Revisiting Tree-Sliced Wasserstein Distance Through the Lens of the Fermat–Weber Problem
Authors: Viet-Hoang Tran
Thanh Tran
Thanh Chu
Trung-Khang Tran
Duy-Tung Pham
Tam Le
Tan Minh Nguyen
Venue: ICLR 2026 Poster (2026)
Keywords: sliced optimal transport
tree-sliced wasserstein distance
tree wasserstein distance
fermat-weber problem
Primary area: learning on graphs and other geometries & topologies
TL;DR: We propose a new variant of Tree-Sliced Wasserstein distance that incorporates positional data more explicitly, inspired by the Fermat–Weber problem.

Abstract: Tree-Sliced methods have emerged as an efficient and expressive alternative to the traditional Sliced Wasserstein distance, replacing one-dimensional projections with tree-structured metric spaces and leveraging a splitting mechanism to better capture the underlying topological structure of integration domains while maintaining low computational cost. At the core of this framework is the Tree-Sliced Wasserstein (TSW) distance, defined over probability measures in Euclidean spaces, along with several variants designed to enhance its performance. A fundamental distinction between SW and TSW lies in their sampling strategies—a component explored in the context of SW but often overlooked in comparisons. This omission is significant: whereas SW relies exclusively on directional projections, TSW incorporates both directional and positional information through its tree-based construction. This enhanced spatial sensitivity enables TSW to reflect the geometric structure of the underlying data more accurately. Building on this insight, we propose a novel variant of TSW that explicitly leverages positional information in its design. Inspired by the classical Fermat–Weber problem—which seeks a point minimizing the sum of distances to a given set of points—we introduce the Fermat–Weber Tree-Sliced Wasserstein (FW-TSW) distance. By incorporating geometric median principles into the tree construction process, FW-TSW notably further improves the performance of TSW while preserving its low computational cost. These improvements are empirically validated across diverse experiments, including diffusion model training and gradient flow. Our code is available at [https://github.com/thanhquangtran/FW-TSW](https://github.com/thanhquangtran/FW-TSW).

Published as a conference paper at ICLR 2026
REVISITING TREE-SLICED WASSERSTEIN DISTANCE THROUGH THE LENS OF THE FERMAT-WEBER PROBLEM
Viet-Hoang Tran'
Thanh Q. Tran"
Department of Mathematics National University of Singapore hoang. tranviet @u edu
Department of Computer Science National University of Singapore thanhtran@u nus edu
Thanh Chu
Trung-Khang Tran
Department of Computer Science National University of Singapore thanh. chu@u. nus. edu
Department of Computer Science National University of Singapore TTKO 6êu nus. edu
Duy-Tung Pham FPT Software AI Center
Tam Le
Department of Advanced Data Science Institute of Statistical Mathematics t tm i sm. ac. jp
Vietnam tungpdl 06 fpt.com
Tan Minh Nguyen
Department of Mathematics National University of Singapore
tanmnênus nus. edu. sg
ABSTRACT
Tree-Sliced methods have emerged efficient and expressive alternative the traditional Sliced Wasserstein distance, replacing one-dimensional projec- tions with tree-structured metric spaces and leveraging splitting mechanism better capture the underlying topological structure of integration domains while maintaining low computational cost. At the core of this framework is the Tree- Sliced Wasserstein (TSW) distance, defined over probability measures in Eu- clidean spaces, along with several variants designed to enhance its performance. A fundamental distinction between SW and TSW lies in their sampling strategies- a component explored in the context of SW but often overlooked comparisons. This omission is significant: whereas SW relies exclusively on directional pro- jections, TSW incorporates both directional and positional information through its tree-based construction. This enhanced spatial sensitivity enables TSW to re- flect the geometric structure of the underlying data more accurately. Building on this insight, we propose novel variant of TSW that explicitly leverages posi- tional information in its design. Inspired by the classical Fermat-Weber problem- which seeks point minimizing the sum of distances to given set of points-we introduce the Fermat-Weber Tree-Sliced Wasserstein (FW-TSW) distance. By incorporating geometric median principles into the tree construction process, FW- TSW notably further improves the performance of TSW while preserving its low computational cost. These improvements are empirically validated across diverse experiments, including diffusion model training and gradient flow, Our code is available https: /ggthub..tt com/ thwnhhhanggtaaaa...... rtt.////n...../////-/... FaNFFWTSS.
"Equal contribution
Co-last author. Please correspond hoang.tranviet w.n.us.euuu edu and tammn @ nus.edu.sg

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Published as a conference paper at ICLR 2026
1 INTRODUCTION
Optimal Transport (OT) (Villani, 2008; Peyré et al., 2019) has established itself as foundational framework for comparing probability measures in way that respects the underlying geometry of the data. By extending ground cost metrics from supports to entire distributions, OT has enabled wide range of applications across machine learning (Bunne al., 2022; Fan et al., 2022), data valuation (Just et al., 2023; Kessler et al., 2025), multimodal data analysis (Park et al., 2024; Luong al., 2024). statisties (Mena & Niles-Weed, 2019; Weed & Berthet, 2019; Wang al., 2022; Liu et al., 2022: Nguyen et al.. 2022; Nietert et al., 2022), and computer vision and graphics (Lavenant al., 2018; Saleh al., 2022; Solomon al., 2015). Despite its theoretical elegance and flexibility, a major practical limitation of OT lies in its computational complexity, which grows supercubically with the number of support points (Peyré et al., 2019).
To address this issue, the Sliced Wasserstein (SW) distance (Rabin et al., 201 Bonneel et al., 2015) has been proposed as a scalable alternative. SW reduces computational cost by projecting high- dimensional probability measures onto one-dimensional subspaces, where closed-form solutions to the OT problem are available. This projection-bas strategy leads to significant computational gains and has inspired large body of research aimed at refining various components of the SW framework. Advances include accelerated sampling strategies (Nadjahi et al., 2021; Nguyen et al., 2024a; 2020), projection direction selection (Deshpande et al., 2019), and extensions generalized integration domains (Kuchment, 2006; Kolouri et al., 2019; Chen et al., 2022; Bonet et al., 2023).
However, the restriction to one-dimensional projections may fail to capture complex geometric or topological features of nigh-dimensional distributions. In response, recent work has proposed using more expressive integration domains for OT, including Euclidean subspaces (Alvarez-Mells et al., 2018; Paty & Cuturi, 2019; Niles-Weed & Rigollet, 2022), tree metric spaces (Le et al., 2019; Le & Nguyen, 2021; Tran al., 2024e), graphs (Le al., 2022), spheres (Quellmalz
al., 2023; Bonet et al., 2022: Tran et al., 2024c), and hyperbolic spaces (Bonet et al., 2023). Among these approaches, the Tree-Sliced Wasserstein (TSW) framework (Tran et al., 2024e; 2025a) replaces directional projections with tree systems. These systems provide a structured integration domain that captures both spatial and directional information. By leveraging efficient OT solvers on tree metric spaces (Indyk& & Thaper, 2003; Le et al., 2019; Le & Nguyen, 2021), TSW retains the computational advantages of SW while offering greater geometric expressiveness.
A key distinction between SW and TSW lies in their sampling strategies. While this aspect has been well studied for SW, is often overlooked in comparative analyses. The difference is significant: SW relies solely directional projections, whereas TSW incorporates both direction and position through its tree-based construction. This added spatial structure enables TSW to more effectively capture the geometry of data distributions. Consequently, the quality of the sampling strategy is crucial to realizing the full potential of TSW.
Standard implementations of SW (Bonneel et al., 2015) rely on uniform sampling sphere. However, this strategy does not distinguish between informative and uninformative direc- tions (Deshpande et al., 2019; Nguyen al., 2024b; Tran al., 2024d; Nguyen & Ho, 2024), which may limit its practical effectiveness. To enhance performance, several studies have proposed data- informed slicing distributions, including both fixed (Nguyen et al., 2024b; Tran al., 2024d: Nguyen & Ho, 2024) and trainable (Deshpande et al., 2019; Nguyen et al., 2020) variants. While trainable approaches yield empirical gains, they often rely on iterative optimization, which is computationally costly and may exhibit instability (Nguyen et al., 2020).
the hyper-
Incorporating similar sampling enhancements into TSW poses additional challenges. Unlike SW, the sampling space in TSW involves both directional and positional components, the latter corre- sponding to the intersection point of the tree system. As I2d is non-compact, there is no canonical uniform distribution analogous to that gd- I This complicates the design of efficient and prin- cipled sampling strategies for TSW. Existing TSW variants (Tran et al., 2024e; 2025a;d) rely on heuristic sampling schemes, which may not fully exploit the positional information encoded in the tree structure.
Contributions. Building on this insight, we propose a novel variant of the TSW framework that explicitly incorporates positional information into its slice distribution. Our approach is motivated by classical problem in location theory- the Fermat-Weber problem- and aims improve upon
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existing heuristic methods by aligning the sampling distribution with the geometric structure of the data. The paper is organized as follows:
1. In Section 2, recall the concepts of the SW and TSW distances, both of which serve as computationally efficient alternatives to the classical Wasserstein distance.
2. In Section 3, we examine the sampling strategies used in the SW and TSW frameworks. We emphasize the role of positional information in TSW, in contrast the purely directional sampling in SW. Furthermore, we revisit the Fermat-Webbr problem and the concept of the geometric median, and explain how these ideas inform the design of improved sampling distributions for tree systems in TSW.
3. In Section 4, formally introduce the Sermat-Weber Tree- Sliced Wasserstein (FW-TSW) framework. We analyze its theoretical properties and computational complexity.
4. In Section 5, we illustrate advantages of the proposed approach on gradient flow and diffu- sion models, and conclude our work in Section 6. The results highlight practical effec- tiveness and computational efficiency across both image-based and distributional learning scenarios.
The Appendix contains all supplementary materials, including theoretical background, detailed proofs, experimental setups with extended tables and figures, as well as table of notation.
SAMPLING PROCESSES IN SLICED AND TREE-SLICED WASSERSTEIN DISTANCES
Let d denote the dimension. Consider two probability measures and v on Rd with correspond- ing density functions fnx and fu. We review the main ideas behind the Sliced Wasserstein and Tree-Sliced Wasserstein distances, which provide efficient alternatives to the classical Wasserstein distance.
2.1 REVIEW ON SLICED WASSERSTEIN DISTANCE
Motivation. A line in Rd uniquely determined by direction E gd-1 and point æ E Re through which it passes. Importantly, the OT problem between two probability measures supported on one-dimensional lines admits dlosed-form solution. Leveraging this property, the SW frame- work projects high-dimensional measures onto one-dimensional lines, computes the Wasserstein distance in each projected space, and aggregates the results via averaging (Rabin et al., 2011: I; Bon- neel et al., 2015). Since the projection depends only the direction of the line, suffices consider projections parametrized by directions in Sd-1
Radon Transform. Consider function f L' (.R). For direction 0 g define the function
Ref: R R, Rof(t)= / f(x) 8(t (x, 8)) dr.
where 8 denotes the Dirac delta distribution. The full Radon transform is the operator
R: (RR)
L'R),
- Raf,
8ES
This construction provides a formal mechanism for projecting measures onto one-dimensional lines.
