Distance-Based Tree-Sliced Wasserstein Distance
Authors: Hoang V. Tran
Minh-Khoi Nguyen-Nhat
Huyen Trang Pham
Thanh Chu
Tam Le
Tan Minh Nguyen
Venue: ICLR 2025 Poster (2025)
Keywords: tree-sliced wasserstein distance
optimal transport
equivariance
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We introduce a novel generalization of Tree-Sliced Wasserstein distance on Systems of Lines, enabling the use of all positional information from input measures.

Abstract: To overcome computational challenges of Optimal Transport (OT), several variants of Sliced Wasserstein (SW) has been developed in the literature. These approaches exploit the closed-form expression of the univariate OT by projecting measures onto one-dimensional lines. However, projecting measures onto low-dimensional spaces can lead to a loss of topological information. Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL) has emerged as a promising alternative that replaces these lines with a more intricate structure called tree systems. The tree structures enhance the ability to capture topological information of the metric while preserving computational efficiency. However, at the core of TSW-SL, the splitting maps, which serve as the mechanism for pushing forward measures onto tree systems, focus solely on the position of the measure supports while disregarding the projecting domains. Moreover, the specific splitting map used in TSW-SL leads to a metric that is not invariant under Euclidean transformations, a typically expected property for OT on Euclidean space. In this work, we propose a novel class of splitting maps that generalizes the existing one studied in TSW-SL enabling the use of all positional information from input measures, resulting in a novel Distance-based Tree-Sliced Wasserstein (*Db-TSW*) distance. In addition, we introduce a simple tree sampling process better suited for Db-TSW, leading to an efficient GPU-friendly implementation for tree systems, similar to the original SW. We also provide a comprehensive theoretical analysis of proposed class of splitting maps to verify the injectivity of the corresponding Radon Transform, and demonstrate that Db-TSW is an Euclidean invariant metric. We empirically show that Db-TSW significantly improves accuracy compared to recent SW variants while maintaining low computational cost via a wide range of experiments on gradient flows, image style transfer, and generative models.

Published as a conference paper at ICLR 2025
DISTANCE-BASED TREE-SLICED WASSERSTEIN DISTANCE
Viet-Hoang Tran'
Khoi N.M. Nguyen FPT Software Al Center khoinnml miiinnn1@@@l @fpt com
Department of Mathematics National University of Singapore hoang. traviett @u. nus edu
Trang Pham
Thanh T. Chu
Qualcomm Al Research 9 tranpham& qt qualcomm.......
Department of Computer Science
National University of Singapore
thanh. chu@u. nus edu
Tam Le
Tan M. Nguyen
The Institute of Statistical Mathematics & RIKEN AIP tam@ 1000ism.. ac ip
Department of Mathematics
National University of Singapore
tanmn nus edu.sg
ABSTRACT
To overcome computational challenges of Optimal Transport (OT), several vari- ants of Sliced Wasserstein (SW) has been developed in the literature. These ap- proaches exploit the closed-form expression of the univariate OT by projecting measures onto (one-dimensional) lines. However, projecting measures onto low- dimensional spaces can lead to loss of topological information. Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL) has emerged as a promis- ing alternative that replaces these lines with more advanced structure called tree systems. The tree structures enhance the ability to capture topological information of the metric while preserving computational efficiency. However, at the core of TSW-SL, the splitting maps, which serve as the mechanism for pushing forward measures onto tree systems, focus solely on the position of the measure supports while disregarding the projecting domains. Moreover, the specific splitting map used in TSW-SL leads metric that is not invariant under Euclidean transforma- tions, a typically expected property for OT on Euclidean space. In this work, we propose novel class of splitting maps that generalizes the existing one studied in TSW-SL enabling the use of all positional information from input measures, re- sulting in novel Distance-based Tree-Sliced Wasserstein (Db-TSW) distance. addition, we introduce a simple tree sampling process better suited for Db-TSW, leading to an efficient GPU-friendly implementation for tree systems, similar the original SW. We also provide comprehensive theoretical analysis of pro- posed class of splitting maps verify the injectivity of the corresponding Radon Transform, and demonstrate that Db-TSW is an Euclidean invariant metric. We empirically show that Db-TSW significantly improves accuracy compared to re- cent SW variants while maintaining low computational cost via a wide range of experiments on gradient flows, image style transfer, and generative models. The code is publicly available https: / thub. com /oom/Fac Ftt.FFffffffffeoeeeee Aet.ttAAATT/TSS..
INTRODUCTION
Optimal transport (OT) (Villani, 2008; Peyré et al., 2019) is a framework designed to compare prob- ability distributions by extending the concept of ground cost metric, originally defined between the
Co-first authors. Co-last authors. Qualcomm Vietnam Company Limited. Correspondence to: www.:rrtttainnssss..cmmm & tanmn @ @nus.edu.s

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supports of input measures, to metric between entire probability measures. OT has found utility across a diverse array of fields, including machine learning (Bunne et al., 2022; Hua et al., 2023; Nguyen et al., 2021b; Fan et al., 2022), data valuation (Just al., 2023; Kessler et al., 2025), mul- timodal data analysis (Park et al., 2024; Luong et al., 2024), statistics (Mena & Niles-Weed 2019;
Weed & Berthet, 2019; Wang et al., 2022; Pham al., 2024; Liu et al., 2022; Nguyen et al., 2022;
Nietert et al.. 2022), computer vision, and graphics (Lavenant et al., 2018; Nguyen et al., 2021a; Saleh al., 2022; Solomon et al., 2015).
OT suffers from a computational burden due to its supercubic complexity with respect to the num- ber of supports in the input measures (Peyré et al.. 2019). To alleviate this issue, Sliced Wasserstein (SW) (Rabin et al., 201 I; Bonneel et al., 2015) leverages the closed-form solution of OT in the one- dimensional case to reduce computational demands by projecting the supports of the input measures onto random lines. The vanilla SW distance has been continuously developed and enhanced by re- fining existing components and introducing meaningful additions achieve better performance. Ex- amples include improvements in the sampling process (Nadjahi et al., 2021; Nguyen et al., 2024a), determining optimal projection lines (Deshpande et al., 2019), and modifying the projection mech- anism (Kolouri et al., 2019; Bonet et al., 2023b).
Related work. Relying solely on one-dimensional projections can result in the loss of essen- tial topologiaal structures in high-dimensional data. To address this, an alternative approach has emerged that replaces these one-dimensional lines with different domains, applying to OT on Eu- clidean spaces (Alvarez-Melis al., 2018; Paty & Cuturi, 2019; Niles-Weed & Rigollet, 2022), tree metric spaces (Indyk & Thaper, 2003; Le & Nguyen, 2021; Tran et al., 2025c:d), graph metric spaces (Le et al., 2022: 2023; 2024), spheres (Quellmalz al., 2023; Bonet et al., 2023a; Tran al., 2024b), and hyperbolic spaces (Bonet et al., 2023b). Specifically, Tran et al. (2025d) introduced an integration domain known as the tree system, which functions similarly to lines with a more ad- vanced structure design. This approach is proven to capture the topological information better while maintaining the computational efficiency of the SW method. However, the proposed Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL) in (Tran et al., 2025d), derived from the tree system framework, fails to meet the Euclidean invariance property, which is typically expected for OT in Euclidean spaces (Alvarez-Melis et al., 2019). In this paper, we address these issues by gen- eralizing the class of splitting maps introduced in (Tran et al., 2025d), simultaneously resolving the lack of invariance and the insufficient accounting for positional information. A new class of splitting maps may encounter challenges, such as verifying the injectivity of the associated Radon transform, resulting in pseudo-metric on the space of measures.
Contribution. In summary, our contributions are three-fold:
1. We analyze the Euclidean invariance of 2- Wasserrstein and Sliced p- Wasserstein distance be- tween measures on Euclidean spaces and then discuss why Tree-Sliced Wasserstein distance on Systems of Lines fails to satisfy Euclidean invariance. To address this issue, we introduce a larger class of splitting maps that captures positional information from both points and tree systems, generalizing the previous class while incorporating an additional invariance property.
2. We introduce a novel variant of Radon Transform on Systems of Lines, developing a new class of invariant splitting maps that generalizes the previous version in (Tran al., 2025d). By providing a comprehensive theoretical analysis with rigorous proofs, we demonstrate how our new class of invariant splitting maps ensures the injectivity of the Radon Transform.
3. We propose the novel Distance-based Tree-Sliced Wasserstein (Db-TSW) distance, which is an Euclidean invariant metric between measures. By analyzing the choice of splitting maps and tree systems in Db-TSW, we demonstrate that Db-TSW enables a highly parallelizable implementation, achieving an efficiency similar to that of the original SW.
Section 2 recalls variants of Wasserstein distance. Section 3 outlines the essential background of Tree-Sliced Wasserstein distance on Sys- tems of Lines and discusses Euclidean invariance. Section 4 introduces a new class of splitting maps and discusses the corresponding Radon Transform. Section 5 proposes Distance-basse Tree- Sliced Wasserstein (Db-TSW) distance and discusses the choices of components in Db-TSW. Fi- nally, Section 6 evaluates Db-TSW performance. A complete theoretical framework of DP-TSW and supplemental materials provided in the Appendix.
Organization. The structure of the paper is as follows:
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2 PRELIMINARIES
We review Wasserstein distance, Sliced Wasserstein (SW) distance, Wasserstein distance on metric spaces with tree metries (TW), and Tree-Sliced Wasserstein on Systems of Lines (TSW-SL) distance.
Wasserstein Distance. Let μl, V/ be two probability distributions on R Let P(u. v) be the set of probability distributions л (R" x A) = ({A) for all measurable sets A. For 2 1, the y- g-Wasserstein distance Wp (Villani, 2008) between μr, defined as:
the product space R R4 such that π ( x IR =(AA) and
(F |I vlli dr dx(x,))
W) = inf TEP(MR)
(1)
Sliced Wasserstein Distance. The Sliced p- Wasserstein distance (SW) (Bonneel et al., 2015) be- tween two probability distributions J4, on I&d is defined by:
Vw)(u,v> = Vorn ((( do) do(())
(2)
where a (((--1) is the uniform distribution on the unit sphere sd-1, operator R (R L' (R x 1) is the Radon Transform (Helgason, 2011) defined by Rf(t, Sge f(z) 5(t - (I, 0)) dr, and Sp. fo are the probability density functions of μr, respectively. The one-dimensional p-Wasserttein distance in Equation (2) has the closed-form Wg(0s,, =
So (F (z) oo (2) (z,, where (-)) and FRF.(-0) are the cumulative distribu- tion functions of (. 0) and (.((-,0) respectively. To approximate the intractable integral in Equation (2), Monte Carlo method is used as follows:
T. Σ (B(RIA()0) 5
where 0,,...,0, Or are drawn independently from the uniform distributions on Sd-1, i.e. (($-11).
