein distance\" [KNSBR '19]\n\"Fast image retrieval via embeddings\" [IT '03]", "essential_references_not_discussed": "I do not know of any essential references which are not discussed.", "other_strengths_and_weaknesses": "Strengths: The authors introduce a new framework for sliced Wasserstein distance which uses projection to systems of lines and uses the associated tree structure on the system of lines to compute an efficient metric between measures. I like that this paper connects sliced-Wasserstein distance to tree-Wasserstein distance, which like 1D OT, is another special case where Wasserstein distance can be quickly computed. At least, it is a new framework for sliced-Wasserstein distance which leverages previous work on tree Wasserstein distance. I feel it is somewhat similar in spirit to the sliced-tree-Wasserstein distance [LYFC '19] as in both cases one samples several different trees and then averages the Wasserstein distance between measures in the tree space.  \n\nWeaknesses: The authors do cite the generalized sliced Wasserstein distance and they briefly mention in their experiments that they will only compare to vanilla sliced-Wasserstein distance. However, I think either (a) the authors should include a comparison to GSW and max-GSW in their experiments or (b) they should provide more justification as to why they do not compare to GSW. I will elaborate more in the questions/suggestions section.", "other_comments_or_suggestions": "I did not immediately see any typos.", "questions_for_authors": "1. Can you elaborate more on why you only compare to SW distance? I think it is not very compelling to just say that TSW-SL is just a simple version of SW distance. In that case, it is unclear the utility of TSW-SL in practice when GSW and max-GSW exist. \n\n2. Could you comment on the connection, if any, between GSW and TSW-SL? It seems that there is a generalized Radon transform also defined in GSW.\n\n3. Is it possible to compare (at least in the gradient flow experiment with the swiss roll and Gaussian datasets) to regular sliced tree-Wasserstein distance? It seems that TSW-SL is very similar (in practice) to the sliced tree-Wasserstein distance as in both cases, one samples a collection of trees and then computes the Wasserstein distance between the two measures on the trees.", "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": "1K1bF5ay5M", "reviewer": "Reviewer_e1vF", "summary": "This paper presents a new variant of the sliced Wasserstein distance, called the tree-sliced Wasserstein distance on systems of lines, or TSW-SL. The main idea is that instead of iteratively projecting the distributions to a random line and computing the average of these 1D Wasserstein distances (as is done in the sliced Wasserstein distance), one can construct a set of randomly selected lines such that each line $\\ell_i$ intersects the lines $\\ell_{i-1}$.  The algorithm then uses a splitter to project the mass of each point to these lines, and then solves the tree Wasserstein distance on these projections efficiently.\n\n\n## update after rebuttal\nI keep my positive score.", "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": "**Q1. Is it true that your algorithm generates a 1D Wasserstein problem instance? ... If I understood correctly, would it be more accurate to use a name other than tree-sliced Wasserstein?**\n\n**Answer Q1.** In the Sliced Wasserstein (SW) framework, each line projection leads to a 1D Optimal Transport (OT) problem. Similarly, in our framework, each tree system projection results in an OT problem defined on a tree metric space—specifically, the tree system itself. When the tree system consists of a single line, the resulting OT problem is identical to that of the SW framework.\n\nThis highlights that our algorithm extends beyond solving a single 1D OT problem—it handles a collection of OT problems on tree metric spaces. Therefore, we believe the naming of our framework is accurate. In the literature (e.g., [1]), the term Tree-Sliced Wasserstein often refers to approaches where different metrics are sampled to compute the final result. In contrast, our method samples different tree structures (i.e., systems of lines), leading to a fundamentally different construction. Moreover, [1] is applied in a different context and to different types of tasks, making the two approaches inherently non-comparable.