Sliced Wasserstein Distance. For p 2 1, the Sliced p-Wasserstein distance (Bonneel et al., 2015) (SW,) between f and v is defined as
(w, = -// WP (Rofa, Refv) do do(0) )o(0));
$
(3)
where (((-111 denotes the uniform probability measure the unit sphere gd-1
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2.2 REVIEW ON TREE-SLAE SLICED WASSERSTEIN DISTANCE
We adopt the formulation of the Tree-Sliced Wasserstein distance introduced in Tran et al. (2024e; 2025a). For complete description, we refer the reader to Appendix A.
Motivation. The OT problem between two probability measures supported on tree metric space (Semple & Steel, 2003; Le al., 2019) admits a closed-form solution, similar to the one- dimensional case used in the SW framework. However, identifying suitable tree metric structures & that permit efficient computation, analogous to projecting along directions 0 E gd-1 in SW, is nontrivial. To address this challenge, Tran et al. (2024e; 2025a) introduced class of structures known as tree systems, which enable efficient computation of OT on tree metrics. Informally. a tree system is collection of k one-dimesiional lines in Rd arranged with fixed tree topology. For sim- plicity, we may, for now, regard a tree system as an element of (R2 xS that is, a collection of lines, without explicitly considering the underlying tree structure. We denote tree system by ' and the set of all such k-line tree systems by T. Leveraging this structure, the TSW framework projects high-dl nigh-dimensional probability measures onto the lines of a given tree system, solves the induced OT problem on this tree system, and aggregates the results, analogous to the averaging process in the SW framework.
Radon Transform on Tree Systems. Define C(R x T, Ak-1) ) as the set of continuous maps from Rdd to the (k- 1)-dimensional standard simplex Ag-1, referred as splitting maps. We fix a splitting map, denoted by a. Consider a function f E I, [..() For each 'I E T, define the function
k
RJ f:
L - R
R9J(4)=
[(v) aly, T) (1: - (y 1yxr;,01 0.)) dy-
(4)
The Radon Transform on Tree Systems is the operator
Ra (RR)
(T),
f- (-9
(5)
TET
This construction provides a formal mechanism for projecting measures onto tree systems. Tree-Sliced Wasserstein Distance. The Tree-Sliced Wasserstein distance between μr, v is defined by
TSW v) - ff (R9 fi) (1) dor(T).
(6)
Here, oT denotes probability distribution over the space of tree systems T. The construction of both the splitting map a and the distribution of is detailed in Section 3. For clarity, we present the formulation for the case p = When p > the Wasserstein distance W Wy(,,)) generally does not admit a closed-form solution as in the led the development of Sobolev Transport (Le et al., 2022) (ST), which differs from Wp. p- While
case. Efforts to derive such expressions for p > 1 have
ST remains a valid metric over the space P(T), we restrict our attention to the case p = 1 in this work, as the generalization to higher p values follows analogously.
3 SAMPLING TREES THROUGH THE LENS OF THE PERMAT-WEBER PROBLEM
We review the sampling processes in both the SW and TSW frameworks, highlighting TSW's posi- tional dependence in contrast SW. We then revisit the Fermat-Webel problem and the geometric median, and discuss how this notion can guide the sampling of tree systems in TSW.
'For brevity, refer collectively to the formulations in Tran et al. (2024e; 2025a) as the Tree-Sliced Wasserstein (TSW) distance. This terminology departs from the original notion introduced in Le al. (2019); Le Nguyen (2021); Yamada Lin et al. (2025), which was primarily developed for static-support measures in settings such as classification and topological data analysis. In contrast, TSW-SL (Tran et al.. 2024e) and Db-TSW (Tran et al., 2025a) are formulated as optimal transport problems over tree systems, specifically designed to handle dynamic-support measures, as commonly found generative modeling tasks.
(2022); Sato al. (2020); Takezawa et al. (2022); Indyk & Thaper (2003);
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3.1 MONTE CARLO APPROXIMATION OF SW AND TSW DISTANCES
Sampling Slices in SW. To approximate the intractable integral in Equation (3) of SW, Monte Carlo method is used as follows:
uw)(u T. (Ro Ro, Ro
where 01....,0, are drawn independently from o. Since the hypersphere gd-1 is compact, or is commonly chosen to be the uniform distribution on S4-1 (Bonneel et al., 2015). While this choice convenient due to its ease of sampling, it fails to differentiate between informative and uninforma- tive projection directions when comparing probability measures (Deshpande et al., 2019; Nguyen et al., 2024b; Tran et al., 2024d: Nguyen & Ho, 2024). The conventional SW distance thus relies on flat prior over directions, which can limit its discriminative power. To address this, alternative formulations propose selecting a from a parametric family of distributions over gd-1, aiming to maximize the expected sliced distance (Nguyen et al., 2020). Although such data aata-adeptive slicing distributions can enhance performance, identifying the optimal a typically involves iterative proce- dures that are computationally intensive and may exhibit instability.
Sampling Slices in TSW. As with the SW framework, the TSW distance can be approximated by randomly sampling tree systems T....., Tr independently from the distribution oy. In this case, the integral in Equation (6) is approximated as:
TSW L W1 (R.
Σ
We first describe the specific tree structure employed in the TSW framework. Tran et al. (2024e) proposed a general and inductive procedure for sampling arbitrary tree topologies. Building on this, Tran et al. (2025a) introduced a simplified construction that retains the representational power of TSW while allowing for efficient implementation. In this formulation, each tree system consists of k lines intersecting at common point. Accordingly, tree system can be represented as a tuple T = (z, 0,, ................................ 02) E gd x (Sd-1)A, where x E IRd denotes the intersection point (or root), and each 0: E g specifies the direction of the 7th line passing through æ. Figure I (left) illustrates these tree structures.
The slicing distribution nents: one distribution over Rd for sampling the root point, and & independent distributions over gd-1 for sampling line directions. While the directional components gd-1, the non-compactness of IRd precludes a uniform distribution. To address this, Tran et al. (2024e; 2025a) propose sampling the intersection point from Gaussian centered at the data mean, which helps prevent the projection of nearby points to distant locations in the tree system.
tree systems is modeled product of k: + independent compo-
be chosen uniform over
However, this sampling strategy inherently constrains the ability to capture positional variability key advantage of TSW over SW. In addition, the splitting maps commonly employed in Tran et al. (2025a) are explicitly position-dependen. For instance, the splitting map a used in Tran et al. (2025a) is defined as
aly, T) softmax (LS d(y, T)))=.....))))
(9)
where d(y, T); denotes the Euclidean distance from the point E IRd to the 4h line in the tree system T, and & E I. is a tunable parameter. Intuitively, under the Radon transform defined on a tree system, the mass at y is distributed among its projections onto the k lines in T, weighted proportionally (or inversely proportionally) to their distances, depending on the sign of
3.2 THE FERMAT- WEBER PROBLEM
The preceding discussion underscores the critical role of sampling in the TSW framework, partic- ularly when positional information is incorporated. Since TSW aims to align a source distribution with target data distribution, it is desirable for the intersection points minimize their average distance to the data. This naturally leads to the classical Fermat-Weber problem in location theory, which seeks point that minimizes the weighted sum of distances set of target points.
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26
#2
23
B5
Figure 1: (Left) Illustration of tree structures used in Tran al. (2024e) and Tran et al. (2025a). The structure in Tran et al. (2024e) is more general, as the framework is applicable to arbitrary tree topologies. In contrast, Tran al. (2025a) focuses trees formed by set of concurrent lines, despite the fact that the underlying framework supports general trees, due to implementation considerations. (Right) Illustration of the geometric median. Given six points Jq, 1,...,6 6 (shown in gray), the geometric median x* (shown in pink) is the point that minimizes the sum of distances all six points, i.e., the total length of the connecting segments.
The Fermat-Weber Problem. Given a probability measure À on Rd, the Fermat-Weber problem defined as the following optimization problem:
argmin E
g ||
(10)
When is approximated via Monte Carlo sampling from n data points {a:) the problem reduces the discrete form:
argmin ZERd
1 x:ll2-
Σ 11
(11)
11
Such optimal point is referred as the geometric median. An illustration of this concept is provided in Figure 1 (right).
Weiszfeld's Algorithm. The Weiszfeld algorithm provides efficient iterative method for approx- imating the geometric median. Starting from initial estimate (0) IRd,, define:
(1+1)
t= 0, 0,1,2, 2,......
(12)
(x(0) s.
| x:112
The iteration continues until convergence, typically measured by the stopping criterion:
||r((++))
(0 112 S,
(13)
for some pre-specified threshold 0. The complete procedure is summarized in Algorithm
Algorithm Weiszfeld's Algorithm for Geometric Median
1: Input: Data points {æ1...., F in Rd initial estimate (0) Rd tolerance
2: Output: Approximate geometric median xº
3: Set iteration counter
4: repeat
5: Compute update: 2((++1)
E 1x00
/ EE Tx
6: Increment / - +1
7: until [|x(((] - (1- 5
8:
return x = (1)
The general formulation for the Fermat- Weber problem is presented in Appendix B.
Application to the TSW Framework. We now define a distribution over the space of tree systems T using the geometric median. Given set of data points -.,...;.f Rdd that are independently

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sampled from A, we apply Weiszfeld's algorithm for fixed number of iterations to obtain ap- proximation of the geometric median x". With slight abuse of notation, we continue to denote this approximation by x². To define distribution over the intersection point Rd, we sample from a Gaussian distribution centered at x² to ensure that sampled points remain close to the geometric median, i.e.,
x N~~ cla),
(14)
where с >> O is small constant and Ia is the identity matrix. The parameter с controls the concen- tration of the distribution around x*: smaller values of produce points more tightly clustered near the geometric median, ensuring that the roots of the sampled tree systems lie close to x². The full distribution over tree systems is then defined as the joint distribution between this Gaussian distri- bution over root points and k independent uniform distributions over directions the unit sphere Sd-1;
M( ela) @ 11$-11))
(15)
Remark 3.1. The constant will be treated as a tuning parameter in our experiments. In practice, since the data is typically normalized, find that setting 1 yields stable behavior and performs well across most datasets.
4 FERMAT- WEBER TREE-SLICED WASSERSTEIN DISTANCE
In this section, we present the Fermat-Weber Tree-Sliced Wasserstein (FW-TSW) framework and analyze its theoretical foundations along with its computational complexity.
4.1 FERMAT-WERT TREE-SLICED WASSERSTEIN DISTANCE
To define the proposed discrepancy, we consider two probability measures μ and y on R with corresponding density functions fp and fu. For given tree system T E T and splitting map a (as in Equation (9)). we apply the Radon transform R (as in Equation (4)) to obtain pushforward densities ft and R. These define new probability measures μT, VT E P(T) supported on the tree metric space T. The OT distance between PTT and PT can then be computed efficiently, due to the existence of closed-form solutions for OT problems on tree spaces (Le et al., 2019).
