Spherical Tree-Sliced Wasserstein Distance
Authors: Hoang V. Tran
Thanh Chu
Minh-Khoi Nguyen-Nhat
Huyen Trang Pham
Tam Le
Tan Minh Nguyen
Venue: ICLR 2025 Poster (2025)
Keywords: tree-sliced wasserstein distance
spherical optimal transport
equivariance
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We propose a novel spherical Radon transform, with tree-structured integration domains, where OT problems admit a closed-form solution.

Abstract: Sliced Optimal Transport (OT) simplifies the OT problem in high-dimensional spaces by projecting supports of input measures onto one-dimensional lines, then exploiting the closed-form expression of the univariate OT to reduce the computational burden of OT. Recently, the Tree-Sliced method has been introduced to replace these lines with more intricate structures, known as tree systems. This approach enhances the ability to capture topological information of integration domains in Sliced OT while maintaining low computational cost. Inspired by this approach, in this paper, we present an adaptation of tree systems on OT problem for measures supported on a sphere. As counterpart to the Radon transform variant on tree systems, we propose a novel spherical Radon transform, with a new integration domain called spherical trees. By leveraging this transform and exploiting the spherical tree structures, we derive closed-form expressions for OT problems on the sphere. Consequently, we obtain an efficient metric for measures on the sphere, named Spherical Tree-Sliced Wasserstein (STSW) distance. We provide an extensive theoretical analysis to demonstrate the topology of spherical trees, the well-definedness and injectivity of our Radon transform variant, which leads to an orthogonally invariant distance between spherical measures. Finally, we conduct a wide range of numerical experiments, including gradient flows and self-supervised learning, to assess the performance of our proposed metric, comparing it to recent benchmarks.

Published as a conference paper at ICLR 2025
SPHERICAL TREE-SLICED WASSERSTEIN DISTANCE
Viet-Hoang Tran'
Thanh T. Chu
Department of Mathematics
Department of Computer Science
National University of Singapore
National University of Singapore thanh. chuêu, nus . edu
hoang. tranviet @u. nus edu
Khoi N.M. Nguyen
Trang Pham
FPT Software Al Center khoi mnmonnnl@ & fpt.pc..m com
Qualcomm Al Research
tranpham@qti qua
qualcomm. comm. com
Tam Le
Tan M. Nguyen
The Institute of Statistical Mathematics & RIKEN AIP
Department of Mathematics
National University of Singapore ta nus mous.edu.d.. edu.sg
tam s.m. c jp
ABSTRACT
Sliced Optimal Transport (OT) simplifies the OT problem in nigh-dimensional spaces by projecting supports of input measures onto one-dimensional lines and then exploiting the closed-form expression of the univariate OT to reduce the com- putational burden of OT. Recently. the Tree-Sliced method has been introduced replace these lines with more intricate structures, known as tree systems. This approach enhances the ability to capture topological information of integration domains in Sliced OT while maintaining low computational cost. Inspired by this approach, in this paper, we present an adaptation of tree systems on OT prob- lems for measures supported form variant on tree systems, we propose novel spherical Radon transform with new integration domain called spherical trees. By leveraging this transform and exploiting the spherical tree structures, we derive closed-form expressions for OT problems the sphere. Consequently, obtain efficient metric for measures on the sphere, named Spherical Tree-Sliced Wasserstein (STSW) dis- tance. We provide extensive theoretical analysis demonstrate the topology of spherical trees and the well-definedness and injectivity of our Radon trans- form variant, which leads to orthogonally invariant distance between spherical measures. Finally, we conduct a wide range of numerical experiments, including gradient flows and supervuue learning, to assess the performance of our pro- posed metric, comparing it to recent benchmarks. The code is publicly available ht thub. com i 1illlytthhhsssSS STt.h gitt..
sphere. As a counterpart to the Radon trans-
INTRODUCTION
Despite being embedded in high dimensional Euclidean spaces, in practice, data often reside on low dimensional manifolds (Fefferman et al., 2016). The hypersphere is one such manifold with various practical applications. The range of applications involving distributions on a hypersphere is remarkably broad, underscoring the significance of spherical geometries across multiple fields. These applications encompass spherical statisties (Jammalamadaka, 2001; Mardia Jupp, 2009; Ley & Verdebout, 2017; Pewsey & Garcfa-Portuguués 2021), geophysical data (Di Marzio et al., 2014), cosmology (Jupp. 1995; Cabella Marinucci, 2009; Perraudin et al., 2019), texture map- ping (Elad et al., 2005; Dominitz & Tannenbaum, 2009), magnetoeneeehhalgrrahyy imaging (Vrba & Robinson, 2001), spherical image representations (Coors et al., 2018; Jiang et al., 2024), omnidi-
Co-first authors. Co-last authors. Qualcomm Vietnam Company Limited. Correspondence to:
trangg..aaviituu.uuu.cooo & tanmn @ @nus.edu.sg

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Published as a conference paper at ICLR 2025
rectional images (Khasanova& & Frossard, 2017), and deep latent representation learning (Wu al., 2018; Chen et al., 2020; Wang & Isola, 2020; Grill et al., 2020; Caron et al., 2020; Davidson et al,, 2018: Liu et al., 2017; Yi & Liu, 2023).
Optimal Transport (OT) (Villami, 2008; Peyré et al., 2019) is a geometrically natural metric for com- paring probability distributions, and it has received significant attention in machine learning in recent years. However, OT faces a significant computational challenge due to its supercubic complexity in relation to the number of supports in input measures (Peyré et al., 2019). To alleviate this issue, several variants have been developed to reduce the computational burden, including entropic regular- ization (Cuturi, 2013), minibatch OT (Fatras et al., 2019), low-rank approaches (Forrow al., 2019; Altschuler et al., 2019; Scetbon et al., 2021), the Sliced- Wasserstein distance (Rabin et al., 2011; Bonneel et al., 2015), tree-sliced- Wasserstein distance (Indyk & Thaper, 2003; Le et al., 2019; Le & Nguyen, 2021; Tran et al., 2025b;c;d), and Sobolev transport (Le et al., 2022; 2023; 2024).
Related work. There has been growing interest in utilizing OT to compare spherical probability measures (Cui al., 2019; Hamfeldt Turnquist, 2022). To mitigate the computational burden, recent studies have focused on sliced spherical OT (Quellmalz et al., 2023; Bonet et al., 2022; Tran al., 2024b). Quellmalz et al. (2023) introduced the vertical slice transform and a normalized version of the semicircle transform to define sliced OT on the sphere. The semicircle transform was also employed in (Bonet et al., 2022) to define spherical sliced Wasserstein. Meanwhile, Tran et al. (2024b) utilized stereographic projection to create a spherical distance between measures via univariate OT problems. However, projecting spherical measures onto a line or circle poses challenges due to the loss of topological information. Furthermore, comparing one-dimensional measures on circles is computationally more expensive, as it requires an additional binary search.
Notably, Tran al. (2025b;d) offer an alternative method by substituting one-dimensional lines in the Sliced Wasserstein framework with more complex domains, referred to as tree systems. These systems operate similarly to lines but with more advanced and intricate structure. This approach is expected to enhance the capture of topological information while preserving the computational efficiency of one-dimensional OT problems. Inspired by this observation, we propose an adaptation of tree systems to the hypersphere, called spherical trees, to develop a new metric for measures on the hypersphere. Spherical trees satisfy two important criteria: (i) spherical measures can be projected onto spherical trees in meaningful manner, and (ii) OT problems on spherical trees admit a closed-form expression for a fast computation.
Contribution. Our contributions are three-fold:
1. We provide comprehensive theoretical construction of spherical trees on the sphere, analogous to the notion of tree systems. We demonstrate that spherical trees, as topological spaces, are metric spaces defined by tree metrics, which ensures that OT problems on these spaces can be analytically solved with closed-fore form solutions.
2. We propose the Spherical Radon Transform on Spherical Trees, which transforms functions on the sphere to functions on spherical trees. We also present the concept of splitting maps for the sphere, key component of this new Spherical Radon Transform, which describes how mass at a point is distributed across the spherical tree. In addition, we examine the orthogonal invariance of splitting maps, which later proves to be a sufficient condition for the injectivity of the Spherical Radon Transform.
3. We propose the novel Spherical Tree-Sliced Wasserstein (STSW) distance for probability distri- butions on the sphere. By selecting orthogonal invariant splitting maps, we demonstrate that STSW a invariant metric under orthogonal transformations. Finally, we derive closed-font approxima- tion for STSW, enabling an efficient and highly parallelizable implementation.