\n\nThis is precisely why we append “on systems of lines” to our framework’s name—to clearly distinguish it from existing lines of work. To the best of our knowledge, our TSW-SL is currently the only tree-sliced framework that can be effectively applied to large-scale generative tasks involving the transportation of a training measure to a target measure in Euclidean space. Other works on Tree-Sliced Wasserstein (TSW), such as [1], are mainly designed for classification, regression, or clustering tasks and are not applicable to generative settings. This limitation stems from their reliance on clustering-based or nearest-neighbor search frameworks for computing slices—a strategy that is theoretically unsuitable (as the clustering must be recomputed each time the training measure is updated, rendering previous clustering results irrelevant) and empirically inefficient (since clustering is significantly more computationally expensive than linear projection methods).\n\n---\n\n**Experimental Designs Or Analyses.** Although the experimental setup and evaluation are informative, I believe there is a lack of comparison with more recent methods. As the authors mentioned, their proposed method is an alternative of SW by proposing the system of lines and they do not expect their method to out-perform recent variants of SW, but I think it would be important to have such comparison.\n\n**Answer.** For the diffusion task, we were unable to include GSW due to time constraints. However, we would like to highlight the promising potential of our tree-sliced approach. We adopt the same experimental setup as in a recent work on SW [2]. The best performance reported in [2] is $2.70$, compared to $2.90$ from the vanilla SW method. Our approach, TSW-SL, achieves a score of $2.83$.\n\nOur method serves as a foundational replacement for the SW framework, introducing a tree-based structure rather than focusing on optimizing specific components like the sampling method, as done in [2]. Therefore, we do not anticipate a significant performance boost. Nonetheless, because our work establishes a solid foundation, a recent follow-up study (see [3]) that builds upon one instance of our tree-based framework (they use concurent-lines tree structures) has achieved a performance of $2.53$ on the same diffusion task. This result highlights the potential and promising direction of future research in tree-sliced approaches.\n\nIn addition, in the Gradient Flow task presented in our paper, we compared our method with several recent sliced Wasserstein (SW) baselines. We also kindly refer the Reviewer to the section **Experiments with GSW** in our response to Reviewer 8fkW’s comments, where we compare TSW-SL with GSW [4] used in Variational Autoencoder.\n\nWe appreciate your comment regarding the inclusion of recent SW methods in our paper and will make the necessary revisions accordingly.\n\n---\nWe thank the Reviewer for the constructive feedback and for pointing out the typos in our paper. We will address them accordingly. If the Reviewer finds our clarifications satisfactory, we kindly ask you to consider raising the score. We would be happy to address any further concerns during the next stage of the discussion.\n\n---\n*References.*\n\n[1] Le, T., Yamada, M., Fukumizu, K., & Cuturi, M. Tree-sliced variants of Wasserstein distances. NeurIPS, 2019.\n\n[2] Khai Nguyen et al., Sliced Wasserstein with Random-Path Projecting Directions. ICML 2024.\n\n[3] Hoang Tran et al., Distance-Based Tree-Sliced Wasserstein Distance. ICLR, 2025.\n\n[4] Kolouri, S. et al., Generalized Sliced Wasserstein Distances. NeurIPS, 2019.", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "Yes, the theoretical claims are reasonable and the proofs that I checked are correct.", "experimental_designs_or_analyses": "Although the experimental setup and evaluation are informative, I believe there is a lack of comparison with more recent methods. As the authors mentioned, their proposed method is an alternative of SW by proposing the system of lines and they do not expect their method to out-perform recent variants of SW, but I think it would be important to have such comparison.", "relation_to_broader_scientific_literature": "The sliced Wasserstein distances and its improved variants have been used in numerous contexts in the machine learning domain. This paper also presents a variant of the sliced Wasserstein distance, which improves the performance generative models such as GANs and diffusion models when trained using TSW-SL instead of SW.", "essential_references_not_discussed": "No.", "other_strengths_and_weaknesses": "**Strengths**\nThe paper presents a new perspective on the sliced Wasserstein distances and presents interesting novel ideas.\n\n**Weaknesses**", "other_comments_or_suggestions": "* Line 82 RC: supports repeated\n* Line 205 LC: it seems like the two sentences are the same sentence with different wordings \n* The definition of $\\mathcal{P}$ is given in line 241 RC but is used before that.