We define our discrepancy the expected transport cost over a distribution of tree systems. Specif- ically, we take the expectation of the OT distance with respect to a data-dependent sampling dis- tribution ,w., (as in Equation (15)). The subscript notation FW..,.v reflects the fact that the distribution is constructed from a set of points derived from μ and v, as discussed in Section 4.2. The resulting expected transport cost defines the Fermat-Wehh Tree-Sliced Wasserstein distance (FW-TSW).
Definition 4.1 (Fermat-Weber Tree-Sliced Wasserstein distance). The Fermat-Weber Tree-Sliced Wasserstein distance (FW-TSW), between j and v in P(Rd) is defined by
FW-TSW(H, = / W1 W1 My (T)).
(16)
4.2 PROPERTIES OF FERMAT-WEBER TREE-SLICED WASSERSTEIN DISTANCE
We investigate several theoretical properties of the proposed FW-TSW discrepancy. Proofs of all results presented in this section are provided in Appendix C.
Constructing the Distribution dpw.u.p- The sampling distribution d pars as defined in Equa- tion (15), is centered at point x², which is the geometric median of a set of data points. We now describe how these points are constructed. practical applications, Optimal Transport aims to align source distribution with target distribution-typice ral to compute x² based on samples drawn from both j and v. Specifically, we sample m points T1,.... F. from the source measure y and m points ......, Um from the target measure The point x² is then computed as the geometric median of the combined set {#1...., Ema VI...., Fm } By construction, this ensures that the distribution satisfies the symmetry property OFw,u. dpw,y.a
the observed data. Therefore, it is natu-
Directional Sampling. The formulation of Frw. in Equation (15) also includes a directional component. Inspired by the data-dependent design of the intersection point distribution, we propose

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analogous enhancement for the directional distribution to go beyond simple uniform sampling over gd-1. To sample informative directions, we randomly select a source point ia and a target point yj, and construct direction vector as follows:
(1b 5 (z, - m3)) (z,-
+ (2, (2--
wi)l 10,-(( E E
(17)
where v ~ 1) is a direction sampled uniformly from sd-1;; ; U({+1]) is random sign; i.j u(f1..... m]) are indices selected uniformly at random, independently; and > scaling parameter that controls how strongly the direction is biased toward the vector (z; Mj ). The resulting directional distribution on gd-1 is denoted by dir,p,. Using this, we define the enhanced sampling distribution on tree systems as:
FW. N(x", Ta) @ (Fdir,...)) @k
(18)
The resulting TSW discrepancy that uses this improved sampling strategy is defined by:
FW-TSW" (R, = J WI(HT, UT) do a. (T).
(19)
Remark 4.2. The random sign U({+13) in Equation (18) ensures symmetry of the directional distribution, i.e., dir, -di, Therefore, the sampling distribution satisfies Tww, Fw.ws
Metricity of FW-TSW. We examine whether FW-TSW satisfies the standard properties of a metric.
Theorem 4.3. Both FW-TSW and FW-TSW* are semi-metrics on the space P(RR). In particular, they satisfy non-negativity, symmetry, and the identity of indiscernibles. Moreover, they satisfy the following quasi-triangle inequality: for any μr, H2. Hz P(R)).
(W1, & FW-TSW pi (1i, Ps) FW-TSW, po (122, 13),
(20)
where the intermediate discrepancy term is defined as
FW-TSW pi (H.V) = / WI(HT, UT) do (T).
(21)
The same properties hold for FW-TSW".
Invariance under Euclidean Transformations. Since the proposed discrepancy operates on prob- ability measures defined over Rd it essential to analyze its behavior under transformations from the Euclidean group E(d). For context, both the classical 2- 2-Wasserstein distance and the Sliced p- Wasserstein distance are known to be invariant under Euclidean transformations. We confirm that this invariance property is preserved in our setting as well.
Theorem 4.4. FW-TSW and FW-TSW" are invariant under Euclidean transformations Rd.
Boundedness. We derive an upper bound related to the proposed FW-TSW discrepancy. Unlike prior TSW variants (Tran et al., 2024e; 2025a), where positional information is uncontrolled and bounds difficult to obtain, the geometric median in FW-TSW allows for tractable bound under mild conditions.
Theorem 4.5. Let μ1, M. P(Rd) be two probability measures. Consider the function
((v) /0)=/11 u12-- 1l
+ [ ||- ul(dd for all v E IR...
(22)
which associated with the joint Fermat- Bermer-Weber problem of μ and v. Let v² = arg mingeRs f(v) be the geometric median of the combined support of and v, and define the sampling distribution Frw,p.v p.u = 8yr 3 (S(SS-1888 where 8 is the Dirac measure centered w², Then, we have:
f( Wi (UT, do (T) S ls W2((,,)) + k(k 1))
2/
f(0*).
(23)
d
2
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Table 1: Average Wasserstein distance between source and target distributions over 5 independent runs on the 25 Gaussians dataset. We use 100 projecting for all methods.
Step
Methods
300
1000
1500
2000
2500
SW Bonneel al. (2015)
3.68e-03 + 1.3e-03
2.42e-03
8.0e
2.13c.03
9.0e-04
1.69e-03 9-8.4--04
1.0le003 = 9.5e
SWOG Maley al. (2023)
7.67e-444 +: 1.46-13
4.85c-06
5.50-66
2.91e-06
2-4e-06
2.72e-06
5.3e-6
2.9e-066 5.7e-06
LCVSW Luong al. (2024)
1.51e-03 +: 1.1e-03
L/40-03
1: 8.7c-01
7.81cc01
1: 5.6c-01
5.73c-01
4: 6.3c-01
6.81c-01
0.9--0117.9:000 7.9c-01
TSW-SI. Tran al. (2024p)
L.12448 + 9.7e-04
1,37e-66
8.7c488
1407c-06
4e8e-08
9.13e.07
5.26-88
8.76--07 1.1
D6-TSW Tran ot al. (2025k)
3.42e-433 1: 7.9c-04
1.55c-06
1.2c-07
1.10e-05
9.20-08
9,50c-07 6.108
8.55-07 5.6c-08
D&-TSW Tran et (2025a)
2,70e-433 + 9.0e-04
1.79-006
2.0e-477
1.25o-00
9.7e-08
1.14e-06
5.6--88
1.05-06
4.84-88
FW-TSW (ours) FW-TSW" (ours)
2,40e-403 + 8.9e-04
1.51e-06
1.4c-077
1.03c-06
1.0e-17
9.18e-07
8.00-00
2.6e-08
2.59e 4: 9.3e-04
1500-06 8.9e-08
1.11e.06 6.6e-08
9.04e407 1.1c-07
8,29c-07 4.7c408
Computational Complexity. Let n and m denote the number of support points in two discrete measures μ4, E P.(R), with > m. The standard Sliced Wasserstein (SW) distance has a computational complexity of OLn log + Ldn), where L the number random projec- tions (Bonneel et al., 2015). More recent approaches, such Tree-Sliced Wasserstein (TSW) and its variants TSW-SL (Tran et al., 2024e) and Db-TSW (Tran et al., 2025a), exhibit a complexity of O(Lkn log + Lkdn); here, L represents the number of sampled trees and & denotes the lines per tree. For FW-TSW, the complexity increases to OLLn logn + Lkdn + Tnd). incorporating an additional Tnd cost for approximating the geometric median via Weiszfeld's Algorithm (where T is the maximum iterations). FW-TSW" further incurs an extra (Ikd) term for generating ran- dom paths. Notably, these additional costs for FW-TSW and FW-TSW" contribute negligibly the overall computation time, as detailed in Appendix D.I.
EXPERIMENTAL RESULTS
In this section, present series of experiments involving Gradient Flows, Topic Modeling and Diffusion Models to assess the effectiveness of FW-TSW and FW-ISW*. Additional experiments on point cloud and MNIST-like images are provided in D.2.
5.1 GRADIENT FLOW
This task employs gradient- based optimization to minimize the discrepancy between a time-evolving source distribution μ4, originating from No. and a fixed target distribution v. The evolution is gov- erned by the differential equation 0,p, -Vp, D(u,, v). In this equation, D(44, v) is a distance metric (e.g., SW, Db-TSW, our FW-TSW and FW-TSW*).
We evaluate our proposed methods, FW-TSW and FW-TSW", on the 25 Gaussians dataset. Table presents the average Wasserstein distance between source and target distributions over five runs, using optimal learning rates for each method (details in Appendix D.2). Performance is tracked at steps 500, 1000, 1500, 2000, and 2500. While SWGG initially exhibits the lowest distance (at step 500), Db- Ob-TSW, Db-TSW FW-TSW, and FW-TSW* demonstrate steady improvement, eventually outperforming SWGG. Notably, from step 2000 onwards, FW-TSW and FW-TSW" yield the best results, with FW-TSW* being the best overall at step 2500.
5.2 TOPIC MODELING
In this experiment, we evaluate the efficiency of our proposed TSW distance for topic modeling (Blei et al.. 2003). Topic models are commonly framed as VAEs (Srivastava & Sutton, 2017), with an objective combining reconstruction and -ll-diveeeeece terms. Following Nan et al. (2019); Adhya & Sanyal (2025), we replace the KI. term with our TSW objective inf p. (Ol)) [CE(x. *)]+ A FFWTTSW(g,, (6), (())) where CE is the cross-entropy between original x and reconstruction & (0), with encoder oo and decoder tb. We compare FW-TSW-TM and FW*.TSW-TM against SW- and bassed baselines. Performance is measured by topic coherence Cv (Röder et al., 2015). As shown in Table 2, our methods achieve higher coherence over SW and TSW variants.
5.3 DIFFUSION MODELS
This experiment investigates training denoising diffusion models for unconditional image synthesis. Inspired by Nguyen et al. (2024b), we integrate Wasserstein distances into the Augmented Gener- alized Mini-batch Energy (AGME) loss function of the Denoising Diffusion Generative Adversarial

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Published as a conference paper at ICLR 2026
Table 3: FID scores and per-epoch training times of DDGAN variants for unconditional generation on CIFAR-I0.
Table 2: Average topic coherence CV across datasets DBLP, M10, and BBC. Higher is better.
Timo/Epoch(s)
Method
DBLP
MIP
BBC
DDGAN Xiso al. (2021)
3.64
72
SW-DD Nguyen (2024h)
IDA (Blci at al., 2003)
0:557
0.457a0 2.
DSW-DD Nguyen et al. (20246)
2.88
498
PrddLDA (Srivastova Sutton, 2017)
0.488-2002
0,088
EBSW-DD Nguyen et al. (2024b)
2.87
76
WTM (Nan sl., 2019)
0.496
0.403gn
RPSW-DD Nguyen
(20244)
2.82
SW-TM (Bonneel et al., 2015)
0.700 0.
IWRPSW-DU DD Nguyen al. (2024b)
2.70
77
RPSW-IM (Sguyen al., 2024b)
0. 0.420
0.47220
0.775
FRRPSW-TM (Nguyen
2024b)
0.416
0.492,0
0.777400m
TSW-SL-DD
(2024g)
DI-TSW-DD Tran et (2025a)
2.601
84
TSW-SL-TM
2024e)
D6-TSW-DD' Tran (2025a)
2.53
Db TT-ISW-TM (Tras et al., 2025a)
0.441 ab.ase
0.468a0mm
a.787
FW: WWW.SSW-IB TM lours)
0,006 g.005
0.79220041 1.