Organization. The rest of the paper is organized as follows: ants in $2. We propose the notion of Spherical Trees on the Sphere with formal construction in $3. We introduce the Spherical Radon Transform on Spherical Trees, and discusses its injectivity in $4. In $5, propose Spherical Tree-Slicer Wasserstein (STSW) distance and derive closed-for form approximation for STSW. Finally, we evaluate STSW on various tasks in $6. Theoretical proofs and experimental details are provided in Appendix.
we review Wasserstein distance vari-
2

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2 PRELIMINARIES
In this section, we review Wasserstein distance, Sliced Wasserstein distance, Wasserstein distance on tree metric spaces and Tree-Sliced Wasserstein distance Systems of Lines.
Wasserstein Distance. Let be a measurable space, endowed with a metric d, and let j4, v be two probability distributions on Ω. Denote P(p, 2/) as the set of probability distributions product space Q x Ω such that (A x Q2) (a(A) and 7 x B) = (BB for all measurable sets A, B. For p 2 1, the p- Wasserstein distance W, (Villani, 2008) between μs, is defined as:
the
( d(z, y)) y) d
Wp(mn)) inf REP(NN)
Sliced Wasserstein Distance. The Radon Transform (Helgason, 2011) is the operator R. (TR4) - (TR x S4-1) defined by: for ʃ E (IR(TR)), we have Rf E I. x st-1) such that Rt(t, 0) = Ja f(x) 8(t - (x, 0)) dx. Note that R a bijection. The Sliced p- p-Wasserstcin (SW) distance (Bonneel et al., 2015) between μ, V E ((R))) is defined by:
(LL.. () 0)) do do do(0))²
SW SWp(,.V)
where 1( is the uniform distribution on gd-1; and fur fu are the probability density functions of μ,, v, respectively.
Tree Wasserstein Distances. Let T be rooted tree (as graph) with non-negative edge lengths, and the ground metric dy, i.e. the length of the unique path between two nodes. Given two proba- bility distributions j and supported on nodes of T. the Wasserstein distance with ground metric dy, i.c., tree- Wasserstein (TW) (Le et al., 2019), yields a closed-form expression as follows:
W (1,
Σ we (a([(()())) (TT(v))), eET
where Ue is the endpoint of edge e that is farther away from the tree root, I((e.) is subtree of T rooted Ues and we is the length e.
Tree-Sliced Wasserstein Distances on Systems of Lines. Tree systems (Tran et al., 2025d) are proposed as replacements of directions in SW. As a topological space, they are constructed by join- ing (gluing) multiple copies of R based on tree (graph) framework, forming measure metric space in which optimal transport problems admit closed-form solutions. By developing a variant of the Radon Transform that transforms functions on Rd to functions on tree systems, Tree-Sliced Wasserstein Distances on Systems of Lines (TSW-SL) is are introduced in a similar manner as SW. The mentioned closed-fore foreddffommmmmo expressions lead to highly parallelizable implementation for TSW-SL. We next extend the tree systems for measures on a sphere.
SPHERICAL TREES ON THE SPHERE
Let be positive integer. Recall the notion of the d-dimensional sphere in IR++1,
gd = (x (oo, ,
I.) E d ||^|| 1} C I=+1
The sphere gd is complete metric space with metric dga defined as dga (a, b) = arecos (a, b) for a. b E gd, where 2241 is the standard dot product in Rd+1, For Sd. denote H,, be the hyperplane passes through 0 E gd+1 and orthogonal to i.c. He {v Id+1; (z, 0}.
We consider the stereographic projection corresponding to x, denoted by Pa. which is a map from gd {x} to Hz defined by: for y gd \ {x), (y) is the unique intersection between the line passes through x, and the hyperplane Hz. In concrete, the formula for yo is as follows
Px
8d (x)
H
{x, v)
1 1 (x, v)
y
I + (x,, v)
y-
(4)
3

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Figure 1: Illustrations of stereographic projection, rays in R and spherical rays in 8
It is rell-known that Pe a smooth bijection between Sd \ { and Hg. Moreover, it convenient to extend Pax to map that we also denote by Pz. from S to H U foo), with Px = oc. Remark. As topological space, He is homeomorphic to ER and H, U {oo} is the one-point compactification of Hg, which is homeomorphic to $4, Also, H2 n $ is homeomorphic to gd-1,
Definition 3.1 (Spherical rays in Rd+1)), For y E Sd, ray in Rd+1 with direction y is defined as the set {t- y >> 0} U {oo}. For x E 8", and E Sd n Hz, the spherical ray with root æ and direction y. denoted by ry. is defined the preimage of the ray with direction y through Yns i.e.,
ry == Pr ((t. t >> 0} U {xo}).
illustration of stereographic projection, rays and spherical rays are presented in Figure/ In words, spherical ray with root and direction ys the great semicirele surface of the hypersphere passes through y with one endpoint x. We have ry is isometric to the closed interval [0, л] via arccos (ar, z), and we also have parameterization of y as (1, r for < L T. In particular,
2 (0) - and Dx (00) --- 0.
Let & be a positive integer, x E gd and ........ Us E Ed H, be i distinct points. We have distinct spherical rays ry with root x and direction Consider an equivalence relation on the disjoint union -l=1.... ro. as follows: For (t.ry) E Th and (t',ri : "0 we have (t, (t, (t', rr if and only if (t, r..) (t',r in Sd, In other words, we identify è points with coordinate on spherical rays ry, &
Denote T as the set of all equivalence classes Li= l,.... Tyk
with respect to the equivalence relation ~., i.e., T Ti.... (U........ 7 /~-
Recall the notion of disjoint union topology and quotient topology (Hatcher, 2005). For 1,... k. consider the injection
f
y:
(t, r
The disjoint union LI= N iee the finest topology - r.. such that the map fa is continuous for all i 1, 1,..., k. Consider the quotient map by the equivalent relation
becomes topological space with the disjoint union topology,
T Vi.....
/~
(t, 1
[tt, (t.r"
Ty -n.... now becomes a topological space with the quotient topology, i.e. the finest topology on T y:..... such that the map π is continuous. In other words, T Ty..... is formed by gluing & spherical rays 3. at the points with coordinate on each spherical rays.
Definition 3.2 (Spherical Trees in sd). The topological space T Tp....... is called spherical tree on
gd. We said that æ is the root and ./..... V/ are the edges of T Ty.......
A visualization for construction of spherical trees is presented in Figure 2a. The number of edges of a spherical tree usually denoted by k. For simplicity, we sometimes omit the root æ and edges and simply denote spherical tree as T. The collection of all spherical trees with k edges
"1,-.. V/ y/k

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6
31
(a) Spherical Tree
(b) Radon Transform
Figure 2: (a) An illustration of 5 spherical rays with the same root e, along with the corresponding spherical tree rooted at Note that, even when endpoints differ from of these spherical rays are all identical to -x on the sphere, the spherical tree treats these as five distinct points, and only identifies the root æ. (b) An illustration of Radon Transform Spherical Trees. Consider a point 2. The hyperplane passing through 2 and orthogonal to x cuts edges of the spherical tree at 5 points. The mass under operator Ra is distributed across these 5 intersections, depending on
on $d is denoted by TV. Since $d n H is homeomorphic to the sphere sd-1, we have the one-to-one correspondence between Ty and the product sd (S n H²)* )a as follows:
TV U
1-1
x E
Sd and (g) (y......yy yy) E (Sa nHa)* (S4-1)A
(5)
From this observation, we can define distribution distribution of distributions on Sd and gd-1, For the rest of the paper, let be the joint distribution of (k - 1) independent distributions, consists of one uniform distributions on S4, i.e. U(S"), ), and k gd-1, i.e. (S$-1)). The topological space T is metrizable by the metric
the space of spherical trees Ty the joint
uniform distributions
dy defined as: For a = (t, r.. )g)) and b = (t', 53,) in T.
( |t t' = and t t if i
dr(a, b)
Moreover, this metric is a tree metric on T. We verify this by showing for every pair of two points a, in T, all paths from a be b in T are homotopic to each other. Then dy (a, b) is the length of the shortest path from to b in T. Moreover, we can define a measure on T that induced from the Borel measure on the closed interval [o, x]. The proof of these properties is similar as the proofs in (Tran et al., 2025d). We summarize our results by theorem.
Theorem 3.3 (Spherical trees are metric spaces with tree metric). T is a metric space with tree metric dr. The topology on T induced by dr is identical to the topology T.
With this design, in the next section, we will define Lebesgue integrable functions on spherical trees.
4 SPHERICAL RADON TRANSFORM ON SPHERICAL TREES
In this section, we introduce the spherical Radon Transform on Spherical Trees, and discuss the injectivity of our spherical Radon transform variant.