Tree Wasserstein Distances. Let 7 be rooted tree (as a graph) with non-negative edge lengths, and the ground metric d7, i.e., the length of the unique path between nodes. Given two probability distributions and supported on nodes of T, the Wasserstein distance with ground metric dy (TW) (Le et al., 2019) has closed-form as follows:
War, = Σ we (eT((l((T))) - ((Tc,)))
(4)
EET
where de is the endpoint of edge that is farther away from the tree root, I(ue) is the subtree of T rooted at and w. is the length of e.
Tree-Sliced Wasserstein Distance on Systems of Lines. Tree-Sliced Wasserstein distance on Sys- tems of Lines (TSW-SL) (Tran et al., 2025d) is proposed as a combination between the projecting mechanism via Radon Transform in SW and the closed-form of Wasserstein distance with ground tree metrics in TW. In TSW-SL, tree systems, which are well-defined measure spaces and metric spaces with tree metric, are proposed to replace the role of directions SW, and the corresponding variant of Radon Transform is also presented. Details of TSW-SL are outlined in Section 3.
THE ORIGINAL TSW-SL AND THE LACK OF IINAAIINA
In this section, we outline the details of TSW-SL in (Tran et al., 2025d) and discuss the invari- ance under Fuclidean transformations of TSW-SL and some variants of Wasserstein distance on Euclidean spaces.
3.1 REVIEW ON THE ORIGINAL. TSW-SL
We recall the notion and theoretical results of the Radon Transform on Systems of Lines and the corresponding Tree-Sliced Wasserstein distance from (Tran al., 2025d) with some modifications
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on the notations. We start with function L' (RR). Usually, d is the dimension of data, and is the distribution of data. line in R is an element in I2 x gd and a system of k lines in IR an element of (R x 1d- We denote system of lines by L. a line in L (also index) by I, and the space of all systems of lines by T.g- The ground set of L is defined by:
L = (2,1 XL: x 2ι + tz 8 for an to E ER).
where + 0,, / E R is the parameterization ofl. By abuse of notation, we sometimes index lines by i = 1,.... and denote the line of index by 4. The source and direction are denoted by ay and 8,, respectively. We also denote the collection of all systems of lines in Fd with & lines by Lg- A tree system is a system lines L with an additional tree structure T. It is a well-defined metric measure space and denoted by (C, T) by L if the tree structure is not needed to be specific. A space of trees (i.e., collections all tree systems with the same tree structure) is denoted by T with probability distribution space of Lebesgue integrable functions on L is
T, which comes from the tree sampling process. For L Lg the
{
f: 5:C- R: I 1111==E E (f(tz.)) < {o) IEC
(2(2)
(5)
Given a splitting map a C(R² Ak- which is a continuous map from I2d to the (k 1)- dimensional standard simplex 1 For f E I. (RR)), we define
Res: L
(6)
(x, J pdd f()) a(y): 8 (y -5-, 01)) dy.
The function R&f is in (2). The operator
R" L L'(R")
II L'(2) DELA
f
(R2f)
is called the Radon Transform on Systems of Lines. This operator is injective. The Tree-Sliced Wasserstein Distance on Systems of Lines TSW-SL between u, E P(Rd) defined by
TSW-SL(M,V)
W (REg, RCv) (C). n
(8)
The TSW-SL is metric on P(Rd) Leveraging the closed-form expression of OT problems on metric spaces with tree metrics (Le et al., 2019) and the Monte Carlo method, TSW-SL in Equation (8) can be efficiently approximated by a closed-form expression.
Remark The Radon Transform R depends on the choice of a E Ax-1). Intuitively, the splitting map represents how the mass at a specific point distributed across all lines in system of lines. In the context the original Radon Transform R, only one line involved, so a is simply the constant function 1. As result, R is meaningful and nontrivial generalization of R.
3.2 E(d)-INVARIANCE IN OPTIMAL TRANSPORT ON EUCLIDEAN SPACES
Consider Rdd with the Euclidean norm, i.e. || ||2. we consider some groups with group actions that preserve Euclidean norm and Euclidean distance between two points in Eed, We then discuss E(d)-invariance of some Wasserstein distance variants.
Euclidean group E(d) and its action on Rd. For E Rd, the translation corresponding to is the map Idd - Rd that - + a. The translation group T(d) is the group of every translations in Idd Note that, T(d) is isomorphic to the additive group Rd The orthogonal group ((d) is the group of all linear transformations of Rd that preserve the Euclidean norm
(d) = linear transformation f: Rd Rd ||~|| = [f) for all x E Rd).
(9)
Note that (dd) is isomorphic to the group of all orthogonal matrices
O(d) Q is an d x d real matrix QQQQ" = - 1=-Qi 1a}.
(10)
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6
7
1
9
Figure I: illustration highlighting the distinction between the old and new Radon Transform on Systems of Lines, specifically focusing on two different definitions of splitting maps. Left: The old splitting map relies solely on the location of points, leading to the same distribution and independent of the position of the line systems. Right: The new splitting map considers the configuration of systems of lines, leading to varied mass distributions depending on each system.
The Euclidean group E(d) is the group all transformations of R that preserve the Euclidean distance between any two points. Formally, E(d) is the semidirect product between T(d) and Ocd), ie., E(d) = T(d) x O(d). An element g of E(d) denoted by pair g = (Q,a), where a Rd y
and Q (dd) The action of IR is y gy
Group actions of E(d) on L and T. The canonical group action of E(d) on Rd naturally induces a group action on the set of all lines Rd, i.c., Rd x gd-1, Given a line / = (ax, 0) Rd x S and y (Q, a) E E(d), we define
gl = -Q (Q.x Q 0) E R x Sd-1
(11)
For L 144 (2,, 0.) E Lgg. the action of E(d) on L is similarly defined
- [gl (Q x, + a. a. Q 0,)) L.
(12)
In (Tran al., 2025d), a tree system system of lines with an additional tree structure, and by design, this tree structure is preserved under the action of E(d). Thus, if L E T is a tree system, then gL also a tree system. The group action of E(d) on T. induces group action of F(d) on T.
E(d) -qqivvariee in Optimal Transport. E(d) -nvaarinee is natural in the context of Optimal Transport on Euclidean space Rd since the distance between two measures should remain unchanged when a distance-preserving transformation is applied to the underlying space. In details, let μ E P(R") be a measure on Rd, For E E(d), denote the pushforward of ju via g: IRd Rdd by gip. canonically defines group action of E(d) P(R"). We have the following result.
Proposition 3.1. The 2- 2-Wasserstein distance and the Sliced p- Wasseesteein distance are E(d)- invariant. In other words, for every μ,v P(R")) and g E(d). have
W2(u, W2(gtu, giv) and v)) SW, SWp(gtp.gth)
(13)
Moreover, for Sliced Wasserstein distance, we have the below result for each projecting direction. Proposition 3.2. Let g (Q,a) E(d). For every μ. v E P(R") and 0 sd-1, have
W Rful(.)) W, RR Q0), R. (,
(14)
The proofs for Proposition 3.1 and Proposition 3.2 can be found in Appendix B.
Remark With the setting of TSW-SL, the E(d)-invariance and the similar property as Proposition 3.2 can not be derived for general splitting map. Assumptions on invariance of splitting maps will be made achieve invariance of TS
RADON TRANSFORM WITH E(d)-INARRIANT SPLITTING MAPS
In (Tran et al., 2025d), the splitting maps are selected as continuous maps from Id to Ag-1, which means that it depends solely the position of points and ignores the tree systems. It is intuitively
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9.
1
66
3
1
.6
h:
Figure 2: An illustration demonstrating E(d) -ivarinnee of splitting maps. Starting with a point and system of lines, two Euclidean transformations applied, resulting in two additional pairs of points and systems of lines. An E(d)-invariant handles all three pairs identically, leading to the same mass distribution from the point to lines within each system.
better for splitting maps to account positional information from both of points and tree systems, than only from points. With this motivation, we introduce larger class of splitting maps with the corresponding Radon Transform and then discuss the injectivity of this Radon Transform variant.
4.1 RADON TRANSFORM ON SYSTEMS OF LINES
Denote C(R² x LLg 1 the space of continuous maps from I2 x LA to A and refer to maps in C(Rd x I. At-1) as splitting mups. Here, we use the same name as in (Tran et al., 2025d) since the new class of splitting maps contains the class of splitting maps in (Tran al., 2025d).
C(R. 1K-1) (ar C(R" Ax-1) a(x,f)
(') for all x E Rhd and L, E L.).
Let L associated to that transforms Lehesgue integrable functions Rd to Lehesgue integrable functions on L. For f E L (IR.), define
be system of lines in Lá and be a splitting map in C(R x IL As-1), we define operator
RBf= E
R
(15)
[ f(v) a00, C), 5 (tz - (y In, 01)) dy.
(x,1)
(16)
where 8 is the 1-dimensional Dirac delta function. We have R2f E L' ('(C) for f L (Rd) and moreover |R2/llc < || In other words, the linear operator R2 L (RR) - (C) is well- defined. The proof of these properties can be found in Appendix B.2. We introduce novel Radon Transform on Systems of Lines, which is a generalization of the variant in (Tran et al., 2025d).
Definition 4.1 (Radon Transform on Systems of Lines). For a E C(R x LLg. Ax-1) 1). the operator
Rs L (IR))
II L'(C)
(17)
LET
(R2f)cceL)
(18)
called the Radon Transform Systems of Lines.
Remark 3. We use the same name and notation for R as in (Tran et al., 2025d). Figure 1 emphasizes the difference between the old and new Radon Transform on Systems of Lines.