FW-TSW-DD (ouns)
2.336 0:003
FW* MNW-TSWTMM (ours)
0.511 L6300
00502
FW-TSW"- MDDSWWWDDOCO (ours)
2,315 0.002
87
Network (DDGAN) (Xiao al., 2021). Our proposed methods, FW-TSW-DD and FW-ISW"-DD. are benchmarked against Sliced and Tree-Sliced Wasserstein-basse DDGAN variants, with results detailed in Table 3. Details can be found in Appendix D.4.
As shown in Table 3, our proposed methods, FW-TSW-DD and FW-TSW*-DD -DD, achieve signifi- cant FID score improvements over all baselines. Notably, they surpass the current sate-of-the-art OT-based competitor, Db-TSW-DD= (Tran et al., 2025a). by substantial FID margins of 0.194 and 0.215, respectively. Furthermore, our methods achieve these improvements with training times com- parable to existing tree-sliced techniques, highlighting their practicality for large-scale applications.
CONCLUSION
In this paper, we introduce the Fermat-Weber Tree-Sliced Wasserstein (FW-TSW) distance, a novel variant of the Tree-Sliced Wasserstein (TSW) framework inspired by the classical Fermat-Weher problem. By leveraging Weiszfeld's algorithm to sample intersection points in the tree structure, FW-TSW captures both positional and directional information through a data-dependent sampling scheme. We analyze key properties of FW-TSW, including semi-metricity, Euclidean invariance, boundedness, and computational efficiency. Empirical results on gradient flow and Diffusion Model training demonstrate improved performance with minimal overhead. A key limitation, shared with other TSW variants, the lack of explicit transport maps. Future work may address this by devel- oping tree-sliced frameworks that produce transport plans.
ACKNOWLEDGMENTS
We thank the area chairs and anonymous reviewers for their comments. TL gratefully acknowledges the support of the JST-BOOST program (FY2025), JSPS KAKENHI Grant number 23K1 1243, and Mitsui Knowledge Industry Co., Ltd. grant. TT acknowledges support from the Application Driven Mathematics Program funded and organized by the Vingroup Innovation Fund and VinBigData.
This research project is supported by the National Research Foundation Singapore under the AI Singapore Programme (AISG Award No: supported by the Ministry of Education, Singapore, under the Academic Research Fund Tier This research / project is also supported by the
This research project is
(FY2023) (A-8002040-00-00, A-8002039-00-000 NUS Presidential Young Professorship Award (A-0009807-01-00) and the NUS Artificial Intelli- gence Institute-See Seed Funding (A- 0-800062-00000),
Ethics Statement. Given the nature of the work, we do not foresce 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 D and the Appendix. All datasets that we used in the paper are published, and they easy to access in the Internet.
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LLM Usage Declaration. We large language models (LLMs) for grammar checking and cor- rection.
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TABLE OF NOTATION
Rd
d-dimensional Euclidean space
12 gd-1
Euclidean norm
(d - -i-dmennsional hypersphere
unit vector
disjoint union
(X(X)
space of Lebesgue integrable functions on X
P(X)
space of probability measures X
M(X)
space of measures on X
p.v
measures
8())
1-dimensional Dirac delta function
U($
uniform distribution on gd-1
C)
space of continuous maps from X to Y metric in metric space tree metric Euclidean group of order d p- Wasserstein distance
d(:,))
dr(:.))
((d)
W,
SW,
Sliced p- Wasserstein distance (rooted) subtree
T
tree system
I.
number of Monte Carlo samples
number lines system of lines or a tree system
Ra
Radon Transform Systems of Lines
Ax-1
(A: -)-dimmnsional standard simplex splitting map tuning parameter space of tree systems
0, dpw, Opw, o.w dir
distributions (components of) space of tree systems normal (Gaussian) distribution
N
u
uniform distribution
8
Dirac delta distribution
e
threshold Weiszfeld's algorithm.
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Appendix of "Revisiting Tree-Sliced Wasserstein Distance Through the Lens of the Fermat-Weber Problem"
Table of Contents
A Background on Tree-Sliced Wasserstein Distance in Euclidean Spaces
17
A.I Tree System Construction
17
A.2 A Generalized Radon Transform over Line Systems
18
A.3 Tree-Sliced Wasserstein Distance in Euclidean Spaces
18
General Formulation for the Fermat-Weber Problem
18
Theoretical Proofs
19
C.I Proof for Theorem 4.3
19
C.2 Proof for Theorem 4.4
20
C,3 Proof for Theorem 4.5
Experimental Details
23
D.I Runtime and memory analysis
23
D.2 Gradient Flow
24
D.3 Topic Modeling
26
D.4 Diffusion Models
29
D.5 Hardware settings
30
Broader Impacts
30
A BACKGROUND ON TREE-SLICED WASSERSTEIN DISTANCE IN EUCLIDEAN SPACES
This section revisits the fundamental components of the Tree-Sliced Wasserstein (TSW) distance, formulated over tree systems embedded in Euclidean spaces. For completeness, we summarize key definitions and core mathematical formulations. Readers are referred to Tran et al. (2024e; 2025a) for detailed proofs and extended exposition.
A.1 TREE SYSTEM CONSTRUCTION
A line in Rd is represented as tuple (a,, Idd x Sd-1, where r denotes reference point and 0 is a direction. The line is parameterized by + L- for l E R. We denote a line by / = (æ), 0z)- point on this line written either as (x., as (tz, depending on whether we refer to the point in ambient space to its parametrization along I, respectivell. A system of & lines in R is an element of the product space (I&. x &d-1)* abbreviated as T. An element E T denotes a specific configuration of & lines. A line system T is connected if the union of all lines in T forms a connected set in Rdd A tree structure can be enforced by removing selected intersection points, so that any two points on the resulting configuration are connected by unique path. The term free system reflects the property that any two points are connected via unique path, akin to trees in graph theory. Using preserved intersections,
build topological tree system by coherently gluing segments of IR via disjoint union and quotient topology (Hatcher, 2005), resulting in a space endowed with a valid tree metric.
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A.2 A GENERALIZED RADON TRANSFORM OVER LINE SYSTEMS
Let L (RR) denote the space of integrable functions on Rd, Given a system of lines T E T, define
(T) as the space of functions such that ||/I|T Σiet SR (1,, 1) dl, 00.
The (k: 1)-dimensional simplex is defined as 4 = {(at)tTT E R a; 20.0, Eier ao = 1}.
Let C(R x T, -1 denote the space of continuous functions, called splittngg maps, from R T to Ak-1- Given splitting map and f E LL' (IR"). define the projection operator:
R7 = S f(v) - aly, T) 8 (tz - (y zı, 01)) dy.
(24)
where (xz, 81) specifies line and ò is the Dirac delta. This operator maps f to function on T, the union of the lines. Extending over all T E 'T define the Radon Transform on Tree Systems by
R : L
(T). f (R4
(25)
TET
If a is invariant under the Euclidean group E(d), then R is injective.
A.3
WASSERSTEIN DISTANCE IN EUCLIDEAN SPACES
Let v v E P(TRO) probability measures. For a tree-structured line system E and an E(d)- invariant splitting map let R μ and RJ be the pushforwards of and 1, respetivelly Equipped with the tree metric dr, we compute the I-Wasscrstein distance:
W W (R Rgv).
(26)
The Tree-Sliced Wasserstein (TSW) distance (Tran et al., 2025a) is defined as:
TSW = War. (R RFv) do
(27)
where a is a probability distribution over T. Though the notation omits explicit dependence on a, T, and ar, the metric depends on all three.
Remark A.1. If tree systems are reduced to single lines, TSW recovers the classical Sliced Wasser- stein distance.
E(d)-Invarrie Splitting Maps. Let E Rd and T T. Define the Euclidean distance from to line as d(z, T) inf || yll2. This function is invariant under E(d). A practical choice for a the softmax:
axx 7) = softmax((E d(a, Dihee),
(28)
with & 0 controlling the sharpness of the distribution over lines.
Remark A.2. Equivariant neural networks (Cohen & Welling, 2016) hard-code task symmetries into model architectures, yielding stronger generalization and improved sample efficiency. They have demonstrated broad empirical effectiveness in trajectory prediction (Walters et al., 2020), robotics (Simeonov et al., 2022), graph learning (Satorras et al., 2021; Tran al., 2024b), Optimal Trans- bost-based approaches (Pham et al., 2026; Tran et al., 2026b; 2025e; 2024f; 2025c), equivariant metanetworks (Tran et al., 2026a; Vo et al., 2025; Tran et al., 2024a; 2025b), and weight-space anal- ysis (Tran et al., 2025f), Overall, exploiting equivariance systematically improves accuracy, data efficiency, and robustness to distributional shifts.
B GENERAL FORMULATION FOR THE FERMAT- WEBER PROBLEM
In this section, we provide the background on the Fermat-Weber Problem and Weiszfeld's Algo- rithm. We begin with the continuous formulation of the Fermat-Weber problem:
arg min ||x 2112 dd(z),
(29)
vER JRd
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Discrete Approximation via Monte Carlo. Let {>r 1コ-1 be i..d. samples from X, with associated weights \(z)). A Monte Carlo approximation of Equation (29) is
v* R arg min wERd
Σ A(x) ||ا v/12-
(30)
Define the weighted geometric median objective
F(N) = Mxx' | w112-
==1
Since F(v) is convex (but non-differentiable the sample points), employ the classic Weiszfeld
iteration to compute its minimizer.
Weiszfeld's Iteration. Denote by u(t) IR the estimate at iteration t. The standard update is
" Σ
X(x)) v
a
X(x²) x 11
(1++1) -T(((o)>
(31)
|1(()) x2 12
(
provided y(() * x for all j. If (+)) = x for some index k, one may either terminate (since F is minimized that sample) or apply the more robust variant below.
Handling Coincident Iterates. To avoid the "sticking" phenomenon when e(9) exactly equals sample x", one introduces a small perturbation or uses the following safeguarded map:
x(x))
ビコ=1 10 2
+ x² x
T(v)
Σラ=
N(w)
(32)
|0-20112
xh provided Ejje 1(x5) 11 x3 112
x
0,
and then define
(1+1) = =
((1)(())))))) -1- (()(0) (( + 3(v(0))
(33)
where B(v) E [0, 1] chosen ensure descent F.
Convergence and Remarks.
Under mild conditions (no three points colinear, positive weights), the sequence {u(!)) converges to the unique geometric median (Weiszfeld & Plastria, 2009).
In practice, when the Monte Carlo weights X(z)) are noisy, Weiszfeld's algorithm can be sensitive. such cases, gradient- methods (e.g. subgradient descent) with a suitable smoothing may be preferred.
Weiszfeld's 's iteration typically converges in O(1/t) rate and is computationally inexpensive per step, making it effective when the sample size n moderate.