4.1 A SPHERICAL RADON TRANSFORM VARIANT
We introduce the notions of the space of Lebesgue integrable functions spherical trees. First, denote L' (Sd) as the space of Lebesgue integrable functions on Sd with norm ||-
T.' (S(( = {1: sd - R 11/1l1 = [f(x)) do < x0)
Two functions f1 E (Sd) are considered to be identical if fi = fz(x) for almost every- where on Sd. Consider spherical tree T with root and A edges ,1...., ........... Mk. a Lebesgue integrable function T is function f: T IR such that || = Σ fo [f , ) dl co.
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The space of Lebesgue integrable functions T is denoted by (T) Two functions f1, fa L' ('(T) are considered to be identical fi(x) = f2(2) for almost everywhere on T. The space L' (C) with norm || ||T is a Banach space.
Let 1= {a)) 0 & ai & 1 and 2== simplex. Denote C (Sd x Ty Ax-1) the space of continuous maps from S T to A and called map in C (St x Ty, Ax-1) by a splitting map, Let T be a spherical with root and edges y1,..... V/- he a splitting map (S4 T Ax-1), we define an operator associated to a that transforms a Lebesgue integrable functions on S to Lebesgue integrable functions on T. For L'LL''''' L'(8),, define
a; =
IR* be the (k-11)dimensiooal standard
RJ T I
(8)
/ [w) aly, 5(1 arccos (2, u)) dy.
(1, (1,7%)
(9)
where the Dirac delta function. We have R E (T) for f E (' and moreover, [R la The operator R9: (S4) (T) well-defined linear operator. The proof of these properties can be found in Appendix A.1. An illustration of R2 is presented in Figure 2b. We next present a novel spherical Radon Transform variant on spherical trees.
Definition 4.1 (Spherical Radon Transform Spherical Trees), For E C (Sd Ty AI-1), the
operator Ra that is defined as follows:
Ra (.')
II (T(T) TETY
I (R9-))TTT)*
called the Spherical Radon Transform on Spherical Trees.
4.2 INJECTIVITY RADON TRANSFORM SPHERICAL TREES
We discuss on the injectivity of our spherical Radon Transform variant. Consider the Euclidean norm Rd i.e. ||- 2/2.
Orthogonal group ((d) and its actions. The orthogonal group ((d) is the group of linear trans- formations of Rd that preserves the Euclidean norm l12.
(dd) linear transformation f: Rd R || = [x) for all x E R
(10)
is well-known that ((d) isomorphic to the group of orthogonal matrices under multiplication,
((d) = {Q E Maxa(R) Q Q = =Q
Q = =1a]-
(11)
The canonical group action of (d) on Re defined by: For g = Q E O(d) and y E IR,, we have Rd+1 canonically induces
y + gy Q y. By the norm preserving, the action of O(d + action O(d + 1) on the sphere Sdt Moreover, the action of ((d) on Rd preserves the standard dot product, the action of O(d + 1) on Sd preserves the metric dgs.
Group actions of Odd+ 1) on space of spherical trees TV. Under E O(d 1), the spherical ray ry transforms to y-g It implies that the action of O(d + 1) on Sd canonically induces an action of O(d + on Tİ. as
T = T
gT == Tg
(12)
991....9
Moreover, each g E O(d+ 1) presents a morphism
gT that isometric.
Odd + i)-invariant splitting maps. Given map f: X - Y and group G acts on X. The map f is called G-invariant fggx) (xx) for all g E and x e X. We have the definition of O(d + ---ivvariance in splitting maps.
Definition 4.2. splitting map a in C(Sd Ty. Ag-1) 1) is said to be O(d + -iinvaaiin if we have a(gu.gT = a(y, T)
(13)
for all (v. T) E gd TV. and g E O(d+1)

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With an O(d + 1) --invariant splitting maps, spherical Radon Transform variant is injective.
Theorem 4.3. R is injective for an O + 1) i)-invariant splitting map a.
The proof of Theorem 4.3 is presented in Appendix A.3. Finally, we present candidate for O(d+1)-
invariant splitting maps. Define the map B: gd TØ R as follows:
0,
if or =2,
Bly, T ::....
arccos
(v. Un)
V (2,9))
if +r.
(14)
(x,, w)²
Remark. The construction of 3 will be explained in Appendix A.2.
The map is continuous and Old + 1)-invariant. The derivation of 8 and the proof for these properties are presented in Appendix A.2. We choose or: 3d x T A as follows:
au: T) = softmax (IS Bly, Detr).......
(15)
Here, Ç E I is treated as a tuning parameter. The intuition behind this choice of a is that reflects the proximity of points to the rays of the spherical trees. As Ksl increases, the resulting value of a tends become more sparse, emphasizing the importance of each ray relative to a specific point.
SPHERICAL TREE-SLICED WASSERSTEIN DISTANCE
In this section, we propose our novel Spherical Tree-Sliced Wasserstein Distance (STSW). We also derive a closed-form approximation of STSW that allows an efficient implementation.
5.1 SPHERICAL TREE-SLICED WASSERSTEIN DISTANCE
Given two probability distributions μl, in P(S"), ), a tree T Ty and an O + 1)-invariant splitting map a C(S Ty A1.)). By the Radon Transform R... in Definition 4.1, l and v tranform to two probability distributions R& and R& in P(T). T is metric space with tree metric dt (Tran et al., 2025d), so we can compute Wasserstein distance W (Rq R&v) between R4μ and RIv by Equation (3).
Definition 5.1 (Spherical Tree-Sliced Wasserstein Distance). The Spherical Tree-Sliced Wasserstein
Distance between μ,, P(Sd) defined by:
STSW(H, = / W War. (R, do( JTg.
(16)
Remark. Note that, the definition of STSW depends on the space Ty the distribution o on TV and the splitting map as in Equation (15). We omit them to simplify the notation.
The STSW distance is, indeed, metric P(SA).
Theorem 5.2. STSW is metric on P(St). Moreover, STSW is invariant under orthogonal trans- formations: For O(d + 1), we have
STSW(H, = STSW
(17)
where gtn, gtv as the fush-forward of μ. via orthogonal transformation g: gd Sd. respectively. The proofs of Theorem 5.2 is presented in Appendix A.4.
5.2 COMPUTATION OF STSW
To approximate the intractable integral in Equation (16), we use the Monte Carlo method as STSW v) = (1/L) Σt_ Wa (Rq,4, R v),, where Ti.... Tr. are drawn independently from the distribution on T, and are referred to as projecting tree systems. We present the way to sample T, and compute War, (R%,,, v).

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Sampling spherical trees. Recall that is the joint distribution of k independent distributions, consists of one uniform distributions on S and & uniform distributions on g This comes from the one-to-one correspondence between Tý and Sd 1 as in Equation (5). In applications, to perform sampling process for T Ty yy..... Ty from o, sample by steps as follows:
1. Sample k + 1 points y1..... yk in R++1 Normalize them to get z, V...... lie on Sd.
2. For each take the intersection of the line passes through 2; yr with Hz i.e. Pax then normalize 2, to get new yi lies on Hg ns
This results in a sampling process based on distribution a.
Computing Way, 1 (RTu, R4v). In applications, given discrete distributions Je and as μ(x) Ej=1 My 0() and v()) = Σi= vy (zxr)))))) We can present and v with the same supports
by combining their supports and allow some My or vj to be O. For spherical tree T = T Mi.....p we want to compute 1 Rgvv). For 1 $ & n, let cj = dga (x. az). Also let co = 0. By re-indexing, we assume that on & eI & .... On. By Radon Transform in Definition 4.1, Jr, transform to Ru, R&V supported on {(c,, r ( 1 of T. with
R&Hle rg,)) ),)- = a(aj. T) Uj and RFvlej r = ala,, T).
(18)
By Equation (3), W dr (R R4v) has closed-form approximation follows
(R.,1 = ( - (9-1)
alap, T)s (Up up))
(19)
The detailed derivation of Equation (19) is presented in Appendix A.5. The closed-form expression
in Equation (19) leads to a highly parallelizable implementation of STSW distance.'
EXPERIMENTAL RESULTS
In this section, we present the results of our four main tasks: Gradient Flow, Self-Supervieed Learn- ing, Earth Density Estimation, and uation for each task, including quantitative metrics, visualizations, and a comparison with relevant baseline methods. Experimental details can be found in Appendix B.
Auto-Encoder. We provide detailed eval-
6.1 GRADIENT FLOW
Our first experiment focuses on learning a target distribution v from a source distribution la by minimizing STSW (t). We solve this optimization using projected gradient descent, discussed in Bonet et al. (2022). We compare the performance our method against baselines: SSW (Bonet al., 2022), and S3W variants (Tran et al., 2024b).
Following Tran et al. (2024b), we use a mixture of 12 von Mises-Fisher distributions (vMFs) as our target v. The training is conducted over 500 epochs with full batch size, and each experiment is repeated 10 times. We adopt the evaluation metrics from Tran et al. (2024b), which include log 2- Wasserstein distance. negative log- likelihood (NLL), and training time. As shown in Table 1, STSW outperforms the baselines in all metrics and achieves faster convergence, as illustrated in Figure 10.