The injectivity R.a will discussed the next part. Surprisingly, the F(d))invariance of a is sufficient condition for the injectivity of R
4.2 B(d)-INVARIANCE AND INJECTIVITY OF RADON TRANSFORM
Variants of Radon Transform usually require the transform to be injective. In the case R since introduce larger class of splitting maps a, the proof for injectivity of the previous variant in

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(Tran et al., 2025d) is not applicable for this new variant. We found that if is E(d)- then the injectivity of the Radon transform holds. We first have the definition of E(d) ((d))innvaiiane of splitting maps.
Definition 4.2. A splitting map a in C(R x Lg. A 1) is said to be E(d)-invariant, if we have
algv.gL) aly, L)
(19)
for all () E RR Lg and g E(d).
A visualization of E(d)-invariant splitting maps is presented in Figure 2. Using this property of splitting maps, we get result about injectivity of our Radon Transform variant.
Theorem 4.3. For an E(d)- E(d)-invariant splitting map E C(TR* x L. Ak-1). R2 is injective.
The proof Theorem 4.3 is presented in Appendix B.3.
Remark 4. Since the action of E(d) on Rd is transitive, i.e. for x. y Rdd there exists g B(d) such that gz = so F((d)-invarian splitting maps a E C(TRé, Ax-1) must be constant. Thus, imposing invariance on previous splitting maps (Tran al., 2025d) significantly limits the class of maps.
Candidates for B(d))invarian splitting maps will be presented in the next section.
DISTANCE-BASHR TREE-SLICED WASSERSTEIN DISTANCE
In this section, we present novel Distance-basee based Free-Sliced Wasserstein (Db-TSW) distance. Let consider the space of all tree systems of lines T and a distribution a on T.
5.1 TREE-SLICED WASSERSTEIN DISTANCE WITH E(d) SPLITTING MAP
For μı, D E PR tree system T. and an -invariatr splitting map a E C(R Ak-1), by the Radon transform R2 in Definition 4.1, probability measures μl and v are transformed to 2 and R2 in P(C). Notice that L is a metric space with tree metrie de (Tran et al., 2025d), so we can compute Wasserstein distance Wa,,1 1 R2²) between R2 and R2 by Equation (4).
Definition 5.1 (Distance-Based Tree-Sliced Wasserstein Distance). The Distance Distance-based Tree-Sliced Wasserstein distance between μ1, P(R.)) is defined by
Db-TSW v) = J/- Wa (REH, R2v) (C).
(20)
Remark 5. Note that, Db-TSW depends on the space of tree systems T, the distribution a over T, and the E(d)-ine invarrint splitting map a. For simplicity, these are omitted from the notation. The choice of a will be discussed in the next part and is the reason for the name Distance-based.
The Monte Carlo method is utilized to estimate the intractable integral in Equation (20) as follows:
Db-TSW( v) = Σ Wae, (R v),
(21)
iml
where Lı,... LI,..., Lr. are drawn independently from the distribution T. The Db-TSW distance is, indeed, a metric on P(Rd)). Moreover, Db-TSW is ((d)-iivaria
Theorem 5.2. Db-TSW is an E(d)-invariant metric on P(Rd).
The proof of Theorem 5.2 is presented in Appendix B.4. We discuss the computation of Db-TSW in details in the next part.
5.2 COMPUTING DB-TSW
We construct the space of tree systems and E(d)-invarian splitting maps that suited for Db-TSW. Choices for the space tree systems. (Tran et al., 2025d), the space of tree systems used in computing Db-TSW is the collection of all tree systems with the tree structure is a chain of

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nodes graph. We discovered that this approach complicates the implementation of D resulting in a considerable increase in computation time. Here, we propose a method for sampling free systems that simplifies the implementation in practice. Let T be the space of all tree systems, which consists lines with the same source. In details, C in TT can be presented as
{ (1. 01), (z, 02). ... (x. 0x)]-
(22)
In other words, tree system L consists of l: concurrent lines with the same root. The distribution o on T is simply the joint distribution of k + 1 independent distributions, which are μ E P(RR)) and 12. Me E P((S-11). In details, to sample a tree system L E T, we
1. Sample x ,4, where μ is a distribution on bounded subset of Fd for instance, the uniform distribution on the d-dimensional hypercube [-1, i.e. 2[[[11, 1] and,
2. For i = l....,k k, sample 0g ~ p4i where μe is distribution on gd-1, for instance, the uniform distribution U (Sd-1).
We also concentrate on subcollection of T, which includes tree systems that take into account the angles between pairs of lines. Assume that k & d, denote T as the space of tree systems consists of k concurrent lines with the same root, and these lines are mutually orthogonal, i.e. L = {(x, ), (xr ........................... (z, 0x)} where (0.,
for all 1 & i < < k. The intuitive motivation for this choice to ensure that the sampled tree systems do not include lines with similar directions. Remark 6. The condition & d arises because (0,,...,0 0z)
forms subset of orthogonal basis of TR.. In practice, this has little impact, as the dimension d is typically large, while the number of directions k is selected to be small.
Choices for E(d)) Eivarinn splitting maps. Since the action of E(d) preserves the distance between two points in R. it also preserves the distance between a point and line in R For E R and Lg we denote the distance between and line of C as d(x, L), given by:
dæ,, L)r = inf || - v/12-
(23)
yEl
We have d(x,,4) is E(d)-invariant. As a result, a splitting map a: Rd - Ax-1 that is in the form
a(x, L)2 = DWhec).
(24)
where 3 is an arbitrary map R - 1 1, is E(d)-invariant. In practice, we empirically observed that choosing 3 to be the softmax function together with scaling by scalar works well in applica- tions. In details, we choose a as follows
a(x, f), softmax (16 d(z, Ch)eec)
(25)
Here, d IR is considered as be tuning parameter.' Intuition behind this choice of is that reflects the proximity of points to lines in tree systems. As Jo] grows, the output of a tends to become more sparse, emphasizing the importance of each line in the tree system relative to a specific point. We summarize the computation of Db-TSW by Algorithm 1.
Remark 7. The above construction of mainly bases the distance between points and lines in tree systems. That is the reason for the name Distance-based Tree-Sliced Wasserstein.
EXPERIMENTAL RESULTS
In this section, we experimentally demonstrate the advantages of our Db-TSW methods over tra- ditional SW distance and its variants across three key domains: unconditional image synthesis, gradient flows, and color transfer. Detailed experimental settings are provided Appendix $C. Our evaluation aims to establish that: (i) Db-TSW and DP-TSW-2 consistently enhance perfor- mance across various generative and optimization tasks; (ii) D--TSW- significantly outperforms not only baselines but also Db-TSW in all tasks, highlighting its superiority; and (iii) Db-TSW and DS-TSW- is universal applicable and can be seamlessly integrated into any Optimal Transport task.
We abuse the notation 8 E R for the splitting map function a.
2.1. stands for using orthogona directions 0.
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Table Results of different DDGAN variants for unconditional generation on CIFAR-I0.
Model
FID
Time/Epoch(s) L
DDGAN (Xiao et (2022))
3.64
136
SW-DD (Nguyen et al. (2024b))
2.90
DSW-DD (Nguyen
(2024b))
2.88
1059
EBSW-DD (Nguyen al. (2024b))
2.87
145
RPSW-DD (Nguyen et al. (2024b))
2.82
159
IWRPSW-DE DD (Nguyen
(2024b))
2.70
152
TSW-SL-DD (Tran et al. (2025d))
2.83
163
Db-TSW-DD (Ours)
2.60
160
Db-TSW-DD- (Ours)
2.525
162
We maintain consistent total number of sampled lines across all methods to ensure fair compari- son. Db-TSW and Db-TSW- offer two important advantages. First, these methods contain richer information compared to SW for the same number of data points, they incorporate positional information from both points and tree systems. This allows for more effective sampling, specifi- cally by sampling trees located around the support of target distributions (ideally near their means). Second, the root-concurrent tree system design enables an algorithm with linear runtime and high parallelizability, keeping the wall clock time of Db-TSW and Db-TSW- approximately equal to vanilla SW and surpassing some SW variants.
6.1 DIFFUSION MODELS
This experiment explores the efficacy of denoising diffusion models for unconditional image syn- thesis. We employ a variant of the Denoising Diffusion Generative Adversarial Network (DDGAN) (Xiao et al., 2022), as introduced by Nguyen et al. (2024b), which incorporates a Wasserstein dis- tance within the Augmented Generalized Mini-batch Energy (AGME) loss function. A detailed explanation of this Optimal Transport-basse DDGAN is provided in Appendix SC.2. We evaluate our proposed methods, Db-TSW-DD and Db-TSW-DD+, against DDGAN and various OT-based DDGAN variants, enumerated in Table 1. All models undergo training for 1800 epochs on the CIFAR1O dataset (Krizhevsky et al., 2009). For vanilla SW and its variants, we adopt the parameters from Nguyen et al. (2024b) and set L 10000. For Db-TSW-DD set T, = 2500, k = 4, = 10. Tree sampling is conducted from a Gaussian distribution ((m, I), where m, represents the mean of all training samples and 0.1. Table presents the Fréchet In- ception Distance (FID) scores and per-epoch training times on an Nvidia V100 GPU for our methods and the baselines. Lower FID scores indicate superior model performance. The results demonstrate that both DD-TSW-DD- and Db-TSW-DD achieve notable improvements in FID compared to all baselines. Specifically, they outperform the current state-of-the-art art OT-based DDGAN, IWRPSW- DD (Nguyen et al.. 2024b), by margins of 0.175 and 0.1, respectively. Importantly, our methods maintain computational efficiency, with only a modest 6.5% increase training time compared to the current state-of-the-an
and Db-TSW-DD DD models,
6.2 GRADIENT FLOWS
The gradient flow task aims to minimize the distance between source and target distributions using gradient descent process. The objective is to iteratively reduce this distance by optimizing the equa- tion 0jf4 = VD(H,, v/) with the initial condition Pv NO 1), where μ4 represents the source distribution at time - denotes the change in the source distribution over time, and VDD v) is the gradient of D with respect f4. Here, 'D refers to our proposed distance metric and the SW baselines. We evaluate our approach Db-TSW-- Db-TSW and several established techniques, including vanilla SW Bonneel et al. (2015), MaxSW (Deshpande et al., 2019), LCVSW (Nguyen & Ho, 2024), SWGG (Mahey et al., 2023), and TSW-SL (Tran et al., 2025d), using the Swiss Roll and Gaussian 20d datasets.