C THEORETICAL PROOFS
In this section, we provide proofs for results stated the paper.
C.I PROOF FOR THEOREM 4.3
Proof. We recall the Fermat-Weber Tree-Sliced Wasserstein distance between y and in P(R is defined by
FW-TSW(H4 WI(T,, (T)
(34)
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Non-negativity. Since W1 is valid distance, we have Wi(HT, uT) 2 for all T T. It implies that
FW-TSW(H, v) W
2 0.
(35)
Symmetry. Given J, V E PP(R). By design, we doww, = doww.w.- Moreover, since W1 is valid distance, have W1 (VT, W1 WI(TT,HT) It implies that
/ (T)), (T)
(N-TSW(A,P)
J WI(UT, (elurrpp)doow...o.
FW-TSW(v.j))
(36)
The identity of indiscernibles. Assume that FW-TSW(H.. (u, 0. Since dwww is continuous distribution on T, we have Wi (UT, (UT) = for all T E T. Since the Radon Transform on Tree Systems is injective (refer to Tran et al. (2025a)), it implies that μ.
Quasi Triangular Inequality. For P1, Ps E P(R"), we have
FW-TSW (1Ss(WM,,A 12)
FW-TSW (110
W1 (i(((r)r, (U2) do
<
Wi ((a)));(ua)7) + (((()) (Wa)T) dow.), (T)
[
Wa (I(()))) (US)T) do (T) J (2((u2)T (1))))) do (T)
FW-TSW, Piges (21, FW-TSW 12. (22)
(37)
Therefore, the proof is completed.
C.2 PROOF FOR THEOREM 4.4
Proof. Note that, the Fermat-Weber problem preserves Euclidean transformations, which means
argmin argmin J(d |||- ||==>/lad(sttt v1(2(((tA)(() #ERd
/ I 9 (()AA)))
#ERd
/ 11g- - dX(y)
argmin
#ERd
/
9
argmin
||
2ll2 (2(()
(38)
mERd
JRd
It implies that drw.u. is equivariant under the action of Euclidean group. Not that, since the Radon Transform on Tree Systems also equivariant under the action of Euclidean group, leads to the induce distance FW-TSW is invariant.
Recall construction of the distribution air, as follows:
1b + S (z, (asa-v)))) x (x+ - 11)la E gd--
(39)
where vi ~ (Sd-11) is a direction sampled uniformly from sd-1, U({+1}) is random sign; j.j ~ U(f1...., m)) are indices selected uniformly at random, independently; and Ç > is a scaling parameter that controls how strongly the direction is biased toward the vector (z, yz). By design, a is equivariant. Thus, the induce distribution Fw on T is equivariant. By the same argument above, FW-TSW" is invariant.
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C.3 PROOF FOR THEOREM 4.5
Proof. Following the definition of Tree-Sliced Wasserstein, we can write
inf / dT (z), TEP TXT
W WI(UT,
(40)
where P denotes the set of couplings on P(T) P(T) and d denotes the distance between two nodes the tree system T. It should be further noted that, due to the way we have constructed T, the distance between two nodes can be explicitly calculated. More specifically, the formula for dr can be derived as follow
x 0g v/ 0.0
for x.y belongs the same edge 0:
dr dz v)
T 0, - u To + by To - 05 for x,y belongs to edge 0g and 0,, respectively
x
(41)
Where is denoted the only vertex of the tree system.
Next, we derive an inequality that upper bounds the infimum over couplings on the tree space defined in Equation (40) by infimum over couplings on Rd thereby simplifying the problem and enabling more tractable analysis in subsequent step.
Claim 1. Denote as the set couplings on Rd x Rd, we show that
f dr ()T TEP
inf
k
< inf rER JERRRdxxe Rdd
Σ 2=1
x 0, y
+
Σ
(|x (lxT0: 0, -,,cl++ 0g,el)
dr(z,
(.E(()
(42)
Proof for Claim 1. From the formula of dr (x, 3) in Equation (41). One can show that
dr v)
x T 0, - +o
Σ
(1x O, - F.ccl + lyT -8j.cl) Var, y E Rd
i=1
i
(])((A]
Hence, for every E R, there exist P such that
[ dr (x(,,))((x.))
x 02 y 00 Σ (lx (lxTO: -0,cl+1yT0, 0z,cl) dr dr (43)
Thus, if we denote 7 as the solution to the right-hand side infimum of Claim 1, it implies inf [
TEP TxT ()(
TXT
<
TT:
501++ Σ
(|x 0, T. + 1y -lv ),cl) di(a.
JRARR
iJE[M]
inf
Σ 0 y TT
0
Σ
(1x 0, 8.cll + 1o,,
+.c1+luTo,
-yell| dr(z.v).
0zel))
FER. RR. Rdd
(
(44)
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This completes the proof for Claim 1.
Additionally, since T a coupling on IRd x R the right hand side of Claim can be further simplified follow
inf FRR
x
y 0,1 +
Σ (lxT0, D,,cl v=0,, og,el) dr(a,
izj
1.jE[k]
inf teñ
Ao
Σ
(1x 1 vT0.1+ 10 - vb) dr(x,y)
JRdxRd
inf reft
An + (k: - 1)
[x - - + +1 - -b00i dr (z(2,))) v)
lx: --- 1s m)
+ U - 0100)ld (1,
inf reft
Ao + (k 1)
R
i=1
[ VTO, ((0gldu(e)) 1
inf reA
Au + (k: - 1)
xx TT - -e---Ialaae -[Blldu(=)
k
+ [/ luTo, u vT0gldu(u)
inf (Ao) + (Is -6-1) Σ
1xc 0, - vZ
reR
1=1
Rd
where
Ao = JRd Rd
LT y 0il
dr dr(x. 8).
(45)
Now, by applying the above calculations, Claim 1 and Equation (40) we can derive an upper bound for FW-TSW below
[ W1 WI )T) dōrw (T)
1
Σ To: 0: 1y.-yool -oil dr(s.)) dopW
inf FER RdXRd
(k 1)
loTo, - 0. x (x) x
RAXT
V RXXT To T TTold(( an)( x
(46)
To further upper bound Equation (46), introduce the following two claims, each providing bound for one of the terms in Equation (46).
Claim 2. It can be shown that
int I
Σ
1o 0,
dr(x, v) dorw... W2( 10-yy T0:
(47)
Proof for Claim 2. is trivial see that 1x To vT, S -11--=-ll v112 for all Thus,
inf J x
Σ
xx To
u vT,l1
dr dr(x.
S k inf /
||x
vll2d((e, (x-lvladre.v)) kW2(u,
(48)
fe RdxRd
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This directly lead to
h inf TE Rdx
Σ
x 0, y v0 dr(z. dorw S k (2(
(49)
which completes the proof for Claim 2. Claim 3. It can be proven that
(k UauXT ToTo O2 --- o.lddd( X
[ RUXT
v oTo,dd(v x Cf(e*),
(50)
where
20/4/2
1
k(k - 1)-
+ 1
2
r
2
Proof for Claim 3. We first do the following transformations to the left-hand side of Claim 3,
A = (k - 1) k i=1
(1er 1o To - vToollda x x x x)
Σ
U BOXT
TT v T,lld(( (x
+
JRd (frr (/pe 100 - v. (T)d(((x))
(1: 1)k
1x To - vTolde(s)) )d0) do
+ Rd
(k: - 1)k f (Jrr (Lee
To -- T0/d0) du((z)
To -- T000 dd(x)) ) do.(v)
+ Rd
(51)
Bd-=
Note that, for an E Rdd, one has:
f Jsan
T
d0
2n 4/2
r
1
||u//2
(52)
d+1
2
It implies that:
2x1/22
1 //c
(1ee || v1(2d)(())
{ 11 vll2du()) do.(v)
A k(k 1)
r
=C((v').
(53)
This completes the proof of Claim 3.
By combining Claim 1, Claim 2 and Claim 3 in addition with the primal formula in Equation (40), yield the result stated in Theorem 4.5.
D EXPERIMENTAL DETAILS
D.1 RUNTIME AND MEMORY ANALYSIS
We analyze the computational and memory complexity of the most expensive operations our proposed distance measures, summarized in Table 4. We also compare the runtime of our method
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Table 4: Complexity Analysis of FW-TSW and FW-TSW".
Distance
Operation
Description
Compatation
Memory
Projection
Matrix maltppicaaiin of poines and lines
O(Lknd)
Of nd) O 4 Lid ud)
FW-TSW
Distanc-buset weight
Distance calculation and softmax
O(Lknn))
splitting
Sonting projected coondintess
O(Ldningg))
O11im)
Weiszfeld's Algurihh
Approsimating geometric
O'Tnd)
([Tad)
Total
O.Lknd + Linloge Tid)
Of + Ltd +d
Projection
Matrix maltipliction of points and lines
O(Lknd]
O(Lki vo))
FW-TSW"
Discance-beee based weight
Distance calculstion and suftmax
O.Lknd)
0. + Lid + nd)
splining
Suring
Sorting projecsed coundinates
O'Linlognn
O[Lkn)
Weiszfeld's Algoritium
Approsinating Generating random patts
OTTui)
O(Tad)
Generating paths
O(Lbd)
OfLkd)
O..kou Mlo log Tod)
0, + Lik nd Tad)
120
1200
100
Memory usage (M4)
100
8C
500
1000
1000
500
1000
800
600
20
20000
33000
20000 30000
40000
50000
Number supports
Number supports
Figure Execution time and memory usage of FW-TSW.
with other approaches. In Figure 4, L 2500 trees and l: lines for the Tree Sliced Wasserstein variants. For the Sliced Wasserstein (SW) method, we set = 10000 projections.
Next, analyze the runtime and memory performance of FW-TSW and FW-TSW* by varying and d, using a single NVIDIA H100 GPU. We use = 2500 trees and = lines for all runs. We select (500, 1000, 5000, 10000, 50000} and E {10, 50, 100, 500, 1000).
Runtime. Figures 2 and 3 show that both FW-TSW and FW-TSW* scale linearly with the number and d. This is consistent with our complexity analysis.
Memory scaling. Figures and 3 present the memory usage of FW-TSW and FW-TSW*. All methods exhibit linear scaling with both the number of supports n and the number dimension d, consistent with the theoretical complexity analysis.
Weiszfeld's Algorithm. We empirically determined that setting the maximum iterations for Weiszfeld's algorithm to T 100 provides substantial performance gains. This value is used as the default in our experiments unless otherwise noted.
D.2 GRADIENT FLOW
Gradient Flow Point Cloud. We perform point cloud interpolation experiment evaluate our method against two baseline approaches: SW (Bonneel et al., 2015) and Db-TSW (Tran et al., 2025a). Both the source and target point clouds are sampled from the ShapeNet Core-55 dataset (Chang et al., 2015), as illustrated in Figure 5. Following the experimental setup in (Nguyen et al., 2023), we use gradient approximation techniques conduct Fuler integration over 500 iterations with a step size of 0.01. Table 5 reports the Wasserstein distances between the interpolated point cloud and the target shape at iterations 100, 200, 300, 400, and 500, averaged over 5 runs. All methods use 100 projections. For TSW variants, use 25 trees and lines. Results shown in Table
Gradient Flow on Images. The experiment synthetic MNIST-like images aims to learn map- ping from noise distribution to a target distribution of 16 ordered digits, concurrently learning both the image content and their correct sequence.