6.2 SULE-SUPERVISED LEARNING (SSL)
Normalizing feature vectors to the hypersphere has been shown to improve the quality of learned representations and prevent feature collapse (Chen et al., 2020; Wang & Isola, 2020). In previous work, Wang Isola (2020) identified two properties of contrastive learning: alignment (bringing positive pairs closer) and uniformity (distributing features evenly on the hypersphere). Adopting
See Algorithm in Appendix $B.1 for pseudo-cade of STSW distance.
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Table 1: Learning target distribution 12 vFMs. We use Ng 30 rotations for ARI-S3W and an additional learning rate LR = 0.05 for SSW.
Method
log W2
NLL
Runtime(s)
SSW (LR=0.01)
-3,21 + 0.16
4980.01 + 1.89
55.20 = 0.15
SSW (LR=0.05)
-3.36 - 0.12
4976.58 + 2.23
55.31 = 0.33
S3W
-2.37 + 0.21
4749.67 + 84.34
1.93 + 0.06
RI-S3W (I)
+
-9964.00 = 27.98
2.03 + 0.12
RI-S3W (5)
-3.47
-498.800 7.32
5.68 0.51
ARI-S3W (30)
4.39+0...
-5020.37 + 6.35
20.25 = 0.15
STSW
-4.69 + 0.01
-5041.13 + 0.84
1.89 + 0.05
Table 2: CIFAR-10 10 linear evaluation accuracy for encoded (E) features and projected (P) features on 8 along with pretraining time per epoch. ARI-S3W and RI-S3W use rotations.
Method
Acc. E(%)
Acc. P(%)
Time (s/ep.)
Hypersphere
79.81
74.64
10.18
SimCLR
79.97
72.80
9.34
SW
74.39
67.80
9.65
SSW
70.23
64.33
10.59
S3W
78.59
73.83
10.14
RI-S3W (5)
79.93
73.95
10.22
ARI-S3W (5)
80.08
75.12
10.19
STSW
80.53
76.78
9.54
the approach in Bonet et al. (2022), we propose replacing the Gaussian kernel uniformity loss with STSW, resulting in the following contrastive objective:
(TSSW(= 1) - STSW(2 v)).
(20)
Uniformity loss
Aligmment losss
where v = >> 0 is regularization factor, AA_ =A E Rnx(d+1) are embeddings of two image augmentations mapped onto Sd. Similar Bonet (2022) and Tran et al. (2024b), we train ResNet18 (He et al., 2016) based encoder on the CIFAR-10 (Krizhevsky et al., 2009) w.r.t C. After this, train linear classifier on the features extracted from the pre-trained encoder.
Table 2 demonstrates the improvement of STSW in comparison to baselines: & Isola, 2020), SimCLR (Chen et al., 2020). SW. SSW, and S3W variants. We also conduct ex- periments with d = 2 to visualize learned representations. Figure 12 illustrates that STSW can effectively distribute encoded features around the sphere while keeping similar ones close together.
Hypersphere (Wang
6.3 EARTH DENSITY ESTIMATION
We demonstrate the application of STSW in density estimation on $ Data used in this task collected by (Mathieu Nickel, 2020) which consists of Fire (Brakenridge, 2017), Earthquake (EOSDIS, 2020) and Flood (EOSDIS, 2020). As in (Bonet al., 2022), employ an exponential map normalizing flow model (Rezende et al., 2020) which are invertible transformations T and aim minimize min7 (1. where y the empirical distribution, and pz is prior distribution g which we uniform distribution. The density for any point g2 is then estimated by f (x) Pz (dxz)))detTr where Jy (x) is the Jacobian of T at r..
Our baselines exponential map normalizing flows with SW, SSW, and S3W variants, and stere- ographic projection based (Dinh et al., 2016) normalizing flows. As seen in Table 3, STSW even with fewer epochs and shorter training time (10K epochs over 2h 1m for STSW versus 20K epochs

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Table 3: Negative log-likelihood on test data, averaged over 5 runs with different data split.
Method
Quake
Flood
Fire J
Stereo
1.94 0.21
1.92 + 0.04
1.31 + 0.12
SW
0.99 0.05
1.47 + 0.03
0.55 + 0.21
SSW
0.84 = 0.06
1.26 + 0.04
0.23 + 0.20
S3W
0.89 = 0.08
1.35 = 0.04
0.34 = 0.05
RI-S3W (1)
0.80 = 0.07
1.25 + 0.03
0.14 + 0.06
ARI-S3W (50)
0.77 = 0.06
1.24 + 0.03
0.08 + 0.05
STSW
0.68 0.04
1.23 + 0.03
-0.07 + 0.05
Table 4: SWAE results evaluated on latent regularization of CIFAR-10 test data.
Method
log W2
NLL
BCE 1
Time (s/ep.)
SW
-3.2943
-0.0014
0.6314
3.4060
SSW
-2.2234
0.0005
0.6309
8,2386
S3W
-3.3421
0.0013
0,6329
4.5138
RI-S3W (5)
-3.1950
-0.0039
0.6354
4.9682
ARI-S3W (5)
-3.3935
0,0012
0.6330
4.7347
STSW
-3.4191
-0.0051
0.6341
3.5460
over 4h30m for ARI-S3W, both on Fire dataset) still outperforms or competitive with SSW and S3W variants.
6.4 SLICED- WASSERSTEIN AUTO-ENCODER (SWAB)
We apply STSW to generative modeling using the Sliced- Wassersetns Auto-Encoder (SWAE) (Kolouri al., 2018) framework, which regularizes the latent space distribution match a prior distribution Let 9 X' Sd and vb S" - X' be the parametric encoder and decoder. The objective of the SWAE ming.v Eyppp [o(æ, (b(((x)))] +-
(F#P, q), where controls reg- ularization, p is data distribution. We use SW, SSW (Bonet et al., 2022) and S3W variants (Tran al., 2024b) as baselines, Binary Cross Entropy (BCE) for reconstruction loss and mixture of 10 vMFs as the prior, similar to Tran et al. (2024b). We provide results in Table 4. We note that STSW has the best results in log 2-Wasserstein and NLL with a competitive training time, though its BCE slightly underperforms the others.
CONCLUSION
This paper introduces the Spherical Tree-Sliced Wasserstein (STSW) distance, novel approach leveraging a new integration domain called spherical trees. contrast to the traditional one- dimensional lines or great semicireles often used in the spherical Sliced Wasserstein variant, STSW ultilizes spherical trees to better capture the topology of spherical data and provides closed-form solutions for optimal transport problems on spherical trees, leading expected improvements in both performance and efficiency. We rigorously develop the theoretical basis for our approach by introducing spherical Radon Transform on Spherical Tree then verifying the core properties of the transform such as its injectivity. We thoroughly develop the theoretical foundation for this method by introducing the spherical Radon Transform on Spherical Trees and validating its key properties, such as injectivity. STSW is derived from the Radon Transform framework, and through care- ful construction of the splitting maps, we obtain a closed-form approximation for the distance.
Through empirical tasks spherical data, we demonstrate that STSW significantly outperforms recent spherical Wasserstein variants. Future research could explore spherical trees further, such as developing sampling processes for spherical trees adapting Generalized Radon Transforms to enhance STSW.
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ACKNOWLEDGMENTS
This research project is supported by the National Research Foundation Singapore under the AI Singapore Programme (AISG Award No: AISGG-TCC-2023-01--5000 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-8002040-00-000 A 4-8002039-00-000 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 are 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 standard dot product
12
(:
Sd
d-dimensional hypersphere unit vector
A
disjoint union
arccos
inverse of cosine function
('(X)
space of Lebesgue integrable functions on X
P(X)
space of probability distributions X
p.v
measures
8())
1-dimensional Dirac delta function
u($*)
uniform distribution on S
d
pushforward (measure)
Y(X
space of continuous maps from X metric metric space
d(..))