To assess the effectiveness of these methods, we employ the Wasserstein distance to quantify average Wasserstein distances between the source and target distributions across 10 runs at iterations 500, 1000, 1500, 2000 and 2500. The results demonstrated in Table 2 show Db-TSW and Db-TSW record the best result almost all steps in Swiss Roll dataset, with 3.78e-8 W2 the last step

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Published as a conference paper at ICLR 2025
Table 2:
Average Wasserstein distance between source and target distributions of 10 runs on Swiss Roll and Gaussian 20d datasets. All methods use 100 projecting directions.
Swiss Roll
Glussian 20d
Iteration
Time/ten(s))
Iteration
Time/Iter(s)
500
1000
1500
2000
2500
500
1000
1500
2001
2500
SW
3.73e.3
2.04e.3
3.23e.3
1.1le.3
1,05e-3 3
0.009
18.24
16.48
14.66
12.60
10.30
0.006
MaxSW
2.47e22
1.03e-2
6.10s.3
4.476-3
3,45e-3
2.46
13:24
13:71
13.46
11.83
2.38
SWGG
3.84c-2
1.53e-2
1.02e-2
4.49e.3
3.57e-5
0,011
8.51
8.71
7.71
8.48
6.45
0.009
LCVSW
7.28e:3
1.40c-3
1.38c3
1.38c-3
1.360-3
0.010
17.26
14.70
12.00
6.15
0.009
TSW-SL
9.1ee33
2.03e.2
9.63e.8
4.44c.8
3.65..8
0.014
3.13
9.67e-3
6.8ee33
6.15e-3
5.71e-3
0.010
D5-TSW
5,47633
8.04-88
5,29--8
3.92x-8
3.01e-8
0.006
3.27
1,126-2
7.21e-3
5.63e-3
4,60e-3
0.006
Db-TSW
7.55e-3
2.65e-7
4.90e-8
4.18c-8
3.78e-8
0.009
2.46
7.96e33
6.11e-3
5.22e-3
5.03e.3
0.009
in comparison with 1.05e-3 of vanilla SW and 3.57e-5 of SWGG. A similar trend is observed in the Gaussian 20d dataset, with Db-TSW- and Db-TSW consistently achieving the second best and best Wasserstein distances respectively, outperforming the next best method SWGG by three orders of magnitude (5.03e-3 and 4.60e-3 vs. 7.34) and vanilla SW by nearly four orders of magnitude (10.30) at the final iteration. Notably, Db-TSW and DD-TSW- maintain competitive computational efficiency, with runtimes equal to good LCVSW variant (0.009 seconds per iteration) while boosting the performance significantly.
6.3 COLOR TRANSFER
The color transfer task involves transferring color from a source image to a target image. Considering source and target color palettes as X and Y. we define the curve (t) = -nVz(o|D(Pz) Py]] where Px and Py empirical distributions over X and Y and D are customizable Wasserstein distance. Here, the curve starts from Z(0) X and ends at Y. We iter- ate along this curve to perform the transfer between the probability distributions Px and Py. We evaluate our method against SW, TSW-SI, MaxSW, and various Quasi-Sliced Wasserstein (QSW) variants (Nguyen et al., 2024a). We follow the settings used in (Nguyen et al., 2024a) for experi- ment settings, detailed in Appendix $C.4. We set L. = 33, k = 3 for Db-TSW and Db-TSW- and L = 100 for all baselines.
In Figure 5 (Appendix), we compare the Wasserstein distances and visualizations of various color transfer methods. Notably, Db-TSW- and Db-TSW achieve near-best performance (0.12 and 0.21 respectively) without randomization overhead of top-performing R-methods like RRQDSW and RQDSW (both at 0.08) and significantly outperforming traditional approaches such as SW and MaxSW (both at 9.58).
CONCLUSION
The paper presents the Distance-based Tree-Sliced Wasserstein (Db-TSW) distance for comparing measures in Euclidean spaces. Built on the Tree-Sliced Wasserstein distance on Systems of Lines (TSW-SL) framework, Db-TSW enhances the class of splitting maps that are central to TSW-SL. This new class of splitting maps in Db-TSW resolves several challenges in TSW-SL, such as the lack of Euclidean invariance, insufficient consideration of positional information, and slower com- putational performance. To address the theoretical gaps regarding these new maps, provide comprehensive theoretical framework with rigorous proofs for key propertics of Db-TSW. Our ex- periments show that Db-TSW outperforms recent Sliced Wasserstein (SW) variants wide range of task, such as gradient flows, or generative models as GAN or diffusion models. In comparison to these SW variants, Db-TSW primarily focuses on refining the integration domain using tree systems and the associated measure projection mechanism through splitting maps, which are not existed in SW framework. Therefore, adapting to Db-TSW several techniques used in proving the SW variant is expected to be beneficial for future research. Additionally, the System of Lines for TSW provides advanced geometric structure to go beyond the lines for SW, which is promising research direction for further investigation, e.g., the concurrent TSW-SL on the sphere (Tran et al., 2025b).
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ACKNOWLEDGMENTS
This research project is supported by the National Research Foundation Singapore under the AI Singapore Programme (AISG Award No: ALSG2-TC-20233012--5000
This research project is supported by the Ministry of Education, Singapore, under the Academic Research Fund Tier This research / project is also supported by the
(FY2023) (A-8002040-00-00, A 1-8002039-00-000 NUS Presidential Young Professorship Award (A-0009807-01-00) and the NUS Artificial Intelli- gence Institute-Seed Funding (A-8003062-000-0))
We thank the area chairs, anonymous reviewers for their comments. TL. acknowledges the support of JSPS KAKENHI Grant number 23K1 1243, and Mitsui Knowledge Industry Co., Ltd. grant.
Ethics Statement. Given the nature of the work, we do not foresee any negative societal and ethical impacts of our work.
Reproducibility Statement. Source codes for our experiments are provided in the supplementary materials of the paper. The details of our experimental settings and computational infrastructure given Section 6 and the Appendix. All datasets that we used the paper are published, and they easy to access in the Internet.
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NOTATION
R
d-dimensional Euclidean space Euclidean norm
12
standard dot product
Sd
(d - 1) dimmennion hypersphere
0
unit vector
disjoint union
(X)
space of Lebesgue integrable functions on X
P(X)
space of probability distributions on X
p,
measures
8())
I-dimensional Dirac delta function
U($
uniform distribution on gd-1
1
pushforward (measure)
Y(X,Y)
space of continuous maps from X to Y metric metric space translation group order d orthogonal group of order Euclidean group of order d element of group p- Wasserstein distance
d(.-))
T(d)
O(d)
E(d)
g
W,
SW, p
Sliced Wasserstein distance
r
(rooted) subtree
e
edge graph
We
weight of edge in graph line, index of line system of lines, tree system ground set of system of lines, tree system topological space of system of lines space of symtems of lines in Rd tree structure in system of lines number tree systems number lines system of lines or tree system original Radon Transform
1
C
E
L
T
R
Ra
Radon Transform Systems of Lines
A
(k: 1)-dimensional standard simplex splitting map
a
8
tuning parameter in splitting maps space of tree systems
T
T
space of orthogonal tree systems distribution on space of tree systems
g
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Supplement for "Distance-Based Tree-Sliced Wasserstein Distance"
Table of Contents
A Background for Tree-Sliced Wasserstein distance on Systems of Lines
16
A.I Tree System
16
A.2 Sampling Process of Tree Systems
17
A.3 Radon Transform on Systems of Lines
17
A.4 Tree-Sliced Wasserstein Distance on Systems of Lines (TSW-SL)
17
B Theoretical Proofs
18
B.1 Proof of Proposition 3.1 and Proposition 3.2
18
B.2 Rg. is integrable
19
B.3 Proof of Theorem 4.3
20
B.4 Proof of Theorem 5.2
23
C Experimental details
24
C.1 Algorithm of Db-TSW
24
C.2 Denoising Diffusion Models
25
C,3 Gradient Flow
26
C.4 Color Transfer
27
C.5 Computational infrastructure
28
C.6 Computation and memory complexity of Db-TSW
28
C.7 Runtime and memory analysis
29
C.8 Ablation study for L., h and
29
A BACKGROUND FOR TREE-SLICED WASSERSTEIN DISTANCE ON SYSTEMS OF LINES
This section provides background for Tree-Sliced Wasserstein distance on Systems of Lines. For completeness, we recall essential definitions and theoretical results. Proofs and further details can be found in (Tran et al., 2025d).
A.1 TREE SYSTEM
line in Rd is determined by a pair (z, 0) E Idd x gd-1 and is parameterized as 3. + I. 0, where t TR. A line in Rd is denoted indexed by (az, 0:) E Rd x $d-1 Here, 31 and 0₁ referred to as the source and direction of I, respectively. For k > 1, a system of k. lines in Rd is set of lines. We denote (Rd Sd-1 by Lİ which represents the space of systems of k lines in Idd and an element of L is typically denoted by L. The system L is said be connected if the points on the lines form connected subset of Rd.. By removing tree system L, where there is unique path between any two points of L. For a quick visualization of tree systems, kindly refer Figure 3.
intersections between lines, obtain
Remark The term tree system is used because there is a unique path between any two points, analogous the definition of trees in graph theory.
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By identifying all remaining intersections, together with the notions of disjoint union topology and quotient topology (Hatcher, 2005), we obtain a topology on a tree system as the gluing copies of R. Analyzing this topology. we find that tree systems metric spaces with tree metric.
A.2 SAMPLING PROCESS OF TREE SYSTEMS
The space of tree systems exhibits significant diversity due to the various possible choices for tree structures (as graphs). (Tran et al., 2025d) outlines comprehensive approach applicable to general tree structures, and focuses on the implementation of chain-like tree structures. The process of sampling tree systems based on this chain-like structure is as follows:
Step 1.
Sample x1 ~ and 01 ~ 01 for E P((S-1) and E P(RR).
Step Sample x; 24-1 + 1. 0,-1 where 4 ~ μ4 and 0; ~ is for μi E P(R) and is E P(S
We assume all j's and v's are independent, and let:
N1 to be a distribution on a bounded subset Rd, for instance, the uniform distribution on the d-dimensional cube [-1, 1| i.e. [1[[1,, 1]
2. for i > 1 to be a distribution on a bounded subset of R, for example, the uniform distribution on the interval [-1, 1], i.e. [([[-,,1])
3. 0n to be a distribution gd-1, for example, the uniform distribution (($d11)).