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120
1400
100
1200
Memory Usage IMB)
500
2000
500
1000
1000
Runtime
80c
600
20000
30000
30000
40000
50000
supports
supports
Figure 3: Execution time and memory usage of FW-TSW".
SW
D8-TSW
00
T5W-51
60
(mos)
20-
Supports
Figure 4: Runtime Comparison of FW-TSW and other methods.
The target distribution consists of 16 unique samples. Each sample represents grayscale image of a digit (0 through 15). These are synthetic 28 x 28 images, flattened to 784-dimensional vectors. A scalar positional encoding. i/16 (where is the sample's index from 0 15), is appended each flattened image vector. This results in each target sample being a 785-dimensional vector (28 x 28 1). The set of these 16 vectors forms the target distribution v. The initial source distribution μoo also comprises 16 samples, each initialized as 785-dimensional vector from Gaussian noise. The discrepancy between the evolving source distribution μ4 and the target distribution minimized using the Adam optimizer with a learning rate of 1 x 10 10-3 applied to all methods.
Wasserstein (TSW) variants, including our proposed FW-TSW-DD and FW- TSW*-DI the number of sampled trees (for TSW variants, T.) is set to 250 and the number of lines set to à: 4 per tree.
For all Tree-Sliced
To evaluate performance, the 16 samples of the current source distribution μ4 are first sorted based on their learned positional encoding values (the last dimension each 785-dimensional sample). After sorting, the pairwise L2 distance is computed between the image-only part (the first 784 dimensions) of these sorted reconstructed samples and the corresponding ordered ground truth target images.
To account for variability and the potential bi-modal nature of the Le metric (due to ordering success or failure), each experimental setup for each method is repeated 100 times with different random seeds. The reported L2 values in Table 6 are percentiles (e.g., P25, P50/Median, P75) derived from these 100 runs.
Table 6 shows that at epoch 3000, FW-TSW and FW-TSW* consistently achieve lower Lg values compared to Db-TSW. For instance, the median L2 for FW-ISW and FW-TSW* is 0.50 and 1.60, respectively, while Db-TSW yields median L of 5.67. Figure visually substantiates this, illus-
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Methods
Step 100
Step 200
Step 300
Step 400
Step 500
SW
2.07e-04
1.39c-04
1.31c-04
1.35c-04
1.28c-04
Db-TSW
4.63e-04
7.54e-05
3.05e-05
1.90e-05
1.37e-05
FW-TSW
4.56e-04
7.17e-05
2.90e-05
1.67e-05
1.30e-05
FW-TSW*
4.93e-04
7.23e-05
2.74e-05
1.65e-05
1.15e-05
Table 5: Comparison of methods across training steps.
SW, w2:
Db-TSW, 155e-55
FWATSW. W2: 1.31-005
FINTSW", w2: 9.55e-06
Target
Figure 5: Point-cloud interpolation with Wasserstein distance at step 500.
trating that FW-TSW and FW-TSW* successfully learn both the image content and their ordering by epoch 3000. contrast, Db-TSW still results in blurry images and incorrect ordering.
Gradient Flow synthetic data. Table presents the performance of our proposed methods alongside various baselines on the 25 Gaussians dataset. The low standard deviation observed for FW-TSW and FW-TSW* highlights their stable and consistent convergence behavior. For the Tree- 25 trees and k lines. We set L = 100 projections for other sliced methods. All models are trained for 2500 steps using the Adam optimizer. The source and target distributions contain 500 samples each.
Sliced Wasserstein (TSW) and its variants, we employ
To ensure fair comparison, perform ablation study over range of learning rates for each method. The results reported the main table correspond to the best-performing learning rate for each method. Complete results of the ablation study shown in Table 7.
D.3 TOPIC MODELING
In this section, we present the details of our Topic Modeling experiments.
Topic Modeling. Topic modeling (Blei et al., 2003) is long-standing task Natural Language Processing that aims to discover latent thematic structures within document corpora. Typically, documents x are represented using a bag-of words model, while the topic proportions @ are modeled discrete distribution over topics. Recent advances utilize variational autoencoder (VAE) to address this task, where an encoder network i estimates the posterior distribution 9,p (0)x), and a decoder network 1 reconstructs documents as & (0). The objective function for training such
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Table 6: Le distances various percentiles (c.g.. P25, P50/Median, P75, where 'P" denotes per- centile) comparing Db-TSW with our proposed FW-TSW and FW-TSW" on the gradient flow task for synthetic MNIST-like images, evaluated selected training timesteps. These L2 values repre- sent the distance computed between pixel data of source (reconstructed) and target (ground truth) images, after both have been sorted using positional encodings. Lower L2 indicate better perfor- mance. For the Step 3000 metrics, our methods demonstrate superior performance.
Steps
Methods
1000
2000
25500
3000
4000)
La (P50)
La (P50)
La (P50)
La (P25)
L2 (P50)
La (P75)
Le (P50)
Db-TSW
12.92
10.37
10.12
1.27
5.67
6,58
0.05
FW-TSW (Ours)
12.91
10.26
10.01
0.26
0.50
4.13
0.05
FW-TSW" (Ours)
1.60)
5.12
0
1
3
2
4
0
1
3
1
2
4
5
6
7
4
3
5
6
4
5
7
4
5
6
8
9
10
11
8
10
11
8
9
10
11
9
10
13
14
12
14
15
12
13
14
15
13
14
15
Ground Truth
Db-TSW (L2 = 5.67)
FW-TSW (I22 = 0.50)
FW-TSW" (I2 = 1.60)
Figure 6: Image reconstruction and ordering by gradient flow methods on synthetic MNIST-like digits (epoch 3000). Ground Truth (far left): numbers 0-15 ordered left-to-right, top-to-bottom. Other panels show reconstructions reflecting each method's median L2 performance. Our proposed FW-TSW and FW-TSW" produce correctly ordered images, unlike Db-TSW's misordered results.
models is usually given by
C = E (((x)((0)x)) [CE(x, x)] + (KL(((()))lI((O)
where CE(.,) is the cross-entropy reconstruction loss and KL(-II-) is the Bailback-Leibblee diver- gence regularizing the posterior to match the prior p(0).
In experiment, we replace the KI. divergence term by FW-TSW and FW*-TSW. We bench- mark these against other sliced Wasserstein methods Fuclidean setting (Nguyen et al., 2024b; Tran al., 2025a; 2024e; Bonneel al., 2015), well classical topic modeling approaches such as Latent Dirichlet Allocation (LDA) (Blei et al., 2003), ProdLDA (Srivastava & Sutton, 2017), and Wasserstein Topic Model (WTM) (Nan et al., 2019).
Datasets. We evaluate our methods on three widely used benchmark datasets for topic modeling:
M10 (Pan et al., 2016): subset of the C digital library, consisting over 8,000 academic documents across 10 research topics.
DBLP (Pan et al., 2016): A bibliographic dataset in computer science, containing more than 50,000 documents from research domains.
BBC (Greene & Cunningham, 2006): A collection of over 2,000 news articles published by the BBC, covering topical categories.
For preprocessing, convert all text to lowercase, remove punctuation, perform lemmatization, filter out short words (fewer than 3 characters), and discard short documents (fewer than 3 words). Detailed statistics of the preprocessed datasets reported in Table 8.
Metrics. A common approach to evaluating topic models involves assessing two key aspects: topic coherence and topic diversity. We adopt the Cy (CV) measure which has been demonstrated to strongly align with human judgment (Röder et al., 2015), as our primary coherence metric. For topic diversity, the IRBO metric (Terragni al., 2021), widely accepted measure capturing the
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Table Average Wasserstein distance between source and target distributions over runs on the 25 Gaussians dataset with different learning rate rj = {0.001, 0.005, 0.01, 0.05, 0.1}. All methods use 100 projecting directions.
Iteration
Method
7
500
1000
1500
2000
2500
0.001
4.20e-01
6.2e-03
1.53e.01 +: 2.3e-03
7.83e-02 +: 03
5.04e-02 = 1.9e-03
3.61e-02 + 4e--03
0.005
4.02c-02
3.0e-03
1.82-02 1.8c-03
1.08c-02 + 1.3c-03
7.9cc-3 9.2c-04
6.53c-03 + 1.2c-03
SW
0.01
1.92c-02
+ 2.0e-3
8.68c-03
= 1.5c-03
6.83e-03 + 9.2c-04
5.93e-03 = 1.5c-03
5,51c-03 # 1.4c-03
6.16e-03
7.3e-04
4.95.03 04
4.48c-03
1.1e-03
3.9%-03 1.4e-03
3,47e-03 1: 1.5e-03
0.1
3.65e-03
+ 1.36-03
2.42o-03 = 8.0c-04
2.13c-03 = 9.0c-04
1.69c-33 = 9.8c-04
1.01c-03 = 9.5c-04
0.001
6.48c
2.6e 02
3.49 4.4c 02
1,73e 1. 5.9e
6.77e 02 4.3e 02
2.32e 2.