O(d)
orthogonal group of order element of group p- W-ssssssttein distance
g
W,
SW,
Sliced p- Wasserstein distance
T
(rooted) subtree
edge graph
We
weight of edge graph
Pe
stereographic projection at
H₂
hyperplane passes through æ and orthogonal to spherical ray spherical tree space of spherical trees edges on Sd
ry
T.
yi
Ty
distribution space of tree systems number of spherical tree number edges in spherical tree original Radon Transform
&
R
R
spherical Radon Transform on Spherical Trees (k: 1)-dimensional standard simplex splitting map
A4-1
tuning parameter in splitting maps
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Supplement to "Spherical Tree-Sliced Wasserstein Distance"
Table of Contents
Theoretical Proofs
16
A.1 Properties of R
16
A.2 Derivation and properties of splitting maps
17
A.3 Proof of Theorem 4.3
18
A.4 Proof of Theorem 5.2
22
A.5 Derivation for the closed-form approximation of STSW
23
Experimental details
25
B.1
Implementation
25
B.2 Evolution of STSW
25
B.3
Runtime Analysis
26
B.4
Gradient Flow
27
B.5
Self-Supervised Learning
28
B.6 Earth Data Estimation
29
B.7 Generative Models
30
A THEORETICAL PROOFS
A.1 PROPERTIES OF R&
Proof. Let f (S4).. We show that [|Q & [[]]]. Note that, arccos (I,z) E [0, n]. so we
have
Σ
( [R)((1)))))
di
Σ 1 1e f(u) aly, T) 5(t arccos x)) dy dt
arecos (z, v}) dy dt (L (((55)) aly, T): -5tt ((())
Σ ge (1," [f(v) aly, T)s 5(t arceos (z, v)) dt dy
Σ gd ((()) alv, ( 8(1 arccos (x.1)) di) dy
Σ I [s(w)) aay, T); dy
3dd
{ (f(r) Ea(sn alv, Ti dy
-10 (f(v) dy
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=If
implies that R7f= (T). which means the operator R3: L' (T) is well-defined. Clearly, R7 is a linear operator.
A.2 DERIVATION AND PROPERTIES OF SPLITTING MAPS
Invariance and Equivariance Properties in Machine Learning. Equivariant networks (Cohen & Welling, 2016) improve generalization and boost sample efficiency by incorporating task symme- tries directly into their architecture. These networks have been notably successful several fields, including trajectory prediction (Walters al., 2020), robotics (Simeonov et al., 2022), graph-based models (Satorras et al., 2021; Tran et al., 2024a), and functional networks (Tran et al., 2025a; 2024c; Vo et al., 2024), among others. The use of equivariance has been demonstrated to enhance perfor- mance, increase data efficiency, and strengthen robustness against out-of-domail generalization.
We recall the construction for splitting map presented in Subsection 4.2. We have map B: Sd Ty - _ defined as follows:
0,
if y x or and
Bly, T Tp.....y
(y) Ma)
(21)
arcoos
V (2, v), if y x.-I.
(x, y)²
Then a: Sd x T
A is defined as follows:
aly T) = softmax (18.9(. Bly T) T)ii=......))
(22)
We will show that
1. Where does 3 come from?
2. a is continuous.
3. o is O(d 1)-inaariant.
Proof. We prove each part.
For (y. T m E gd x Ty. let Ny be the hyperplane passes through Y and orthogonal x. Then Ny intersects the spherical ray oi at a single point a, and intersects the vector æ at a single point b. The B(y, T); is the length of the small are from y to on the circle centered at passes through y and a. Indeed, y x and y = -r, this length is equal to 0, the same as the definition of B. If y * x, =x, let is the intersection of the line passes through moreover, let be the unique intersection of the segment with endpoints 0, c, and the hyperplane Hy In details, we have
and the hyperplane H,;
(y and ||c||2" Note that, the condition y # x. - guarantee that # 0, co. We compute e details as follows:
(23)
(z, w)
(x,, y) y.
(24)
(xr, v)
1
SO
(r,y)
x +
y
(I, y)
1
(x,, yr
d
(25)
|
(I, 3)
1
(xr, v) V
(I, y)
(x,, y)
1
(I,
20 +
(x,, y) y
(26)
(x, v)
1
(x. 1 +
1
(x, 3)
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(2, v)
1 y
+
(x. y)
(x,, y)
(27)
(a, y)²
(x, v)
||x:1/
11
(r, 1)
(1 (Ir, 1))
(1 - (c,, 0))2
(1 (r, 13
(2r0)) 2 +
1
(0r, y) y
(or v)
(28)
(r, ui²
(1
(2, y²
(x. y) + y (x: y)²
(29)
Note that, since b is the projection of on vector x, so we have
b = (I, y) x.
(30)
By similarity, we have
length of are from y to a on the circle centered at passes through y and a
ly b||2
(31)
length of are from d to yı on the circle centered at 0 passes through d and ||d - Oll
Note that, the length of are from d to yi on the circle centered at passes through d and ys is
dga (d, y1) arccos (d, yi)
(32)
So
length of are from to the circle centered at b passes through and
(33)
arccos (d, vi) | bll2
(34)
||d 0|2
= arccos (d, vi) 1ly - bll2
(35)
(x, u) x +
arccos
Me
1ly - (x, v) - x1122
(36)
(x, y) 2
(u. Mb
arccos
V Ilull + (x, 110111 (x, (x.1)
(37)
(ar, 21
(y Ms)
V1 (x,
= arccos
(38)
(ar, v)
Bly T Wi.....
(39)
We finish the derivation of B. In context of splitting maps, this is reasonable choice, since it relates to evaluate distances from point to spherical ray.
2. The derivation of 3 clearly implies that 3 is continuous. We can also check the continuous of directly from the formula of B. Since 3 is continuous, we have a is continuous.
3. We have 3 is O(d ii-invariant since orthogonal transformations preserve the standard dot product. Since 3 is O(d + 1)- invarrian,,, we have a is O(d + -)-invariant.
A.3 PROOF OF THEBREM 4.3
Proof. Recall the notion of (vertical) Radon Transform (Quellmalz et al., 2023). Let od be the collection of all spherical rays on Sd. i.e.
Te (1 E ESS . E Hz).
(40)
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Note that, this is the same as the collection of all spherical trees with edge, i... Tí. For f L' L'()), consider the map (Rrg) f: ry [0, x - R defined by
f(t) (RT))(() / f(z) 8(t arccos (z, 2))
(Rr
(41)
Similar to Appendix A.I, we can show that (Rr²) (rr))
We have an operator
R: L
L'(rg))
(42)
r EEGR
f- (Rriy) (R))))
(43)
rjeda
This is exactly the (vertical) Radon Transform for Lebesgue integrable functions on S as in (Quell- malz et al., 2023). This is proved an injective linear operator, so if (Rr ) f
for all
Ty E dd, then f = 0.
Back to the problem. Recall that T is the space of all spherical trees of ls edges on So
TV (T . ... rT)
gd and y1. .i...... Uk Hz).
(44)
For an 1 i < k and r E Φd, define
(i) = (T
y/
(45)
In words, (i, ri)) is a subcollection of TØ consists of all spherical trees with root r and the th spherical ray is ry.. It is clear that To is the disjoint union of all (ú(i, rz) for ry E Φd.
TØ
(46)
We have some observations subcollections ri))
Result 1. Each orthogonal transformation g E O(d + 1) define a bijection between ri, and (i rg)) In details, the map bg defined by
0g
T r T Vi....uu
T (i, rg
(47)
gg 99 .9..."
(48)
well-defined and is bijection. This can be verified directly by definitions.
Result 2. For & & ls and ry ry E कh. we have
S alz, T)z,T) T) dT
S ,(2 dT
(49)
for all 2,2 E gd such that dga(z, = dga(x', 2²). Note that, the intergrations are taken over T (i, and (i(i, r with measures induced from the measure of Lgg To prove Equation (49), we first show it in two specific cases as follows:
Case 1, Assume x = x and y = y.
Case 2. Assume lies on r and z' lies on rig.
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If show that Equation (49) holds for assumptions in case and 2, then Equation (49) holds for all x, y, 2, x', V. 2'. Indeed, assume that Equation (49) holds for assumptions in case and 2. Then for all x, Us 2, x', v/,==, we consider t E ry and t' E rig such that
(a, 1) = dga (z,, =) = de d3 (e',t)
(50)
Then from the results in case and we have
Jegannz al2, Ti dt by 1 beeenr alt', ((',T))d dT
by cuse
1 SeLanED) x(t,T)dd dT
(51)
H a(z', T) dT. (52)
So Equation (49) holds for all 2;, y, 2, x', y/ z'. Now prove it holds for case and
For case 1, from the transitivity of orthogonal transformations on s there exists E O(d+ 1) such that
gz z. y.g2 = z'.
(53)
From Result 1, there is a corresponding bijection o, from (g) to (i(), We have
/ Jegarg T(cTT))dT
Jmgarg) a(s' d(gT)
(change of variables)
(54)
I a(gz,gT); d(gT)
(since z' = =gz)
(55)
SAgUuE alz, T)s d(gT)
(since a is O + 1)-invariant)
(56)
= S al2, T)s d(T)
(since det(g)) = 1)
(57)
(58)
So Equation (49) holds for case 1. A similar proof can be processed for case 2. From the transitivity of orthogonal transformations on Sd there exists h E O(d + 1) such that
hæ = x', by 1/ hæ = z'.