We derive distribution, denoted by a, over the space of all tree systems that can sampled in this way, denoted by T. The tree system shown in Figure an example of tree system with chain-like structure.
A.3 RADON TRANSFORM ON SYSTEMS OF LINES
Denote L (IRd) as the space of Lebesgue integrable functions on Rd with norm -1 Let L be a system of lines. A Lebesgue integrable function on L function f: L IR such that:
111
llfllc
Σ
(f(te dt,, TEC.
(26)
Denote L' (C) as the space of Lebesgue integrable functions on L.
Denote C(Rd A 1) as the space of continuous maps from Rd the (k - -)-dimensional standard simplex A {Ax)te a > 0 and Σiec = 1] C R A map in C(TR A is called splitting map. Let C Lg C(Rd A+-1), we define linear operator R2 that transforms function in L (12") to function in (C). For f L (RR),, detine:
Ref:
C
R2
(x, (1,1)~ J fly) a(y): (to (y xι, 0()) dy
(27)
The (old) Radon Transform for Systems of Lines RD in (Tran et al., 2025d) is defined as follows:
R": L (RR)
II (C)
CELŞ
(RE
As shown in (Tran et al., 2025d), R is injective for all splitting maps E C(Rd, Ab-1).
A.4 TREE-SLICED WASSERSTEIN DISTANCE ON SYSTEMS OF LINES (TSW-SL)
Given the space of tree systems T. distribution a on T. E C(Rd, A . the Tree-Sliced Wasser- stein distance Systems of Lines (Tran et al., 2025d) between in P(Rd) is defined by
TSW-SL(U,u)- / W We (R2 R2v) do (L(.
(28)
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Figure 3: An illustration of constructing tree system: Starting with bunch of lines with structure (left), we consider the intersections of all pairs of lines (middle), then removing some of the intersections obtain tree system (Right). There exists unique path between any two points, since we only allows the pass through the remained intersections.
TSW-SL is indeed a metric on P(R)). The proof (Tran et al., 2025d) primarily relies on the injectivity of the (old) Radon Transform Systems of Lines R
B THEORETICAL PROOFS
B. PROOF OF PROPOSITION 3.1 AND PROPOSITION 3.2
Invariance and Equivariance Properties in Machine Learning. Equivariant networks (Cohen & Welling, 2016) enhance generalization and improve sample efficiency by embedding task symme- tries into the model architecture. They have shown considerable success in a range of domains such as trajectory prediction (Walters et al., 2020), robotics (Simeonov et al., 2022), graph- bash models (Satorras et al., 2021; Tran et al., 2024a), functional networks (Tran et al., 2025a; 2024c; et al., 2024), etc. Utilizing equivariance has been found boost performance, increase data efficiency, and enhance robustness against out-of-domain generalization.
Proof. For Wasserstein distance, recall that, for μl, v E P(RR),, we have
(Feee ||00 v12 (I, u)
W2(u, v) = inf xEP(uN)
(29)
We need show that: for g E E(d),
W2(u, = (o(etH,gE));
(30)
For (Q,a), a). the map : Rd - Rd is affine map that preserves Euclidean norm det(Q) = By applying change variable, we have
||2 and
Ueer xe || -v122 dz
inf We(gin, xEP(oti.
(31)
inf xEP(u.) U 1|0- w1l2 d(g
(32)
Upr 11x0 v1l2 dr t,,, g g-
inf =EP(ux)
(33)
inf =EP(UN) ( llge gull dr dx (x. v)
$
(34)
( || vlla dr v)
inf xEP(HN)
(35)
=W2(uV).
(36)
We finish the proof for 2-Wasserstein distance. For Sliced passssesttiin distance, first show that, for g (Q,a) E E(d), u, ((IRd) and E Sd-1
Wp (Rf((.0) 0), (),(((()) Wp (Rfoau(: Rf)
(37)
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Indeed, we have
Rt,uult, Q0) = | (z) 8(t - (a,, Q0)) dr
(38)
Ja Fulg (z) -8(t (x) Q8)) dr
(39)
fu(a) 8(t (gx, Q0)) dr
(40)
Fu(z) 8(t (Qz + a, Q8)) dz
(41)
(u(T) 6(1 (ar + Q a, 0)) dr
(42)
JRd fu(x) 6(t - (Q-'a,,
((t-(Q- aa 0)
(x, 0}) dx
(43)
Rfy(tt - (0- a, 0).
(44)
Using the closed- of one dimensional Wasserstein distance, we have
wp Rootull, Q0), R. R. (,,00)
Wr(Rfal-- (. Rf: 01)
(45)
w, (.((RA((0)
(46)
where leverage the translation invariant properties of the ground cost nnrmorm of Wp for the above second row.
So Equation (37) holds. For the rest of the proof, we show that
SW swp(u,v) SW (gtpu.
(47)
Indeed, we have
swplstu,gtu) = (Lo. WB(Rfatu(s:)) R. do 1
(48)
(Lr (K do (QF)
(49)
(fr ( do(
(50)
(K( (())((()())))))))))))
(51)
(52)
We finish the proof.
B.2 R&J INTEGRABLE
Proof. Let f E L' (RR.). We show that || & [1f Indeed,
|R2fllc= Σ
R2f(ta, dt,
(53)
IEC
Σ
f(y) aly, C): 8 (tx (y xx, 01) dy de
(54)
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/ (LL
Σ
((w)l-auu (12 - (v - xr, 01)) dy
dlx
(55)
LEL
Σl
(f11()) alv, C)r (tz (v - Iı, 01)) dtx
dy
(56)
EER Ba
Σ/
[f()] aly. C)i 0tt (y - Iz, 02) dt
dy
(57)
IEC
Σ IEC
|s(W) aly: L)i dy
(58)
[ ((()
Ea(v: L)x dy
(59)
IEC
[5()))
dy
(60)
JBd
= ||5'|| <0.
(61)
So, we have R2f E (C). It implies the operator R2: L' (R())) - L' L'( is well-defined.
Clearly, R2 is linear operator.
B.3 PROOF OF THEOREM 4.3
As we nontrivially expand the class of splitting maps from C(R AR- to C(IR L AR-1), the proof presented (Tran et al., 2025d) is not applicable to this larger class. By introducing an invariance condition the splitting maps, we provide new proof for the injectivity of R as follows.
Proof. Recall the notion of the original Radon Transform. Let R ('Rdd) L'(R s be the operator defined by: For f E L (RR), we have
Rf 0) S f(y) 5(t 6)) dy
(62)
well-known that the Radon Transform R a linear bijection (Helgason, 2011) and its inverse R is defined as
RR)=)--(R[(...
(63)
= =/c- (R]((ss, 0) n) ((r,, 0)) d0,
(64)
where the convolution kernel satisfies that its Fourier transform f(w)) = Ww/. Back to the problem. Recall that L is the collection of all systems of k lines in Rdd Sd-1
L = { c [1,,, 0,), : a, E Rdd 0, E sg = (IRd x zd-1)²
(65)
For an index i such that < i < k and a direction 0 E 8d-1 define
L 0) = { c = (1,,,
(Rd x sd-1)) 0, = o).
(66)
In words, (i, 0) is a subcollection of L consists of all systems of k lines with the direction of the 1th line is equal to 6. It is clear that g is the disjoint union of all Li 0) for d E gd-1
L
(Z, 0).
(67)
0ES
We have some observations these subcollections.
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Result 1. Each 9 (Q.a) E(d), define bijection between (i, 8) and Lg(i, Q 8). More precisely, for the map Øg. defined by
Φg
I 8)
(1, Q-0)
(68)
C= [T 03) - gC [Q 2j + 0,)
(69)
it is well-defined and is a bijection. This can be directly verified by definitions.
Result 2. For all < k. y. E Rd and 0, E Sd-1 we have
legao aly, L)s dc = SugciOn alv', L); dc.
-
(70)
Note that, the integrations taken over Lg 0) and [i 6*) with measures induced from the measure of Li- We show that Equation (70) holds. Note that, for 0, o' e S there exists an orthogonal transformation Q E O(d) such that Q-O = = Let = QQ- and g = (Q, a) E E(d).
By this definition, have
gy = Q. y Q y + Q. v'-
(71)
From Result 1, we have a corresponding bijection og from [i, 0) (i, We have
SoU alv', .), dc
[ Jugao , d(gL)
(change of variables)
(72)
Jugaann alwv,9L); d(gf)
(since s' =gy)
(73)
S Ql,, d(gL)
(since a E(d))invariian)
(74)
h (gga f), dc
(since det(()) = 1)
(75)
We finish the proof for Result 2.
Result 3. From Result 2, for all < k. we can define constant e; such that
4 /
aly, C)2 dC.
(76)
l1((1,0)
for all y E Rd and E gd-1 Then
Ci +
+ 1.
(77)
In particular, there exists 1 & & k such that is non-zero. To show this, first, recall that L is the disjoint union of all (i, 0) for 0 e gd-1
L
(1,
(78)
BE8
so we have
1 aly, aly, C), C)x dc - - (ASSNO) alv, dc) dB
(79)
f
ei d0
(80)
C₁
(81)
Then
C1 + 02 + ... Ch Σ
a)y, dC
(82)
j=1 l
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k
Σ
aly, alus L)s j=l
de
(83)
1 dc
(84)
1.
(85)
We finish the proof for Result 3. Consider a splitting map a in C( x Li Ax-1) 1) that is E(d)-invariant. By Result 3, let 1 < i < is the index that
4: = = alv, C); dL + O.
(86)
Now, for a system of lines C in Lgg, we denote the hh line by lov and its source by Tza- For a
function f E L' (IR),, define a function E ('R x gd-1) as follows
g : Rx Exga-1
R
(87)
(1,0)
RBS(L (F 0) (2x)) dc
(88)
2L((1))
From the definition of R2J,
R2f: L - R
(89)
(x,1) - Jsc fly) aly, C)2 (tz - (v Iı, 01) dy,
(90)
have
g(t,0) - Isga. (t- (cxx) dC
(91)
Sagaes (Fe (ou)-alfe aly, L); 5 (t (1 0) (y - x2 07) dy
dC
(92)
- -haacn, (po
aay, aly, C): 8 (t (1 + y xe 03) du dc
(93)
-hegao Spe (lo) aly, L); 8 (t - (v. 0)) dy df
(94)
- -/or ( f(1) aly, L); - 8 (t (v, dl dy
(95)
J ((v) 8. (()))((-(
(y, (LESCO alv, de dy
(96)
c. / f(0) 5(t ()) dy
(97)
= c. Rt((t
(98)
Let f E Ker RD, which means R2f = 0 for all L E L. So g 0 E L'(RR x sd-1), and since # 0, it implies Rf = 0 E L (R x gd-1). Additionally, recall that the Radon Transform R. is bijection, we conclude that f = 0 E (.R)).).