SWGG
0.005
2.06e-02
1.fe-02
8.26e-04 1.6e-03
4.50c-06 +: 4.2e-06
5.29c-06 8.6e-06
3.42e-06 + 065
0.01
7.67e
1.4e
4.85e 5.5c
0.91e 1. 2.4e 06
2.72e 06 5.3e 06
2.91e 06 1 5.7e 06
0.05
2.26e-05
8.5e-06
2.93e-05 1.3e-05
4.61e-05 + 1.7e-05
5.81e-05 2.2c-05
9.15e-05 + 2.5e-05
0.1
5,39c-05
+ 1.2c-05
7.03e-05 = 2.4c-05
1.38c-04 # 1.6c-05
1.98c-04 = 3 3.4c-05
5.33c-03 # 8.8c-03
0.001
3,47c-01
5.0c-03
6.99e-02 - 2.5c-03
2.35c-02 + 2.0c-03
1.25e-02 = 1.7e-03
9.04c-03 + 1cc-03
0.005
8,07c-03
1.5c-03
5.02e-03 = 1.0c-03
4,45c-03 = 9.1c-04
4.08c-03 9.2c-04
4.17c-03 = 9.2c-04
LCVSW
0.01
4.15e-03
+: 1.4e-33
3.77e-03 = 0.3
3.75e-03 1: 94
3.83e-03 1.0c-03
3.81e-03 1 9.2e-04
0.05
2.33c-03
1.6c-03
2.11e-03 = 1,3c-03
2.18c-03 = 1.4c-03
2.13c-03
1.3c-03
2.36c-03 = 1,4c-03
0.1
1.54c-03
1.1e-03
1.40e-03 = 8.7c-04
7.84c-04 1: 5.6e-04
5.73e-04 = 6.3e 04
6.84e-04 1 7.9c-04
0.001
3.50e-01
8.122-02 038 03
1.09 9.0e-04
2.73e-03 8.9e.04
2.16e.04 + 3.3e-04
0.005
1.12e.03
04
1.37e 1 8.7e 08
1.07e 1 4.8e 08
9.13e 07 5.2c 08
8.76 1 1.1e
SL
7.73e-06
07
4.73e-06 2.0-07
4.49e.06 2.6
4.11e.06
1.4e-07
3.76e 0.7
0.05
9,25c-05
4.5c-06
8.72c-05
= 4.8
9.03c-05
+ 2.1c-6
8.70c-05
4. 1c-06
8.76c-05
+ 9.2c-06
0.0c-05 3.87e-04
3.20e-04 3.8e-05
3.53e-04 + 5.4e-5
3.29e-04
3.3e-05
3.52e-04 + 2.0e-05
0.001
3.68c-01 + 5.0-03
L06x-01
3.4c-03
1.91c-02 + 1.1c-03
4.14c403 = 9.0c-04
3.44c-04 : 6.7c-04
Db-TSW
0.005
3.42e-03
+: 7.9c-04
1.55-066 = 1.2e-U7
1.10e-06 +: 9.2e-08
9,50c-07 6. 10
8.55e-07 1: 5.6e-08
0.01
8.49c-06
5.50-07
5.52x006 1.7c-07
4.90c-06 3.6c-07
4.50c406 = 2.5c-07
4,28c-06 + 3.3c-07
0.05
9,98e-05
+ 7.3e-06
9.75e-05 = 5.1
9.54e-05
+ 6.0e-06
1.00-044 1.1e-05
9,82e-05 1 9.3c-06
05
3.9e
2.9e.05
3.69e
3.63e 1
0.001
3.72e-01 1. 4.6e--03
1.06e 1 4.0
1.88c-02 1. 9.9c-04
3.40e 03 6.5e-04
3.61e-04 1 5.4e-04
0.005
2.70e-03
9.0e-04 04
1.79e-06 + 0.7
1.25e.06
+: 0.
1.14e 06 5.6e-08
1.03e-06 + 4.8e-
D6-TSW
0.01
1.36c-05
1.0c-06
7,95--06 7.3c-07
6.89c-06 6.5c-07
6.20e-06 3. Ic-07
6.89c-06 + 7.0c-07
0.05
1.22e.04
+: 06
1.13..04 + 5.2c-06
1.1
+ 063e
1. 8.3e 06
1.22e-04 + 1.3e.05
0.1
4.44c-04 4.7e-0
4.44o-04 4.0c-05
4.53c-04 + 7.1c-05
4.00c-04 4.9c-05
4.33c-04 8.1c-05
0.001
3.68c-01
:: 4.9e-33
1.06--01 = 3.6c-U3
1.96e-02 : 8.4c-04
5.09c-03 6.5e-04
6.98c-04 # 5.0e-04
FW-TSW
0.005
2.40c-03
8.90004
1.5c-00 = 1.4c-07
1.05c-06 = 1.0c-07
9.18c-07 = 1.-008
8:40c-07 + 2.6c-08
0.01
8.66e-06
6.0e077
5.40e-06 1.8c-U7
4.84c-06 : 4.0c-07
4.55e-06 2.3e-07
4.33e-06 = 3.3e-07
0.05
1.03e
1.0e 05
9.67e 5.6z
9,93e 6.9e 06
1.0le 04 8.0e 06
9.70e-05 1 0.
0.1
3,77c-04
4.0e-05
3.68c-04 = 4.6e-U5
3,57e-04 : 4.9c-05
3.72e-04 3.6e-05
4.04c-04 : 2.9c-05
0.001
3.68e 01 1: 03
1.06e :: 4.2e 03
1.90e 1: 1.6e 03
4.06e 03 1.3e-03
0.1 1: 1 1.6e
FW-TSW"
0.005
2.59c-03
9.3e-04
1.50=-06 8.9e-08
1.1 6.6e-08
9.04c-07 = 1.le-07
8.29e-07 + 4.7c-08
0.01
8.66e
1: 0.
5.79 06
: 1.9e-07
4.83e 06 1: 4.2c
4,27e 06 2.7e-07
4.10e:06 1: 1.5e-07
9,80--05
2.5c-06
9.50e-05 1.
1.03e-04 1.1e-05
9.09c-05 2.2c-06
9.73e-05
+ 688-06
0.1
4.29c-04
+ 5.9c-05
3.53c-04 = 2.6c-05
3.89c-04 = 2.9c-05
3.74c-04 5.2c-05
3.80c-04 = 2.3c-05
Table 8: Dataset statistics and hyperparameters.
Dataset statistics
Hyperparameters
Dataset
#Docs
#Labels
#Words
#Projections
Batch size
Dropout rate
DBLP
54595
1513
1000
512
0.2
M10
8355
10
1696
2000
64
0.5
BBC
2225
2949
8000
256
0.05
distinctness among topies. Intuitively, topic coherence quantifies how frequently the top words of topic co-occur within the same documents across the corpus, while topic diversity reflects the degree to which the topics are well-separated and capture different themes. We report the mean and standard deviation across 10 independent runs.
Training, We employ OCTIS (Terragni et al., 2021), a widely adopted framework for training and evaluating topic models. We adhere to the experimental settings outlined in (Adhya & Sanyal, 2025), employing a Euclidean latent space with a Dirichlet prior. Each model is trained for 100 epochs. The weighting hyperparameter is systematically varied over the interval [0.5, 10) in increments of
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Table Topic diversity by IRBO (1) on DBLP, MIO, and BBC.
Method
DBLP
M10
BBC
LDA (Blei et al., 2003)
0.837 =0.023
0.915
0.91 2.91720.11
Prodl DD (Srivastava & Sutton, 2017)
1.000
0.997 10.002
1.000
WTM (Nan et al., 2019)
0.953_g.003
0.81210.060
0.997
SW-TM (Bonneel al., 2015)
0.984
0.971
+0.008
0.998
RPSW-TM (Nguyen al., 2024b)
0.992 10.007
0.969 0.969.1.010 10.010
0.994_0..010
EBRPSW-TM (Nguyen al., 2024b)
0.991 10.007
0.96310.013
0.99910.000 10.003
TSW-SL-TM (Tran al., 2024e)
0.989=0.014
0.981
0.99922.00
Db-TSW-TM (Tran al., 2025a)
0.988_20.012
0..054
1. 1.000.20.00
FW-TSW-TM (ours)
0.944
0.812+0.088
0.998
FW*-TSW-TM (ours)
0.96 10.003
=0.033
0.9 0.913
0.998
0.5. For tree-based approaches, the number of trees is fixed at 100. All other training parameters detailed Table 8.
Topic Diversity. We provide topic diversity result in Table 9.
D.4 DIFFUSION MODELS
Diffusion Models. A prominent category of generative models, diffusion models (Sohl-Dickstein al., 2015; Ho al., 2020), recognized for their capacity synthesize high-fidelity data. This section outlines their fundamental mechanisms and sets the stage for the enhancements introduced by our work. The core idea involves a forward process where an initial data sample g(rr) system- atically degraded by the incremental addition of Gaussian noise across T discrete timesteps. This transformation formally described by:
T ( II (()))=1-))
with each step (=x in this sequence being a Gaussian transition defined as:
( = Na,;; / 1,
The parameters in this equation correspond to predetermined noise variance schedule.
Conversely, the generative aspect of these models lies in learning the reverse process: to denoise corrupted sample and recover the original data structure. This involves parameterizing the reverse transitions Po(2t-1lan) (xz) using a neural network with parameters O. The complete reverse process is given by:
T
po(zorT) p(ar) II po(
t=1
where each individual reverse step is also modeled Gaussian:
(0(#1_110;) = (2 (0( l), a)
The training objective is generally to maximize the Evidence Lower Bound (ELBO). This is tanta- mount to minimizing the Kullback-Leinh (KL) divergence between the true, but intractable, poste- rior q and the model's learned approximation (0( (xt), summed over all timesteps:
T Σ
L
C.
where denotes a constant term and KL(-II-) is the KL divergence.
Denoising Diffusion GANs. A primary limitation of standard diffusion models is their consider- able sampling latency, which can hinder their use in time-sensitive applications. Denoising Diffu- sion GANs (DDGANs) (Xiao al., 2021) were developed to mitigate this inefficiency. DDGANs
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Published as a conference paper at ICLR 2026
reframe each denoising step conditional generation task handled by multimodal Generative Adversarial Network (GAN). This architecture permits larger individual denoising steps, drastically cutting down the total number of required steps as few four. Consequently, DDGANs can achieve sampling speeds over 2000 times faster than their traditional counterparts, without a sig- nificant compromise in the quality diversity of the generated samples. The implicit denoising distribution within DDGANs is expressed as:
Pe (Tz-1 = / po( Go e)de, E~ NO,,1).
Originally, Xiao et al. (2021) trained the model parameters via an adversarial objective:
min Σ E Be(z Padc:
with Dude signifying the adversarial loss. However, Nguyen et al. (2024b) proposed an alternative by substituting this adversarial loss with the Augmented Generalized Mini-batch Energy (AGME) distance. For two distributions, l and v, and mini-batch size 1, the AGME, when employing a Sliced Wasserstein (SW) kernel and nonlinear function is given by:
IRd TR to define f(x) = (z, g(r),,
AGME (ss, v; GME (p, i²),
where jı = ful and E = faV. The underlying Generalized Mini-batch Energy (GME) distance (Salimans al., 2018) is formulated as:
GME (w, v) DE Py) [ED E Pt-),,
where X, X f.i.d. 1 @m and Y, i.i.d. yam The empirical distributions Px and Py are constructed from mini-batches, e.g., Px 1 Ee 82, for X -x1,,..., Fm) The metric 1) in the GME formulation can be any valid distance. In this work, we explore the use of Sliced Wasserstein (SW) variants and our proposed Tree-Sliced Wasserstein (TSW) variants as choices for D.
Setting. Our experimental configuration largely mirrors that of Nguyen et al. (2024b) and Tran et al. (2025a) in terms of model architecture and foundational hyperparameters. All models are trained for 1800 epochs. For Tree-Sliced methodologies, including novel techniques, ure I. = 2500 sampled trees and k = 4 lines per tree, following Tran et al. (2025a). In contrast, for vanilla SW and its associated variants, L = 10000 projections used, consistent with Nguyen et al. (2024b). Learning rates are also adopted from Nguyen et al. (2024b), specifically Ira 1.25 x 10-4 and lr 1.6 x 10-4, For FW-TSW" method, we used к scheduling scheme in Nguyen et al. (2024b). The standard deviation for tree sampling is 0.1, as per Tran et al. (2025a). Runtime evaluations are conducted using batch size of 128 on two NVIDIA H100 GPUs. Our results for FW-TSW and FW-TSW" are averaged over 10 runs while other results are obtained from previous results.
config-
D.5 HARDWARE SETTINGS
All experiments utilized Intel Xcon Platinum 8580 CPU. Gradient flow experiments were per- formed on a single NVIDIA H100 GPU, while denoising diffusion experiments were exccuted in parallel across two NVIDIA H100 GPUs.