(59)
From Result 1, there is a corresponding bijection Φi from (i, to Ti rg)). We have
Heang al'', (),,T)),T dT=
( a(E',1T); d(hT)
(change of variables)
(60)
Leltomy a(hz, a(hz,hT); hT), d(hT)
(since z' h2)
(61)
Descur a(2,T); d(TT)
(since is O(d + 1) 1)-invariant)
(62)
f alz, d(T)
(since det) = 1)
(63)
(64)
We finish the proof for Result 2.
Result 3. From Result 2, for all < ls and [0, TT. we can define a constant (t) such that
(t = / T((
2(2.)).d dT
(65)
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for all ry E &d and S such that t dga 2) arccos (z, 2). ei (t) + c2(() +... + c* (1) = 1.
Then for all t [0, T], have
(66)
To show this, first, denote To the collection of all spherical trees with root on Sd We have
(x)x)
Tg(i, r.),
(67)
vEHan
so we have
-SEANO (FEENNE) alz, alz,); T)s dT dy
Sngoon alz, T)s dT = Honga
(68)
J Jnnne carccoss ()) dy
(69)
carccos 2)).
(70)
Then
Σ f
Ci (arccos (x, ... + Ck (arccos (z, 2),
alz, T)s dT
(71)
Tx((x)
k
Σ CT2 \i=1
dT
(72)
(eg(a)
1TT
(73)
1.
(74)
We finish the proof for Result 3. Consider a splitting map a in C(Sd TV 11-1) that O(d + 1)-invariant. For function f (S*) for each i < k, define function gi (=0 π] x Φd) as follows
9: :
[0, r] x Th
R
(75)
S
(t,r
Rts(t,r dT.
(76)
From the definition of R.
R& T R
(77)
(t, (t, - I f(y) ooy,T): 5(t arcoos (x, v)) dy.
(78)
we have
gi(t.rg) Segung) RFf(t,rg) dT
(79)
- (L alz, T)s ô(t arccos (z, z)) (x) dz dT
(80)
= =for (Frriap f(z) a(2, 8(t arccos (I, dT dz
(81)
= - f(z) 5(t - arecos (x, x))
(Frgang alz, (5,,7s dT
dz
(82)
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-Sh f(z) -(( - -arcoos (z, 2)) (arccos (z, 2)) dz
(83)
gd
c(tt) { f(z) .5(t - arccos (x, 2)) dz
(84)
= c(t) (Rrÿ)) f(t). ((t).
(85)
So
k
Σ
gi m = Σ ci(t) (Rrÿ) f(t) f(t)
(86)
i=1
k
Σ ca(tt) ((t)) =1 (Rrī) f(t) = (Rr;) f(t)
f(t)
(87)
Let / E Rer which means R4 ʃ 0 for all T E Ty. So g 0 E L' m| x Φअ) for all
1 i k. It implies (Rrz) Transform is injective, conclude that f = 0 L' (S*). R is injective.
/ 0 E L (rr)) for all ry Φd. So, from the (vertical) Radon
Remark. To formalize the proof above, the notion of Haar measure for compact groups is required. However, simplify the explanation goes beyond the scope of this paper.
A.4 PROOF OF THROREM 5.2
Proof. We want to show that
STSW(H, / JT War, RSv) (T)
(88)
is a metric on P(St).
Positive definiteness. For p. V E P(S"), it clear that STSW p)) and STSW v) > 0. If STSW = O, then W (RT p, Rgv) = 0 for all T E Ty Since W is metric on P(T), we have Ry = RRv for all T E TØ. By the injectivity of our Radon transform variant, we have =.
Symmetry.
For E P(8d), we have:
/ W RFV) (T
STSW(H,))
(89)
T (Rr,1 R71) dT = STSW(u, n).
(90)
So STSW v) = (V.).
Triangle inequality. For (2, Ua P(S)), we have: STSW (2) + STSW 13)
(91)
W 1 R2) (T + / W R&us) do(T)
(92)
Ty
/ (Wor R + War 1 R&Hs) dT
(93)
TV.
War.1 (R+ 43) (o
(94)
STSW( pa)-
(95)
So the triangle inequality holds for STSW.
We conclude that STSW is metric the space P(Sd).
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(d + -)-inaariance of STSW. For g E O(d + 1), show that STSW(H,, STSW gtu),
(96)
where g&u, giv as the pushforward of μ. v via orthogonal transformation g: 3 - sa respectively.
For T T T we have gT = -T 9y........" Note that det(g)) = 1, so
/ gdu(t)) a(y.gT); 8(t arccos (gx, v)) dy
Rgy (gip)(t, rgg.))
(97)
[ plg '1) aly. (T); 8(1 - arcoos (gx, y)) dy
(98)
JsL وlg gg) algy,gg gT); o(t arccos (وx, gyl) d(gy)
(99)
plv) .oly, T): 5(t arccos u)) d(y)
(100)
(t, r.).
(101)
Similarly, we have
RT (gtt)) rg,, 99h) = Rov(t, )t))
(102)
Since g induces an isometric transformation T 9T, so
War, (R9 Rgv)
W W (R Rut Ri Rotgtu).
(103)
We have
/
STSW stv)
W W (R9 Tt
(T)
(104)
W dg 1 (R (RT R Rag dg(gT)
(105)
W 1 R7v) (T)
(106)
T
ST,W(,,
(107)
So STSW is O(d 1)-invariant.
A.5 DERIVATION FOR THE CLOSED-FORM APPROXIMATION OF STSW
We derive the closed-form approximation of STSW for two discrete probability distributions μ and given as follows
(xx)= Σ Ug 8(x a²) and v(x)=
uj 8(x aj).
(108)
We can write μ and v in these forms by combining their supports and allow some uj and fy to be 0. Consider spherical tree -T= For 1 < j & n, let ej = dga (I, az), and also let Go = 0. By re-indexing, we assume that the sequence co, ... Cn is increasing.
co
C2 & ... & Cn-
(109)
For & j < n and 1 < i & k. consider all points r = (c,, r on the spherical tree T. Since
co 0, we have
(1) = 0
Z0
0
(110)
and for 1 E n. .r and orthogonal to x, and the spherical ray r We compute R4 For L [0, π] and <isk k.
(1) is exactly the unique intersection between the hyperplane passes through az
R9 (1)) = 1 p(v) aly. T) 5(t - arccos (r. dy
(111)
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/
Σ uj 8(y - az))
aly, T): 8(1 - arccos (a) dy
(112)
{
Σ
Uj
aly, T)- (8(y az) 8(t arccos (x) dy.
(113)
j=1
So,
1. If é fci,...,cch Cn then Rt rg)) 0; and,
2. If = e for some j. then utt rg,) = (((. = R = ala,, T)..u)
Similarly, we have
1. Ift & (ci,..., 3. then R.vu(t, rgg)) 0; and,
2. If ey for some j, then Rt r.. RRv (c) = =RF((a)))) = ala,, uy.
For < < and 1 < i < k. let
( = a((aj T). " = a(aj, T) Uj and 0j
(114)
Consider T as graph with nodes s for & & & Note that 2/6 for all i, and we assign this is the root of TT Two nodes adjacent is the shortest path on T does not contain any other nodes. In other words, the set of edges in T are efy) (z - for 1 < i k. < n, (xx -2)--)) has length Cj Cy-1- For an edge ej its further endpoint from the root is x)) Also, for node zf6) with > 0, the corresponding subtree I(x))) contains all nodes (i)
and ej (
(t)
Tp with j < p n. From these above observations, can μ and transform R$μ and (t) RyV supported on nodes of T, where the mass at node zj is uj and u respectively. So, from Equation (3), we have
a RRv)
Σ We -
(115)
eET
k
(cj - -g-1) -
(116)
k
(g - -g-1)
Σ -
(117)
Σ (es ---1
u
(1)
Σ
u(att))
(118)
F=1
Σ
(cj C--1)
Σ
(
100
(119)
J=1
i=1
Σ(g (c ---1)
Σ
Σ
up
(1)
(120)
1=1
p=j
Σ (cg - ---1)
Σ
(aps T): (up - up))
(121)
1=1
p=
(122)
This is identical to Equation (19). We finish the derivation.
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Wasserstein
Sinkhorn
100
SW₂
SSW;
Seconds
25W BS
2SW,, Unif
10
53W₂
RI-53W,
10
ARI-S3W,
STSW2
10
10a
10*
10a
Number of samples in each distribution
Figure 3: Runtime Comparison, averaged over 15 runs.
B EXPERIMENTAL DETAILS
All our experiments were conducted on single NVIDIA H100 80G GPU, For all tasks, if specified, hyperparameter Ç in STSW is set to its default value of 2.
B. IMPLEMENTATION
We summarize a pseudo-code for STSW distance computation in Algorithm 1.
Algorithm Spherical Tree-Sliced Wasserstein distance.