Hence, Ra is injective. The proof is completed.
Remark 9. A formal proof would require Haar measure theory for compact groups, but we simplify this for brevity and relevance to the scope of the paper.
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Remark 10. The injectivity still hold if we restrict Lg to non-empty subset of Lg that is closed under action of E(d). In concrete, let A be non-empty subset of LL satisfies that gf E A for all g E E(d) and L A. Let f E (IRd)) such that R 0 for all L A. Using the same argument, we can demonstrate that f - O. In particular, for T and T- are introduced in Subsection 5.2, since both TT and T are closed under action of E(d), we see that a function f E I.- (TB()) is equal to 0, R2 0 for all L E T, for all C T-.
B.4 PROOF OF THEOREM 5.2
Proof. We show that
(1, W R3v) da (C),,
(99)
metric P(RR).
Positive definiteness. For μ., E P(R"), is clear that Db- (p. (4) = 0 and Db-TSW(H,,) > If Db-TSW(H.,)) (H.v) then Wde.1 (R2H, R3v) = 0 for all C E Since Wae.1 metric on P(C), we have R2g RRU for all L E T. By the remark at the end of Appendix B.3, it implies that μ
Symmetry.
For u, E P(R"), have:
DD-TSW((L J W R2v) (L
(100)
S W R2p) do(() Db-TSW(y,,p))
(101)
So Db-TSW(((,)) TSW-SL(V,R))
Triangle inequality. For μ1, P2, Ha E P(R"), we have:
(O-TSW(11:PP) 12) + DD-TSW 143)
(102)
J. W R2))) do (f.) + / Wae. R2g do (f)
(103)
/ (R R2 d + W W (R2
R&us) (R
do dC(
(104)
> J (Re R2) do
(105)
Db-TSW( (mi, p3).
(106)
So the triangle inequality holds for Db-TSW.
In conclusion, Db-TSW is metric on the space ((RR)).
Db-TSW is B(d) -(d))invariant. We need to show that Db-TSW is E(d)-invariant, which means for all g E E(d) such that
Db-TSW (1,,))
(107)
where g&μ, gtv as the pushforward of μr, v via Fuclidean transformation g: Rdd Rd. respectively. For tree system L E T such that f = {1: (Ir, 0²))*
1= we have
gC {gl, (Q v. a, Q 0.)
(108)
For g aQ, note that det(
have
(g g() = (o)))(s) alv. gC): 5 (tzg - (y Fgl. 0g1)) dy
(109)
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1 plg ")) aly gC) 8 (tx Igt 0gi)) dy
(110)
[ pío gy) alov g()1 o (tz (gy zl 8 ggt)) d(gu)
(111)
[ ply) aly. Ch 8 (t. (gv at 0p))) dy
(112)
{ p(v) aly, L) 8 (12 (Q y + Q Q 01)) dy
(113)
Rd
ply) aly. [) 6 (tz - (Q y Q 21, Q 0() dy
(114)
Rd
p(v) aly, L) 8 (lz (y 31, 01)) dy (2,,111)
(115)
(116)
Similarly, we have
R Rgc( (gtt)) gl) = R2,(x.1).
(117)
Moreover, y induces an isometric transformation - gC, so
W Wde.1 1 RBv) W W (R st R Ricgtv).
(118)
Finally, we have
D OD-TSW(otu.gtt))
= (R2gt, do(
(119)
J.
/ W Wa (R Roootn Recotv) da (gL)
(120)
[ W W R2v) (C(
(121)
= Db-TSW(
(122)
We conclude that Db-TSW is E(d)-invariant.
Remark 11. We will omit the "almost-surely conditions" in the proofs, they are simple verify and their inclusion would only introduce unnecessary complexity.
C EXPERIMENTAL DETAILS
C.1 ALGORITHM OF DB-TSW
Algorithm Distance-based Tree-Sliced Wasserstein distance.
Probability measures μ and v in P(Rd), number of tree systems L, number of lines in tree system k; space of tree systems T or T splitting maps a with tuning parameter å E IR. for i = to I. do
Input:
Sampling E dg and 0 01,..., O2 4 U(S-11).
if space of tree system T then Orthonormalize 01, .1,....6 8r.
end if
Contruct tree system L; ((1, 01) 01),... (sr, 0k)). Projecting μ and v onto L, to get R2,11 n and R2.u.
Compute Db-TSW(H,P) v) (1/L) Wae, (R2, v)
end for
Return: Db-TSW DD)
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C.2 DENOISING DIFFUSION MODELS
Diffusion Models. Diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020) have gained significant attention for their ability to generate high-quality data. this experiment, we explore how these models work and the improvements introduced by our method. The diffusion process starts with a sample from distribution ((2)) and gradually adding Gaussian noise to data zo over T steps, described by (("122/00 (o) = [[==1 ((Fe where g(<e|=r-1) = W((;; v1 1, with a predefined variance schedule 5t.
The denoising diffusion model aim to learns the reverse diffusion process to effectively recon- struct the original data from noisy observations. The training process of the denoising diffusion model aim to estimate the parameters 0 of the reverse process, which is defined by po (o = p(xT)) IT=1 proach uses maximum likelihood by optimizing the evidence lower bound (ELBO), E & po(x)), which aims to minimize the Kullback-Leiberr divergence between the true posterior and the model's approximation the reverse diffusion process across all time steps:
where po (2 (xxz) (2 po (), I). The standard training ap-
C - -
Σ Bglr)) (LLggla-==z))lll00=>-aaa +
where KL refers the Kullback-Leibler divergence, and represents a constant term.
Denoising Diffusion GANs. Original diffusion models, while producing high-quality and diverse samples, limited by their slow sampling process, which hinders their applications in real-world scenarios. Denoising diffusion GANs (Xiao et al., 2022) address this limitation by modeling each denoising step with multimodal conditional GAN, enabling larger denoising steps and thus signifi- cantly reducing the total number of steps required to 4, resulting in sampling speeds up to 2000 times faster than traditional diffusion models while maintaining competitive sample quality and diversity. Denoising diffusion GANs introduce an implicit denoising model:
(1(==11<=)) = f |Pee pe((==-1 e G) e)de, E E~ NO,,
Xiao et al. (2022) employ adversarial training to optimize the model parameters 0. Its loss is defined by:
T
min
E By(zr
where Dadv the adversarial loss. Nguyen et (2024b) replace the adversarial loss by the aug- mented generalized Mini-batch Energy distance. For two distributions y and v, with mini-batch size n >> 1, the augmented generalized mini-batch Energy distance (AGME) using a Sliced Wasser- stein (SW) kernel is expressed as:
AGME vi = GME (ū, i)),
where ž = ft and i = fae with f(x) = (a. g(x)) for a nonlinear function g RR - R. GME is the generalized Mini-batch Energy distance (Salimans et al., 2018), defined
GMF 2E Pr) (D (Px Pk) - EID Pb)],
where X, X" kddd fmo and Y, Y fidd emmm with Px 1 En 8z, X (.1,... Fm ). and
m D are any valid distance. Here we replace D by TSW variants (including our proposed Db-TSW) and SW variants.
Setting We use the same architecture and hyperparameters as in Nguyen et al. (2024b) and train models over 1800 epochs. For all our methods, including Db- OD-TSWDDD and Db-TSW-DD, use = 2500, l = 4, 8 = 10. For vanilla SW and SW variants, we follow Nguyen et al. (2024b) to I. = 10000.
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Log Wasserstein Distance over Runs (gaussian 20d small
W2 Distance (log scale)
SW
TSW-SL
-2
Db-TSW
Db-TSW
LCVSW
SWGG
500
1500
2000
2500
Iterations
Figure 4: Logarithm of Wasserstein Distance over runs on Gaussian 20d dataset.
Table 3: Average Wasserstein distance between source and target distributions of 10 runs on 25
Gaussians dataset. All methods use 100 projecting directions.
Iteration
Methods
Time/Iter(s)
500
1000
1500
2000
2500
SW
1.61e-1
9.52e-2
3.44e-2
2.56e-2
2.20e-2
0.002
MaxSW
5.09c-1
2.36c-1
1.33e-1
9.70e-2
8.48e-2
0.144
SWGG
3.10e-1
1.17e-1
3.38e-2
3.58e-3
2.54e-4
0,002
LCVSW
3.38e-1
6.64e-2
3.06e-2
3.06e-2
3.02e-2
0.001
TSW-SL
3,49e-1
9.06e-2
2.96e-2
1.20e-2
3.03e-7
0.002
DR-TSW-DD
3.84e-1
1.13e-1
2.48e-2
2.96e-3
1.00e-7
0,002
DD-TSW-DD-
3.82e-1
1.11e-1
2.73e-2
1.45e-3
9.97e-8
0.003
C.3 GRADIENT FLOW
Additional Results on Multi-modal Synthetic Dataset. Table 3 showcases the performance of methods the Gradient Flow task using the 25 Gaussians (multi-modal distribution) dataset. Our approach consistently achieves the smallest Wasserstein distance, demonstrating robust perfor- mance across various dataset types, including non-linear, multi-modal, and sigh-dimensional cases. These results indicate that our methods outperform baseline models, further highlighting their effec- tiveness different challenging scenarios.
Hyperparameters. For D and Db-TSW, we use L 25, k 4,8 10 for all our experiments. For the baselines of SW and SW-variants, we use L = 100. The number of supports generated for each distribution in all datasets is 100.
We follow Mahey et al. (2023) to set the global learning rate for all baselines to be 5e-3 for all datasets. For our methods, we use the global learning rate of 5e-3 for 25 Gaussians and Swiss Roll datasets and 5e-2 for Gaussian 20d.