\n* Section 5 and subsection 5.1 have the same title.", "questions_for_authors": "* Is it true that your algorithm generates a 1D Wasserstein problem instance? More specifically, your algorithm generates a set of line segments connected together, and projects the distributions over these line segments. So although the general framework works for trees, your algorithm only deals with 1D instances. If I understood correctly, would it be more accurate to use a name other than tree-sliced Wasserstein?", "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": "5L2NqFFGXs", "reviewer": "Reviewer_NG1f", "summary": "The paper introduces a novel approach to **projected Optimal Transport (OT) computation**, termed **Tree-Sliced Wasserstein (TSW) Distance**. The key contributions include:  \n\n1. **Tree Systems**, a generalization of straight-line projections that incorporate hierarchical structures.  \n2. **Radon Transform on Tree Systems**, along with its **injectivity property**, ensuring meaningful projections.  \n3. **Tree-Sliced Wasserstein Distance**, which retains a **closed-form solution** and satisfies **metric properties** for OT computation.\n\n# update after rebuttal \nMost of my concerns have been explained, and I raised my score to 3 (weak accept).", "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": "**Answer W2.** All the lines in a tree system are infinitely long. In practical applications, empirical measures have bounded support. As a result, when these measures are projected onto the lines of a tree system, the resulting measure on the tree system also has bounded support.\n\n*It is worth noting that TSW-SL is a more general framework than SW [...] in any prior sliced Wasserstein variant, as those methods operate strictly with single-line projections in each slice*. (Kindly refer to Answer **Q1 + Q2** in our response to Reviewer 8fkW for the full details)\n\n**Answer W3.**  In practical applications, we aim to position the tree root and sources in proximity to the empirical data. Therefore, as noted in line 320, “the tree systems will be sampled such that the root is positioned near the mean of the target distribution,” i.e., near the data mean. We simply write the distribution of $x_i, t_i$ as stated to emphasize that the sampling process naturally induces a distribution over the space of trees.\n\nWe thank the Reviewer for suggesting a grid search strategy to optimize the sampling method. However, we consider this beyond the scope of the current paper and view it as a promising direction for future work. This is analogous to how SW was initially developed with randomly sampled lines, and subsequent works later refined the sampling process for improved performance.\n\n**Answer W4 + Q3.** For $p>1$, the proposed approach can be extended. However, the Tree-Wasserstein distance with $p>1$ lacks a simple closed-form approximation (see [1]). A meaningful alternative is provided by Sobolev Transport [2], which offers a closed-form approximation and has been applied in the tree-sliced framework, as discussed in Eq. (13).\n\nAlthough we do not mention the case of $p >1$ in the paper, our implementations support arbitrary values of $p$. Indeed, all experiments in the paper are conducted with $p=2$, as it serves as the default setting. We believe this way of writing simplifies the presentation by avoiding the complexities of the Sobolev Transport literature, while still preserving flexibility in implementation. For this reason, we have chosen not to include it in our paper.\n\nDue to space constraints, we strongly encourage the Reviewer to refer to the Sobolev Transport literature. Our extension to the $p>1$ case still satisfies the theoretical guarantees discussed in the paper.\n\n**Answer Q1.** The full tree system concept requires rigorous derivation, that is why we emphasize that Section 3 offers only an intuitive and concise overview, and a careful reading of Appendix A is necessary for full mathematical rigor. While some notations may seem redundant at first glance, they are ultimately essential for defining the concepts precisely.\n\n**(1.1)** The sentence mentioned in Section 3.2 is mathematically correct, with a rigorous explanation in Appendix A.3.\n\nAs the Reviewer suggested, we can visualize the system of two lines as follows: Let $l$ and $l'$ be two lines that intersect at a point $x$. The points $(x,l)$ of $l$ and $(x,l')$ of $l'$ — representing the same point $x \\in \\mathbb{R}^d$— are identified. By taking the quotient topology, the resulting tree system formed from  $l$ and $l'$ resembles the shape of the letter \"X\".\n\n**(1.2)** The Reviewer may have missed the connectedness condition noted in line 153. A tree system is formed only when the set of lines is connected—a property we define rigorously in Appendices A.1 and A.2. Overlooking this condition may also contribute to the confusion in the following discussion.