Input: μ4, E P(84) u(x)
Σ}= "j 5(x ag). v(x) Ej=1 by 0(x az), number of
spherical trees I., number of rays in spherical trees k. splitting maps with weight 5 R. for / = to L do
Sample (!)), V1 ..... 3k iid NO (O,,Idu+1) Ids+1).
Compute (1) - (11) /||x (4) 12 and y/ - Pais (1, Ille (v, lle.
Contruct spherical tree Ti = T
v1
Compute War, (R R4j /) by Equation (19).
end for
Compute STSW (p, v) (1/L) E/ Way, (R9, R9. Return: STSW (p,
B.2 EVOLUTION OF STSW
In this section, examine the evolution of STSW well as different distances when measuring two distributions. line with (Bonet et al., 2022; Tran et al., 2024b), we select source distribution vMF(-. 0) a.k.a uniform distribution and target distribution vMF(u, R). We initialize 500 samples in each distribution. We use kappa & 10, L 200 trees, k = 10 lines for STSW, L = 200 projections for other sliced metrics, Nz 100 rotations for RI-S3W, ARI-S3W, and pool size of 1000 for ARI-S3W in all experiments unless specified otherwise. Results are averaged over 20 runs.
Evolution w.r.t Figure 4 shows the evolution of various methods w.r.t to As expected, STSW aligns with the trends in S3W and SSW, decreasing with higher dimensions, unlike KI. divergence. Here, use derived form for KL divergence (Davidson et al., 2018; Xu & Durrett, 2018) as follows:
14a-11/2(k)
d (d+1.
d+1 -1 log K log(2m)
[L)
=MTWH1)/-1(A)) + 2
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0m
300
150
206
190
100
$360
213
322
-13
185
100
150
200
130
Figure 4: Evolution between (MF(p, x)) and vMF(., 0)) w.r. on gd across various methods. We
use E {1,5, 10, 20, 30, 40, 50, 75, 100, 150, 250}
d+1 1
d (+11 1
log ((+1))2-1(5)) +
log + log log l'
2
2
539
0-3
d+3
d-3
0-
0-30
01
30
R1-Y3/V
53.8
0
3-
Figure 5: Evolution between rotated vMFs distributions, averaged over 100 runs. d denotes data dimension.
Evolution w.rt rotated vMFs. Next, we evaluate a fixed vMF distribution and its rota- tion along great circle. Specifically, compute metric between vMF((1, 0, O........ k) and vMF((cos f, sin 8, O,...), for E {kkT)/6]12 {
We plot results in Figure 5
Evolution of STSW w.r.t Number of Trees, Number of Lines and Ç. Next, we study the effect of the number of trees and lines STSW. If not specified, we fix ş 10 and d = 3. We present the results Figure 6, Figure and Figure 8
B.3 RUNTIME ANALYSIS
Runtime Comparison. We now perform runtime comparison with other commonly used dis- tance metrics, including the traditional Wasserstein, Sinkhorn (Cuturi, 2013), Sliced-Wasserstain (SW), Spherical Sliced- Wasserstein (SSW) (Bonet et al., 2022) as well as Stereographic Spherical Sliced Wasserstein (S3W) (Tran et al.. 2024b) and its variants (RI-S3W, ARI-S3W). For a fair com- parison, we also include SSW, with binary search (BS) and Unif when a closed form available for uniform distribution. We set I. 200 projections for all methods. For STSW, we use L. 100
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Roed fun
Poot lire Unaon 2009
300
09 3000
Foet
Rumber Thc
Vumber 2002
Figure 6: Evolution of STSW between the source and target distributions when varying number of trees {1, 10, 50, 100, 200, 400, 500, 750,900, 10003.
F-dd Sar Teas
Foad hurt Thees
Foad Hon Toee 230
240
0- 3000
Tumber 1imm
Proet
Reed 1630
Linder Sturet
Figure 7: Evolution of STSW between two distributions w.r. number of lines A
{1, 10, 50, 100, 200, 400, 500, 600, 700, 750).
trees and & = 10 lines. The runtime of applying each of these methods on two distribution on S is illustrated Figure 3.
To further assess STSW performance, we conduct a runtime analysis to un- derstand the computational cost associated with different configurations. We again choose uniform distribution and vMF(u, k) where к = 10 as our source and target distribution and use STSW to measure distance between these two probabilities. All experiments are repeated 20 times with de- fault parameters set to L = 200 trees, k 10 lines and N = 500 samples, unless otherwise stated.
Runtime Evolution.
We vary the number trees 9a. adjust the number of lines & across (5, 10, 25, 50, 100, 150, 200, 300, 500, 750, 1000} in Fig- ure 9b and change the number of samples N within (500, 1000, 3000, 5000, 7000, 8000, 10000) in Figure 9c. We note that the runtime of STSW scales linearly with these parameters.
{200, 400, 500, 750, 900, 1000, 1250, 1500, 1750, 2000] in Figure
B.4 GRADIENT FLOW
The probability density function of the von Mises-Fisher distribution with mean direction μ E gd given by:
f(; p, k)) = Ca(k) exp(kpT x)
dd/2-1
where 6 > 0 concentration parameter and the normalization constant Ca(k) (2T)p/² (
Our target distribution, 12 vMFs with 2400 samples (200 per vFM), have 50 and
/4 = (-1, 0, 0). μ/ (0, o),
/12 = (1, 1. 0),
(-1, 66 0). H4 (0,-1. -4).
μ4 = (1, 10-bb10 0)
A4 (0, 1, (),
μĄ (0. -0) A12 = (-0,0,
(Ø, 0, - #10 (ó, 0,),,
= (-ò, 0, -1),
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1-
6-1
575
de
6010
d-300
6-100 100
d= 1000
03
Dela
Dalta
Figure 8:
Evolution of STSW
between
tWO
distributions
W.r.t
5
{1. 2,5, 10, 20, 25,50, 50, 75, 100, 150, 200}
d=3
d=3
0.0
300
d= 300
500
d- 500
0.00
Number
Number of lines
(a)
(b)
3.04
Seconds
de 10
2.50
de 1030
3.50
1300
4000
5000
1903
20000
samples.
Figure 9: Runtime of STSW w.r.t number of trees, lines and samples
where o- 1+ V5
The projected gradient descent described in (Bonet et al., 2022):
((++1) = (k) vV (n()) ( STSW (l++1) ((t+1) [
Setup. fix L, 200 trees and & lines. For the rest, we use I, 1000 projections. As in the original setup, ARI-S3W (30) has 30 rotations with pool size of 1000 while RI-S3W (1) and RI-S3W (5) have and 5 rotations respectively. We train with Adam (Kinga et al., 2015) optimizer Ir 0.01 over 500 epochs and additional Ir 0.05 for SSW.
Results. As seen from Table and Figure 10, STSW provides better results in log 2- 2-Wasserstein distance and NLL., while also being efficient in terms of both runtime and convergence speed.
We perform additional experiments on the most informative sliced methods including MAX-STSW, MAX-SSW, and MAX-SW. We present Table 5 the results after training for 1000 epochs with a learning rate LR = 0.01. Each experiment is repeated 10 times. Figure 11 illustrates the log 2- Wasserstein distance between the source and target distribution during training. We observe that MAX-STSW performs better than others.
B.5 SUPERVIPE LEARNING
Encoder. Consistent with the setup in (Bonet et al., 2022; Tran et al., 2024b), we train a ResNet18 (He et al., 2016) CIFAR-10 (Krizhevsky, 2009) data for 200 epochs using batch size of 512. We use SGD as our optimizer with initial Ir = 0.05 a momentum 0.9, and a weight decay 10-3 The standard data augmentations used to generate positive pairs similar prior works (Wang & Isola, 2020: Bonet et al., 2022; Tran et al., 2024b) which include resizing, cropping, horizontal flipping, color jittering, and random grayscale conversion.
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SSW (LR=0.01)
-15
SSW (LR=0.05)
-2.
S3W
RI S3W (1)
1og10 (W2)
RI RI_S (5)
ARI_S3W (30)
STSW (k=5)
-35
-4.0
-4.5
100
200
300
400
500
Iteration
Figure 10: Log 2- Wasserstein distance between source and target distributions
Table 5: Learning target distribution 12 vFMs, trained for 1000 epochs and averaged over 10 runs.
Method
log W2 J
NLL J
MAX-SW
-3.10 + 0,06
-4959.14 + 12.22
MAX-SSW
-2.76 0.02
-4868.78 + 60.51
MAX-STSW
-3.19 + 0.03
-5007.72 + 16.34
We set L 200 trees and Ao 20 lines for STSW and fix L 200 projections for all other sliced distances. NR 5 and a pool size of 100 are used for RI-S3W and ARI-S3W as in Tran et al. (2024b). For settings of the regularization coefficient, please refer Table 6.