Gaussian 20d detail results. We show the detail result of Db-TSW and D in Table 4 and visualize the result in Figure 4. Figure reveals distinct performance patterns among different meth- ods, with TSW variants showing faster convergence rate. Notably, Db-TSW- achieves the fastest convergence, followed by Db-TSW and TSW-SL, all exhibiting sharp performance improvement around iteration 1500. While these methods show some fluctuation mid-process as indicated by the shaded variance regions, they maintain remarkable stability in the final iterations (iteration 2000- 2500). These results are further supported by the quantitative mean and standard deviation values provided in Table 4.
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Table 4:
Wasserstein distance between souree and target distributions of 3 runs on Gaussian 20d dataset. All methods use 100 projecting directions.
Iteration
Methods
500
1000
1500
2000
2500
SW
17.57 +: 2.4e-1
15.86 1 3.1e-1
13.92 1 3.7e-1
11.70 1 4.2e-1
9.22 + 3.8e-1
SWGG
16.53 + 1.2c-1
16.64 + 1,4e-1
16,65 + 1.7c-1
16.63 + 1.6c-1
16.64 + 1.5e-1
LCVSW
16,86 + 4e-1
14.36 + 3.4e-1
11.68 + 3.3e-1
8.80 + 4e-1
5.91 + 2.3e-1
TSW-SL
12.61 + 3e-1
5.68 + 3.8e-1
6.90e- + 8e-2
6.83e-4 + 2.6c-5
4.22e-4 1.2e-5
Db-TSW
12.46
4.6e-1
5.12 + 3.9e-1
4.8e-1 3.3e-1
6.08e-4 6.7e-5
3.84e 2.20-5
DSW
12.15 + 3.5e-1
4.67 + 3.6e-1
1.22e-1 + 1.6e-1
5.74e + 2e-5
3.78e + 1.4e-5
Source
06-T8N, Hy 022
D5-TSN M2
TSN-SL W3 - 2.41
SW. Mss 99.8
MarSel Ws 9.50
G69W. W. m2
RODOWN W; 0.06
RRGOSW N; 0.00
EDSW. Ws 105
My
brget
Figure 5: Comparison of color transferred image.
C.4 COLOR TRANSFER
Color transfer. Given source image and target image, represent their respective color palettes as matrices X and Y, where each matrix characterized by dimensions n (with n denoting the number of pixels). The source and target images employed in this analysis are derived from the work of Nguyen et al. (2024a). This curve is initialized at (() reduce the number of distinct colors in both images 1000
X and terminated
Z(T) Y. Subsequently, utilizing K-means clustering. We then trace the trajectory between the empirical distributions of colors in the source image (Px) and the target image (Py) through an approximate Euler integration scheme. Given that the RGB color values are constrained within the range 255}. rounding procedure is implemented at each Fuler step during the final iterations.
Hyperparameters. The total number of iterations for all methodologies standardized at 2000. A step size of 1 is utilized for all baseline methods, while step size of 17 is adopted for TSW-SL, D and Db-TSW-SL. For the sliced Wasserstein variants, we set L 100, whereas in TSW-SL and proposed methods, we utilize L 33 and k to ensure consistent and fair comparison.
Qualitative results. Figure shows the qualitative results of our methods, including Db- DD-TSW-DD and -
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(a) Generated images of Db-TSW-DD
(h) Generated images of Db-TSW-DD
Figure Comparison of generated images from Db-TSW-DD and Db-TSW-DD- methods.
C.5 COMPUTATIONAL INFRASTRUCTURE
The experiments of gradient flow and color transfer were carried out on a single NVIDIA A100 GPU. Each experiments for gradient flow take roughly 0.5 hours. The runtime for color transfer experiments is 5 minutes.
In contrast, the denoising diffusion experiments were executed in parallel on NVIDIA A100 GPUs, with each run lasting around 81.5 hours.
C.6 COMPUTATION AND MEMORY COMPLEXITY OF DB-TSW
Since GPUs are now standard in machine learning workflows due to their parallel processing ca- pabilities, and Db-TSW is primarily intended for deep learning tasks like generative modeling and optimal transport problems which are typically GPU-accelerated, we provide both complexity analysis and empirical runtime/memory of Db-TSW with GPU settings.
Computation and memory complexity of Db-TSW. Assume n > m, the computational com- plexity and memory complexity of Db-TSW are O(Lknd + Lkn Lknlog n) and O + Lkd + nd), respectively. In details, the D--TSW algorithm will have there most costly operations as described in Table 5:
Table 5: Complexity Analysis of Db-TSW
Operation Projection
Description
Computation (LLnnd)
Memory
Matrix multiplication
O(Lkd + nd)
of points and lines
Distance-based weight splitting Sorting
Distance calculation ((Lknd)
O(Lkn + Lkd nd)
and softmax
Sorting projected co- ordinates
(LLnlogg))
O(Lkn)
Total
OLLknd + logn) O(Lkn + Lkd + nd)
The kernel fusion trick: the table, the distance-basse weight splitting operation involves: (1) finding the distance vector from each point to each line, (2) calculating the distance vectors' norms, and (3) applying softmax over all lines in each tree. The first step costs O(Lnnd) computation and O (Lkkdd) memory. Similarly, the second step costs O(Tkdn) computation and ([.knd) memory.
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Lastly, the final step costs (Lkn) computation and O(Lkn) memory. Therefore, this operation theoretically costs ((Lknd) terms of both computation and memory. However, we leverage the automatic kernel fusion technique provided by torch.compile (Torch document) to "fuse" all these three steps into one single big step, thus contextualizing the distance vectors of size Lkn x d in a shared GPU memory instead of global memory. In the end, only need to store two input matrices of size Lk x d (a line matrix) and n x d (a support matrix), along with one output matrix of size Ln k (a split weight), which helps reduce the memory stored in GPU global memory only O(Lkn + Lkd + nd).
C.7 RUNTIME AND MEMORY ANALYSIS
In this section, we have conducted runtime comparison of Db-TSW with respect to the number of supports and the support's 's dimension in a single Nvidia A100 GPU, We fix L = 100 and k = 10 for all settings and varies N (500, 1000, 5000, 10000, 50000) and E {10, 50, 100, 500, 1000).
Runtime evolution. In Figure 7a, the relationship between runtime and number of supports demon- strates predominantly linear scaling. particularly beyond 10000 supports, though there's subtle non- linear behavior in the early portions of the curves for higher dimensions. The performance gap be- tween high and low-dimensional cases widens as the number of supports increases, with 1000 taking approximately twice the runtime of d 500, suggesting linear relationship between dimen- sion and computational time.
Memory evolution. In Figure 7b the memory consumption analysis of Db-TWD reveals several key patterns which align with the theoretical complexity analysis and suggest predictable scaling behavior. Firstly, there is clear linear relationship between memory usage and the number of samples across all dimensions. Secondly, in higher-dimensional cases (d 100, 500, 1000), the memory requirements exhibit proportional relationship with the dimension, evidenced by the approximately equal spacing between these curves. Finally, for lower dimensions (d = 10, 50, 100), the memory curves nearly overlap due the dominance the Lkn term in the memory complexity formula OLLn++ (id++nd where the number of supports components (Lkn) has greater impact than the dimensional components (L.d and nd) when T. 100 and k- 10.
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Figure 7: Runtime and memory evolution of Db-TSW with respect different dimension.
C.8 ABLATION STUDY FOR L, k AND
In this section, we include the ablation study for I,, k. and 8 on the Gradient flow task. We chose the Gaussian 100d dataset. When not chosen vary, used the default values: number of projections n = 100, learning rate (r = 0.005, number of iterations Nitee = 5000, number trees L = 500, number of lines per tree I 2, and coefficient 5 10. We fixed all other values while varying k E {2, 4, 16, 64, 128), E {100, 500, 1000, 1500, 2000, 3000), and 8 E (1,2, 5, 10, 20, 50, 100).
Ablation study for k. From Figure 8a, the ablation analysis of the number of lines & reveals an interesting relationship between the number of lines per tree (k) and the algorithm's 's performance. When fixing the number of trees (T.), configurations with fewer lines per tree demonstrate faster
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Published as a conference paper at ICLR 2025
convergence to lower Wasserstein distances. This behavior is reasonable because increasing ex- pands the space of tree systems, which in turn requires more samples to attain similar performance levels. For instance, = 2 and & configurations reach lower Wasserstein distances compared to k = 64 and = 128. Additionally, as demonstrated in the runtime and memory analysis section C.7, configurations with higher number of lines per incur greater computational and memory costs while yielding suboptimal performance.
Ablation study for L. From Figure 8b, the convergence analysis demonstrates clear relationship between the number of trees (L) and the algorithm's 's performance. When fixing the number of lines per tree (k), configurations with more trees achieve significantly better convergence, reaching much lower Wasserstein distances. This behavior is expected because increased sampling in the Monte Carlo method used Db-TSW leads more accurate approximation of the distance. Specifically, comparing T. 3000 and 7. 100, we observe that I, 3000 achieves final Wasserstein distance of 10-5, while I 100 only reaches 102. However, there a computation-acee accurrcy trade-off since more trees involves more computation and memory.
Ablation study for 8. From Figure 8c, the analysis of different values indicates variations in algorithm's performance. Moderate values of delta (6 = 10) enhance the convergence speed compared to = as they effectively accelerate the optimization process. However, when delta becomes too large (ò 20 100), the converged speed decreased.
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Figure 8: Ablation study for k. L, and d on Gaussian 100d dataset.