\n\n**(1.3)** In Fig. 1, when viewed as lines in $\\mathbb{R}^2$, the five lines intersect pairwise, making the system connected. Once connected, a tree structure can be imposed by selecting only four adjacent pairs, removing certain geometric intersections. This results in a tree structure where some lines still intersect in space but are not connected in the tree.\n\nThis also justifies our use of the notation $(x,l)$ for a point on a line, rather than simply $x$.\n\n**Answer Q2.** We acknowledge that a clarification on an abuse of notation is indeed missing. For simplicity, we denote a function \n$f \\in L^1(\\mathcal{L})$ as a function defined on the ground set of $\\mathcal{L}$, denoted by \n$\\bar{\\mathcal{L}}$. We will clarify it.\n\n**Answer Q3.**  We kindly refer the Reviewer to the section **Experiments with GSW** in our response to Reviewer 8fkW’s comments.\n\n---\nWe thank the Reviewer for the constructive feedback, as well as for pointing out typos and missing references, which we will address. If the Reviewer finds our clarifications satisfactory, we kindly ask you to consider raising the score. We would be happy to address any further concerns during the next stage of the discussion.\n\n---\n*References.*\n\n[1] Tam Le et al., Tree-Sliced Variants of Wasserstein Distances. NeurIPS 2019\n\n[2] Tam Le et al., Sobolev Transport: A Scalable Metric for Probability Measures with Graph Metrics. AISTATS 2022", "claims_and_evidence": "Yes.", "methods_and_evaluation_criteria": "Yes.", "theoretical_claims": "The paper presents several key **theoretical claims** that establish the foundation of **Tree-Sliced Wasserstein (TSW) Distance**:  \n\n- **System of Lines (Definition 3.1)**: Introduces a **set of $k$ lines**, referred to as a **system of lines**, which forms the basis for tree-based projections.  \n- **Tree Structure Construction (Algorithm 1)**: Provides a **procedure for constructing tree systems**, ensuring connectivity and well-defined hierarchical structures.  \n- **Radon Transform on Systems of Lines (Definition 4.1) & Injectivity (Theorem 4.2)**:  \n  - Defines a **Radon Transform** adapted to tree systems.  \n  - Proves **injectivity**, ensuring that distributions mapped onto tree systems retain distinct information.  \n- **Tree-Sliced Wasserstein Distance (Definition 5.1) & Metric Property (Theorem 5.2)**:  \n  - Formalizes **TSW distance** as an extension of Sliced Wasserstein using tree projections.  \n  - Establishes **metric properties**, proving that TSW is a **valid distance function** for probability measures.", "experimental_designs_or_analyses": "Yes, I've reviewed the experiment designs and analyses.", "relation_to_broader_scientific_literature": "### **Relation to Prior Work**  \n\nClassical **Optimal Transport (OT)** methods project data onto a **single straight line**, as seen in the **Sliced Wasserstein (SW) Distance** and its variants. The proposed **Tree-Sliced Wasserstein (TSW) Distance** generalizes this by **projecting data onto a system of $k$ lines**, forming a hierarchical structure that better preserves geometric and topological properties.  \n\n#### **Related References**  \n1. **Carriere et al. (2017) [1]** – Introduced the **Sliced Wasserstein Kernel** for persistence diagrams, leveraging one-dimensional projections to improve OT computations.  \n2. **Liutkus et al. (2019) [2]** – Proposed **Sliced-Wasserstein Flows**, using 1D projections for generative modeling, demonstrating the efficiency of SW-based transport.  \n3. **Kolouri et al. (2019) [3]** – Developed **Generalized Sliced Wasserstein Distances**, modifying the projection mechanism but still relying on **one-dimensional lines**.  \n\nThe **TSW approach extends these works** by incorporating **multi-line tree-based projections**, which retain computational efficiency while improving structure preservation.", "essential_references_not_discussed": "Recommend to add the following: \n\n[1] Carriere, M., Cuturi, M., & Oudot, S. (2017). Sliced Wasserstein kernel for persistence diagrams. *In ICML 2017 - Thirty-fourth International Conference on Machine Learning*, pp. 1–10.\n[3] Kolouri, S., Nadjahi, K., & Simsekli, U. (2019). Generalized Sliced Wasserstein Distances. *Advances in Neural Information Processing Systems (NeurIPS)*.", "other_strengths_and_weaknesses": "## Weaknesses  \n\n1. **Unclear tree structure/tree topology**  \n   - One of the key concepts of tree structure and tree topology is not clearly explained. See Question 1 for details.  \n\n2. **Combination of Tree Random Transform (Eq. 9) and Closed-Form Wasserstein Tree Distance (Eq. 13)**  \n   - The tree-sliced Wasserstein distance appears to project data into the union of joint line segments, where the first and last line segments have infinite length, and all other middle line segments have finite length.  \n   - If this understanding is correct, I do not see why this slicing approach retains more information than simply projecting onto these \\( k \\) lines.  \n   - Specifically, suppose each line system \\( \\mathcal{L} \\) contains \\( k \\) lines. I cannot see how the projected tree Wasserstein distance in \\( \\mathcal{L} \\) contains more information than the standard sliced Wasserstein distance projected onto these same \\( k \\) lines, given that their computational costs are identical.  \n\n3. **Dependency on the tree construction algorithm (Algorithm 1)**  \n   - The performance of the tree Wasserstein distance seems to heavily rely on the tree construction method.  \n   - In particular, choosing \\( x_1 \\sim [-1,1] \\) and \\( t_i \\sim [-1,1] \\) may not always be optimal, especially when the data scale is too large or too small.  \n   - I recommend that the authors discuss the potential impact of these hidden hyperparameters and explore a grid search strategy to optimize them.  \n\n4. **Limitations in extending to \\( L^2 \\) cost**  \n   - The tree Wasserstein metric appears to be difficult to extend to \\( L^2 \\) cost due to the constraints of the tree metric structure.  \n   - In contrast, the classical sliced Wasserstein (SW) distance can naturally incorporate an \\( L^2 \\) cost.  \n   - The authors should discuss this limitation in detail and consider potential ways to address it.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "## 1. P3, Line 127  \n- **Sentence unclear:** *\"quotient at the intersection of these copies.\"* Could the authors clarify this statement?  \n- **(1.1) Quotient space clarification:**  \n  Given a set \\(X\\) and an equivalence relation \\(\\sim\\), the quotient space \\(X/\\sim\\) consists of the disjoint union of all equivalence classes in \\(X\\) determined by \\(\\sim\\). I recommend including a simple example to illustrate the concepts of equivalence relation and equivalence class in this context.  \n- **(1.2) Defining the tree structure in higher dimensions:**  \n  If \\(d \\geq 3\\), then, in general (with probability 1), randomly selected lines \\((x^1, l^1)\\) and \\((x^2, l^2)\\) do not intersect. In this case, how is the tree structure defined? Additionally, if such a scenario occurs, how is the tree distance in Equation (6) determined?  \n- **(1.3) Understanding Figure 1:**  \n  I find Figure 1 difficult to interpret. The caption states that *\"only four pairs of lines are adjacent.\"* I assume the authors mean pairs \\((1,3), (1,4), (1,5), (2,4)\\). However, intersections also occur for \\((2,3), (2,5), (2,1), (3,4), (3,5), (4,5)\\). Could the authors clarify why these additional intersections on the left are not counted as adjacent nodes in the tree structure?  \n\n## 2. P4, Line 252  \n- The statement *\"We can show \\( R_L^\\alpha f \\in L^1(L) \\)\"* appears inconsistent. However, in Equation (9), the domain of \\( R_L^\\alpha f \\) is \\( \\tilde{L} = (\\mathbb{R}^d \\times L) \\), not \\( L \\). Could the authors clarify this?  \n\n## 3. Experiment 6.2  \n- **(3.1) Missing baselines:**  \n  - I recommend adding \"generalized sliced Wasserstein\" as a baseline in at least one experiment.  \n  - Additionally, I suggest including the \\( L^2 \\) cost for \"SW\" as a baseline. Furthermore, for a fair comparison, always set \\( L(SW) = L(TSW) \\times k(TSW) \\).", "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": "N/A"}}]

Author rebuttal:

Meta-review: This paper proposes a new sliced variant of Wasserstein distance that can be applicable to diffusion models, where the tree sliced Wasserstein distance was not used for diffusion tasks except for sliced Wasserstein distance (a special case of TWD). The approach is new and an interesting approach. However, there are also some concerns and weaknesses (see below).

Two main weaknesses:
1.  Some reviewers requested authors to add the comparison with GSW. The concerns were partly addressed. However, it would be better to include more comparison with GSW. 

2. In general, the tree Wasserstein was proposed by Indyk et al. 2003 with QuadTree and the first version of the paper completely ignores the facts. This is very confusing and not fair to the authors. So, I strongly request authors to give more credits to the authors who invented the Tree Wasserstein distance and add more detailed discussion in related work. My understanding is that Le et al. 2019 rediscovered the tree Wasserstein distance and proposed the tree-slicing method. Thus, Indyk et al. 2003 should be cited. [1] Piotr I., Nitin T. Fast Image Retrieval via Embeddings, 2003.

Overall, this is a nice paper with some merits. For the camera-ready version, I strongly request authors including the comparison with GSW and cite Indyk et al. 2003 as a reference of tree Wasserstein distance.
Decision: Accept (poster)