E BROADER IMPACTS
The FW-TSW method introduced in this paper has significant societal implications by improving the accuracy and flexibility of optimal transport techniques across a wide range real-world ap- plications. It has the potential to advance fields such as realthcare-wher improved image pro- cessing can support more precise medical diagnostics-and the arts and entertainment industry, by enabling more refined and creative generative models. Additionally, its ability to operate effec- tively in dynamic environments unlocks new opportunities for real-time data analysis and informed secision-making in domains like finance, logistics, and environmental monitoring. In essence, FW- TSW enhances the practicality and reach of modern computational tools, promoting innovation and contributing to overall societal advancement.
30


Reviews:
[{"review_id": "LSaOu6mp66", "reviewer": "Reviewer_7to6", "summary": "This paper proposes Fermat–Weber Tree-Sliced Wasserstein (FW-TSW) and FW-TSW* — new variants of the Tree-Sliced Wasserstein (TSW) distance that integrate positional information via the geometric median (Fermat–Weber point).\n\nThe main idea: previous TSW approaches sample tree intersection points (roots) heuristically (typically from a Gaussian centered at the data mean). In contrast, FW-TSW replaces this with a data-adaptive distribution centered at the geometric median of the combined source–target supports, computed via Weiszfeld’s algorithm.\n\nThis modification yields:\n\nBetter alignment between positional sampling and data geometry;\n\nTheoretical guarantees: semi-metricity, Euclidean invariance, and boundedness;\n\nEmpirical gains across gradient flows, topic modeling, and diffusion model training — with comparable runtime to TSW and Db-TSW.", "questions": "1. Could you clarify whether the FW-TSW framework can be extended beyond the star-shaped tree assumption?  \n   In particular, is it possible to construct tree systems with multiple intersection points or hierarchical branches under the same theoretical formulation?\n\n2. How many iterations are typically required for the Weiszfeld algorithm in your experiments?  \n   Did you observe any stability or convergence issues for high-dimensional datasets?\n\n3. In particular, what is the benefit of \\sigma_{dir,mu,\\nu}  compared with U(S^{d-1})?", "limitations": "", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "### **Strengths**\n\n- **Novel conceptual link:**  \n  Clever integration of *location theory* (Fermat–Weber / geometric median) into OT sampling, yielding a clean geometric interpretation that grounds the stochastic tree construction in data geometry.\n\n- **Sound theoretical guarantees:**  \n  The paper rigorously proves semi-metricity, symmetry, Euclidean invariance, and boundedness of the proposed FW-TSW distance. These properties ensure mathematical consistency while extending classical TSW theory.\n\n- **Efficiency preserved:**  \n  Despite adding geometric-median computation via Weiszfeld’s algorithm, the complexity remains $O(Lkn\\log n + Lkdn)$, with only minor $O(Tnd)$ overhead — maintaining the practical efficiency of TSW.\n\n- **Strong empirical performance:**  \n  FW-TSW and FW-TSW consistently outperform SW, TSW, and Db-TSW baselines on gradient flow, topic modeling, and diffusion model tasks. Results show improved convergence, stability, and lower FID without sacrificing runtime.", "weaknesses": "### **Weaknesses**\n\n- **Limited generality of the tree structure.**  \n  The proposed FW-TSW and FW-TSW* are defined only on *star-shaped* tree systems, where all branches share a single root point (Eq. 15 / 18).  \n  More general tree geometries—e.g., unions of multiple disjoint lines or hierarchical branching—cannot be represented within this framework.  \n  This limits the flexibility of the model and its ability to capture more complex spatial relationships.\n\n- **Unclear computational overhead.**  \n  The complexity of Weiszfeld’s algorithm (used to compute the geometric median) is not analyzed or included in the overall runtime discussion.  \n  While each iteration is \\(O(nd)\\), the number of iterations and convergence behavior can vary depending on data geometry, which may affect scalability in high dimensions.\n\n- **Loss of metric property.**  \n  FW-TSW and FW-TSW* do not satisfy the triangle inequality because the sampling distribution \\(\\sigma_{\\text{FW},\\mu,\\nu}\\) depends on the data pair \\((\\mu,\\nu)\\).  \n  As discussed around Eq. (20), the distance is only a *semi-metric* when the sampling distribution is fixed.  \n  This data-dependent design improves adaptivity but sacrifices the strict metric property of the original TSW.\n\n- **Minor issue:**  \n  Line 349 contains a small typo: “Equation equation 15.”", "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": "085U0blKdy", "reviewer": "Reviewer_cvC1", "summary": "This paper proposes the Fermat-Weber Tree-Sliced Wasserstein (FW-TSW) to solve a geometric inconsistency in the standard tree sliced Wasserstein (TSW) framework. While TSW is an $L_1$-based metric ($W_1$) and its tree-sampling is position-dependent, existing methods sample this position (the tree root $x$) from a Gaussian centered at the $L_2$ data mean. FW-TSW corrects this mismatch by centering its sampling distribution $\\sigma_{FW}$ at the $L_1$ geometric median ($x^*$) which is the solution to the Fermat-Weber problem. This scheme is claimed to better capture data geometry and improve performance, adding only a negligible upfront cost to compute the median.", "questions": "- In the 25 Gaussians experiment (Table 1), TSW and FW-TSW methods achieve a better final error compared to SWGG, which converges faster initially. What properties of tree-slicing give this better final convergence, and does the Fermat-Weber centering improve the global optimization landscape beyond the standard TSW's mean-centering?\n\n- The geometric median computation adds an $O(Tnd)$ cost (with $T = 100$) per batch, compared to the baseline's $O(nd)$ mean calculation. Is this iterative cost truly negligible for large $n$ and $d$?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- This paper’s proposal to center the sampling distribution at the $L_1$ geometric median (the Fermat-Weber point) is the principled solution, aligning the geometry of the sampling space with the geometry of the metric.\n\n- Experiment results, particularly in generative modeling (Table 3), are significant. The FW-TSW* variant achieves FID score (2.315) by making both the position ($x^*$) and directions ($\\theta_i$) dependent on the measures being compared. This provides strong evidence that for high-dimensional generative tasks, data-agnostic slicing (like uniform sampling) is suboptimal, and a data-dependent discrepancy is superior.\n\n- The cost of solving for the geometric median ($O(Tnd)$) is paid once per batch and does not dominate $O(Lkn \\log n + Lkdn)$ TSW computation. The paper shows that this significant performance gain (e.g., in Table 3) is achieved with a negligible increase in wall-clock time, making it a drop-in replacement.", "weaknesses": "- FW-TSW is no longer a metric.\n\n- The paper introduces two ideas: (1) data-dependent *positional* sampling (the core FW insight) and (2) data-dependent *directional* sampling (the FW-TSW* variant, eq 17). The paper’s narrative is built around the principled Fermat-Weber contribution, but the best empirical result (FID in Table 3) relies on the FW-TSW* variant, which includes the more heuristic directional sampling. This makes it difficult to attribute the performance. It is not clear if the central insight (the geometric median) is the key driver, or if the (less-justified) directional sampling is doing most of the work.", "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": "uZakwHEdB0", "reviewer": "Reviewer_jHMb", "summary": "While Optimal Transport (OT) allows to compare data in a 'meaningful' way, it is computionally costly. Slicing strategies allow for the definition of more efficient OT-based metrics.\nIn particular, this paper introduces a new variant of the Tree-Sliced Wasserstein (TSW) distance, called the Fermat-Weber Tree-Sliced Wasserstein (FW-TSW). It is presented as an efficient alternative to standard Sliced Wasserstein (SW), utilizing tree-structured metric spaces to better capture data geometry while maintaining low computational cost. The key insight leveraged is that TSW incorporates both directional and positional information distinguishing it from the purely directional SW. A key question for TSW is the generation of random trees. The authors propose FW-TSW, which explicitly integrates the geometric median (based on the Fermat-Weber problem) into the tree construction process to enhance this positional sensitivity. They claim that this leads to further performance improvements over TSW, which is empirically validated in experiments involving diffusion model training and gradient flows.", "questions": "1. Scaling in (14): In Equation (14), why is the identity matrix $\\mathbf{I}_d$ not scaled according to the data's covariance or variance?\n \n2. Subscripts in (15): The measure dependence should be made explicit: the subscripts $\\mu, \\nu$ of $\\Sigma$ should already appear in Equation (15).\n \n3. Choice of Central Point: Why was the geometric median chosen over simply using the barycenter of the measures in the tree construction? Is it because of Wasserstein-1 instead of Wasserstein-2? Did you try using the barycenter?\n \n4. $k$-Disjoint Union: In Equation (6), what is the precise definition and calculation of the distance on the $k$-times disjoint union of $\\mathbb R$?\n \n5. Typos and Formatting: Please correct the formatting error \"Equation equation 4\" (bottom of p. 6 and elsewhere). Ensure distinct notation is used for the exact distance (6) versus its approximation (8). Also, correct Equation (28) by removing the spurious subscript $l$ on the left-hand side.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "Novelty: Sampling strategies for sliced Wasserstein and kernel distances are an active field. Beyond projections onto the line, it is valuable to study sampling strategies for TSW, as the combined use of directional and positional information has been relatively underexplored. \n \nStructural Design: The integration of the geometric median into the tree construction mechanism is an intuitive apprach for sampling meaningful tree projections.\n \nTheoretical Contribution: Since there are limited theoretical results for TSW, the bound presented in Theorem 4.5 is an interesting result that provides new theoretical insight into the properties of Tree-Sliced distances.\n \nEmpirical Validation: The experiments are targeted at typical applications of sliced OT and provide good support for the proposed FW-TSW method. There are extensive ablation studies in the supplementary material.", "weaknesses": "Limited Impact: While the question for 'smart' tree sampling is interesting and deserves further studiy, the contribution seems rather incremental.\n \nTheoretical Complexity: Most of the theoretical results are assessed are  straightforward with the exception of  Theorem 4.5.\n \nDoubts about empirical validation: Much of the experiments focus on comparing SW and TSW for a predefined number of projections. However, SW is faster, so it might be fairer to compare the two for a predefined run time. After all, the number of projections is very important for the Monte Carlo estimates.\n \nBound Limitation: The utility of the Theorem 4.5 bound is questionable due to a potentially large additive constant, especially for a high number of splits $k$. \n \nApplicability: The practical relevance of the demonstrated applications is limited outside of a dedicated Computational Optimal Transport research area. To my knowledge, TSW is not widely used for practical application (unlike standard OT or SW).\n \nClarity and Notation: Section 2.2 suffers from inconsistent and confusing notation. Key terms like $\\mathbb T_k^d$ are undefined, and the subscript $l$ in Equation (4) is unexplained, hindering comprehension. The rather central closed-form Wasserstein formula for trees is not provided in the paper, but only in the references. It is also confusing that the supplementary material around (24) uses partly different notation.\n \nReferences: Relevant references on sampling strategies for sliced divergences, e.g. (Hertrich et al., 2025) and (Sisouk et al., 2025) are missing.", "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)