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Reviews:
[{"review_id": "iydIW6EaLs", "reviewer": "Reviewer_xP5V", "summary": "The paper introduces the Distance-based Tree-Sliced Wasserstein (Db-TSW) distance, which addresses limitations in existing Tree-Sliced Wasserstein on Systems of Lines (TSW-SL) by improving Euclidean invariance and preserving positional information in optimal transport (OT). Unlike traditional Sliced Wasserstein (SW) distances, which can lose topological information, Db-TSW incorporates tree structures and a novel class of E(d)-invariant splitting maps to maintain metric consistency under Euclidean transformations. This approach also includes a new tree sampling method for efficiency, enabling GPU-friendly implementations. Through rigorous theoretical analysis and empirical experiments, Db-TSW is shown to outperform recent SW variants on tasks such as gradient flows, image style transfer, and generative modeling, while remaining computationally efficient​.", "questions": "* The root-concurrent structure is computationally efficient, but it may affect the transform's injectivity, especially in cases with overlapping or closely aligned lines. How does this design ensure that unique positional information is retained across sampled trees, and are there any trade-offs in injectivity or stability in high-dimensional data?\n\n* How does Db-TSW handle cases where the Euclidean distance between points and tree lines is nearly uniform across lines, potentially reducing the distinctiveness of the splitting map?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* Paper is very-well written and clear.\n* Through extensive experiments, authors show the effectiveness of Db-TSW and compare it with existing benchmarks.\n* Authors propose a novel variant of Radon transform on Systems of Lines which generalizes the invariant splitting mapping of the previous works.", "weaknesses": "* Although Db-TSW is optimized for GPU implementation, the method’s reliance on high-dimensional Euclidean distances for each point-line relationship could still pose challenges for scalability in very high-dimensional settings. This could result in increased computation times or memory usage for large datasets, particularly when fine-grained positional information is necessary. I would like to see a thorough runtime comparison of Db-TSW with respect to the number of samples and dimensionality of the data.\n\n* The root-concurrent structure for sampling tree systems, though efficient, limits Db-TSW’s flexibility in capturing diverse spatial arrangements. I am curious about how effectively Db-TSW captures these spatial variations. Is there a way to demonstrate this? Could you provide an example where Db-TSW might fail to capture them?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "a3CPhL1tHh", "reviewer": "Reviewer_2G1e", "summary": "The paper introduces a distance-based variant of the tree-sliced SW on a system of lines (TSW-SL) [1]. The core contribution lies in generalizing the class of splitting maps used to project measures onto tree systems while preserving Euclidean invariance. The authors provide theoretical results for their method, including proofs of metric properties and injectivity of the associated Radon transform. They also introduce a simplified tree sampling process that with efficient GPU implementation. Various experiments including gradient flows, image generation, and color transfer show competitive or better performance in certain hyperparameter settings.\n\n[1] Tran, Viet-Hoang, et al. \"Tree-Sliced Wasserstein Distance on a System of Lines.\" arXiv preprint arXiv:2406.13725 (2024).", "questions": "Q1: Can any component of this framework be made learnable? That would probably enable various interesting extensions.\n\nQ2: Are there instances where prior knowledge on the data allows a good/optimal tree design?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "S1: Generalization of splitting maps addresses a limitation in previous work [1].\n\nS2: Efficient implementation thanks to the proposed simplified tree sampling process.\n\nS3: The writing is well-written and easy-to-follow. \n\nS4: Good empirical results within the given setups.", "weaknesses": "W1: It would be nice to have an explicit discussion of computational/memory complexity.\n\nW2: For setups that aggregate results from multiple runs (e.g., gradient flow), the paper should include both mean and std/variance (instead of just the means). Additionally, it may be a good idea to use plots to better present the results in Table 2,3 (visually). Both of these are good practices in general, and also used in [2], which is cited in the paper.\n\nW3: The paper would benefit from a detailed discussion on various instances of tree systems and especially splitting maps (which are directly related to the primary contributions). For instance, using $k$ concurrent lines with shared root is fast but what are we sacrificing? Are they quantifiable? Softmax works well as an E(d)-invariant splitting map but what are some other instances (along with pros and cons)?  Along those lines, a stronger ablation section would be helpful. It should clearly demonstrate that any performance gain is due to the novel technical contributions to support the paper's claims throughout (is it the invariance or the proposed tree structure that gives better performance?). How much is the gain if one adjusts $L,k, \\delta$ for datasets of varying complexity? What about root sampling? Since the design space includes various components, that would help practitioners in making sensible design choices for their applications.\n\nW4: The paper would be strengthened if explicitly discussing sample/projection complexity, and also error bounds, convergence rate for MC approximation.\n\nW5: What are the authors' rationales for not using a consistent set of baseline methods to evaluate relative performance across tasks? Moreover, is it a good idea to compare against non-linear slicing methods like GSW, ASW? Why or why not?\n\n---\n[2] Mahey, Guillaume, et al. \"Fast Optimal Transport through Sliced Generalized Wasserstein Geodesics.\" Advances in Neural Information Processing Systems 36 (2024).\n\n[3] Kolouri, Soheil, et al. \"Generalized sliced wasserstein distances.\" Advances in neural information processing systems 32 (2019).\n\n[4] Chen, Xiongjie, Yongxin Yang, and Yunpeng Li. \"Augmented sliced Wasserstein distances.\" arXiv preprint arXiv:2006.08812 (2020).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4zadxQIiOH", "reviewer": "Reviewer_vezg", "summary": "Paper proposes aims at enhancing the Tree-Sliced Wasserstein on a System of Line (TSW-SL) by defining a specific splitting maps (that are parts of the radon transform on systems of lines) that takes into account the distance to the lines. The method is coined Dd-TSW and it can be shown that it preserves some invariances (that are also preserved in $W_2$ or sliced-Wasserstein). By construction, it allows taking into account the distance to the lines and provides varying mass distributions. A variant that samples orthogonal lines, allowing to better cover the space, is also provided. Experiments show improved performances in several settings with similar computational timings.\n\n\n**Note**: I have also reviewed the paper that introduces TSW-SL.", "questions": "- The trees are recommended to be sampled close to the means of the target distributions. Does this relate to numerical instabilities that can occur for large values of $\\delta$, such as those considered in the paper ($ \\delta = 10$), to avoid excessively large distances?\n- The $\\delta = 10$ parameterization should lead to a very sparse splitting map. Could you provide an ablation study on this parameter and discuss why large values should be considered?\n- Does the new method of sampling lines affect the performance of Dd-TSW, or is its definition justified only by computational issues?\n- It seems (on the Gaussian 20D dataset) that Dd-TSW performs better in higher-dimensional cases than other SW variants. Is this result also true in higher-dimensional settings, and can you provide an explanation for this interesting behavior?\n- It is stated in the introduction that Dd-TSW enables a highly parallelizable implementation, and in Section 6, it mentions \"*that enables an algorithm with linear runtime and high parallelizability, keeping the wall-clock time of Dd-TSW and Dd-TSW⊥ approximately equal to that of vanilla SW and surpassing some SW variants*.\" Do the reported timings in section 6 relate to the parallel implementation?\n- In the conclusion, it is mentioned that improved performance is expected by adapting recent advanced sampling schemes or techniques from SW to TSW-SL. However, it seems that the extension is not straightforward: can you provide concrete examples that could be adapted to Dd-TSW?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Dd-TSW is an extension of TSW-SL, in which the splitting map is refined and a more efficient method for sampling lines is provided. As such, the originality is fair. The definition of Dd-TSW is well-grounded and the experimental section shows that it yields competitive results in several scenario, even in one specific challenging scenario of high dimensional setting (GF experiment in 20d).", "weaknesses": "The main weaknesses of the paper lie in: \n\n- Unsufficient analysis of its various parameters. The settings of some parameters are not discussed (e.g., the number of lines $k$ and the value of the inverse temperature parameter $\\delta$).\n- In the gradient flow experiment, differences in ground costs (e.g., $L_2$​ and tree metric) between methods make it challenging to compare results at a fixed number of iterations for a fixed LR.\n- Competitors are not consistent between the different experiments.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0XzKscQjA6", "reviewer": "Reviewer_YmcA", "summary": "This work proposes a variant of the TSW-SL distance. It is observed that the latter is not invariant w.r.t translations and rotations (Euclidean transformations), and that the splitting map used in TSW-SL does not take into account the trees but only the positions of the samples. Thus, the authors propose to take these two observations into account in order to define a new projection (Radon transform) on trees with a splitting map which is invariant w.r.t translation and rotations, and which takes into account the trees. Their new Radon transform is shown to be injective provided that the splitting map is invariant w.r.t Euclidean transformation. Then, they propose a splitting map which satisfies this, as well as two new more efficient ways to sample trees. Finally, they validate their new distance on several tasks such as diffusion models, gradient flows and color transfer.", "questions": "In equation (2), $Rf_\\mu$ is not a probability. So I think there is an abuse of notation. In equation (8), $R_\\mathcal{L}^\\alpha\\mu$ is used, but $R_\\mathcal{L}^\\alpha$ is only defined on $f\\in L^2$ in Equation (7).\n\nIn Section 5.2, when suggesting to sample orthogonal directions, it is stated that it is \"to ensure that the sampled tree systems do not include similar directions\". While I understand what is meant here; the goal is to have different directions to better capture the topology of the data; the sentence is not very clear as in practice, we would have $\\theta_i\\neq \\theta_h$ almost surely. Note also that it was proposed e.g. in [1] to sample orthogonal directions for Sliced-Wasserstein and in [2] for a max-Sliced-Wasserstein variant, but these references are missing.\n\nI do not think I saw the step size used for each variant of SW in the gradient flow experiment. I believe that using the same step size might not always be fair as they do not necessarily have the same scale/smoothness properties (although I would agree that it is complicated to use a different step size for each method).\n\n\nTypos:\n- In legend of Figure 2, the two last sentences are almost the sames.\n- Line 371: \"$\\mathcal{L}$ is a tree system consists\"\n- Line 476: \"Db-TSW and $\\mathrm{Db-TSW}^\\top$ is\"\n\n[1] Rowland, M., Hron, J., Tang, Y., Choromanski, K., Sarlos, T., & Weller, A. (2019, April). Orthogonal estimation of Wasserstein distances. In The 22nd International Conference on Artificial Intelligence and Statistics (pp. 186-195). PMLR.\n\n[2] Dai, B., & Seljak, U. (2020). Sliced iterative normalizing flows. arXiv preprint arXiv:2007.00674.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "This paper first identifies some drawbacks of the TSW-SL distance, and proposes to overcome them with a splitting map which depends on the trees sampled and which is Euclidean invariant. These new choices are well motivated and make sense.\n\nThen, the authors provide the new Radon transform to project measures on trees using these splitting maps. They show that it is injective, and that the counterpart TSW distance is well an invariant distance. They also introduce a suitable splitting map, which they use in practice, and propose a new way to sample trees.\n\nFinally, they experiment on challenging tasks such as generative modeling on high dimensional images with Diffusion, and demonstrate that their distance outperforms all previous Sliced-Wasserstein variants.", "weaknesses": "There is no statistical analysis of the new method. One could wonder how the distance evoluates w.r.t. the number of samples or the number of projections.\n\nAnother weakness is the lack of an ablation study, which would help us to see what really make it better than TSW-SL. Is it the new way of sampling trees which makes a big difference? Or is it because the distance is $E(d)$-equivariant? Or is it because the splitting map depends now on the distance to the lines?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)