Linear Classifier. A linear classifier is then trained on feature representations from the pre-trained encoder. Similar Bonet et al. (2022), train it for 100 epochs using the Adam (Kinga al., 2015) optimizer with a learning rate of 10-3, a weight decay of 0.2 at epoch 60 and 80.
Results. We report in Table the best accuracy of the linear evaluation features taken before and after projection on gd where = 9. The visualizations of learned representations when be found in Figure 12.
B.6 EARTH DATA ESTIMATION
Similar to Bonet et al. (2022) and Tran et al. (2024b), we use exponential mapping normalizing flows model consisting of radial blocks with 100 components each, totaling 24000 parameters. The model is then trained with full batch gradient descent via Adam optimizer. Dataset details provided in Table 7.
Setup. Our settings for STSW in this task are L 1000 trees, A 100 lines, and Ç 100. We use Ir = 0.05 for STSW, S3W, RI-S3W and ARI-S3W and 0.1 for SW and SSW, We train other sliced distances for 20,000 epochs as in the original setup while our STSW is only trained for 10,000 epochs.
Results. Table 3 highlights the competitive performance of STSW compared to the baseline meth- ods. To further evaluate the efficiency of our approach, we compare the training time of STSW with that of the second-best performer, ARI-S3W, using the Fire dataset. Our findings show that STSW (2 hours 10 minutes) is twice as fast as ARI-S3W (4 hours 30 minutes). We also present in Figure 13 the normalized density maps of test data learned.
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-122
MAX_SW SW
-150
MAX SSW
MAX_STSW STSW
-1.75
10g10 (W,)
-2.00
-2.25
-2.50
-2.5
-3.0
-3.25
200
400
600
800
1000
Iteration
Figure 11: Log 2- Wasserstein distance between source and target distributions.
Table 6: Regularization coefficient À across various methods w.r. projection on g in Self- Supervised Learning task.
STSW
SSW
SW
S3W variants
d 9
X == 10.0
=
== 1.0
d 2
A = 10.0
20.0
x= 1.0
À = 0.1
B.7 GENERATIVE MODELS
Setup. We use Adam (Kinga et al.. 2015) optimizer with learning rate Ir = 10 10-3, We train with batch size of 500 over 100 epochs using BCE loss as our reconstruction loss. We choose L 200 trees and A: 10 lines for STSW. Following the same settings in Tran et al. (2024b), we fix L 100 projections for others, NR rotations for RI-S3W and ARI-S3W, and pool size of 100 random rotations ARI-S3W. We use prior 10 vMFs, X = 1 for STSW, À = 10 for SSW, and = 10 for SW and S3W variants.
Additional Results MNIST. For quantitative analysis, train the SWAE framework on MNIST and report the FID score Table 8, along with the generated images in Figure 14. We follow the same settings in Tran et al. (2024b), which the latent prior U(52) and train the model with batch size of 500 over 100 epochs. For STSW, we fix L = 200 trees and A = 10 lines L 100 projections and learning rate LR 10-3, 33 as described in Tran et al. (2024b). The FID scores are computed using 10,000 samples from the test set.
with learning rate LR 0.01 and X = For other sliced methods,
We use the same model architecture as specified in Tran et al. (2024b).
CIFAR-10 Model Architecture.
Encoder:
3x3x2233
Conv2dg2 ReLU Conv2dg2 - ReLU
Conv2d61 ReLU Conv2d61 - ReLU
Conv2d ReLU Conv2dr2s Flatten
FC512 - ReLU - FC3
l normalization $2
Decoder:
2
FC512
FC2048 ReL.U
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airplane
automobile
75
60
bird
45
cat
15
30 60 so
deer
0
dog
15"
30
frog
horse
60
75"
ship
truck
(a) STSW
75
75
60
60
30
30
15
15a
1501
90'
601 s0: 05 3 32 OF 20
30 30"
40
30 on 60 DO0
30
-5
45
60
60
75"
-75
(b) ARI-S3W
RI-S3W
75
75
60
60
45
45
15
250 90100990000000000 50 yet 01. OF 120
300 1122 60'
30
30
o0
30 120 150
33
45*
60'
-60
-75*
-75*
(d) S3W
(e) SSW
75
75
60
60"
45"
15
230 200 90 400 or
oe 600 90% 1201250
150
30t our so 2or
-30
60
-0
-75*
-75*
(f) SimCLR
(g) Hypersphere
Figure 12: Distributions of CIFAR-10 validation
3² after pre-training.
Reshape x 4) Conv2dT122 128 ReLU
Conv2dTes - ReLU Conv2dTe4 ReLUU
Conv2dT32 + ReLU Conv2dT22
ReLU
Conv2dT2 Sigmoid
MNIST Model Architecture.
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(a) Ground Truth Fire.
STSW Fire.
(c) Ground Truth Flood.
(d) STSW Flood.
Ground Truth Earthquake.
(f) STSW Earthquake.
Figure 13: Density estimation on earth data. The left figures (ground truth) represent training data estimated with KDE. The right ones depict the normalized log likelihood of the trained models on test data.
Table 7: Earth datasets.
Earthquake
Flood
Fire
Train
4284
3412
8966
Test
1836
1463
3843
Data size
6120
4875
12809
Encoder:
I 28
Conv2dg2 ReLU Conv2da2 ReLU
Conv2de4 - ReLU - Conv2d6t ReLU
Conv2d12s - ReLU - Conv2d12s Flatten
FC512 - ReLU - FC3
l normalization E g
Decoder:
2 2
FC512 - PC512 - ReLU
Reshape(128 x 2) Conv2dT 128
ReLU
Conv2dTe1 + ReLU - Conv2dTe1 ReLU
Conv2dT - ReLU Conv2dT ReLU
- Conv2dT, - Sigmoid
32

---

Published as a conference paper at ICLR 2025
Table 8: Average FID of runs on MNIST.
Method
FID |
SW
73.35 + 2.01
SSW
76.14 + 2.73
S3W
75.55 = 2.80
RI-S3W (10)
72.80 + 3.39
ARI-S3W (30)
70.37 + 2.58
STSW
69.16 2.74
/
6
9
5
8
9
3
5
7
2
5
0
7
1
8
3
6
0
4
0
O
2
6
8
0
9
7
7
0
9
4
2
0
7
6
/
7
7
3
3
8
8
2
9
9
6
O
8
5
4
5
3
9
9
9
0
8
9
2
4
2
4
4
5
9
5
5
6
4
(a) SW.
(b) SSW.
(c) S3W.
7
9
7
1
3
3
8
4
2
3
9
3
8
9
9
7
9
3
8
3
9
9
0
9
9
8
5
2
9
7
7
7
3
9
6
0
0
3
3
4
3
9
7
9
2
2
9
0
2
8
4
7
3
9
9
8
O
9
1
7
8
1
2
2
7
3
O
3
9
5
6
(d) RI-S3W (10).
(c) ARI-S3W (30).
(f) STSW.
Figure 14: Generated images of different methods on MNIST of SWAE.
33


Reviews:
[{"review_id": "sttlMOpvUZ", "reviewer": "Reviewer_oCQ3", "summary": "The paper is a natural extension of sliced spherical OT to incorporate tree systems. The authors propose a topological space on spheres called spherical trees by connecting spherical rays with a common root. They then adapt Radon transform onto the spheres and slice the trees which is equivalent to slicing the spheres with trees. After showing that spherical tree are metric spaces, the authors followed classic approach of adapting the OT computation in the tree-sliced spheres. They provided comprehensive theoretical and empirical results to support their theories.", "questions": "419: \"STSW outperforms the baselines in all metrics and achieves faster convergence.\" Why is that? Is this result theoretically predictable? \n\n466: \"We also conduct experiments with $d=2$ to visualize learned representations.\" Why don't we directly project the learned features in the original image space to a sphere, rather than redoing the experiments on the sphere?\n\n443: The variances from STSW are quite small. What's the explanation of that? Why doing tree-slicing on spheres is more robust (against different sampling?)?\n\nWe have seen spherical trees in the following article, although it's for a different application and the construction of the trees is different as well. Is there any connection between this work and that? \n\nMeng, Yu, Yunyi Zhang, Jiaxin Huang, Yu Zhang, Chao Zhang, and Jiawei Han. \"Hierarchical topic mining via joint spherical tree and text embedding.\" In Proceedings of the 26th ACM SIGKDD international conference on knowledge discovery & data mining, pp. 1908-1917. 2020.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The motivation is justified and the narrative follows standard ones when substituting simpler structures with more sophisticated variants for solving specific OT problems. Claims have been proved by theoretical results and verified by empirical results.", "weaknesses": "While I didn't find major weaknesses in the paper, the authors didn't provide sufficient theoretical explanation in several critical places. The proposed STSW outperforms baselines in all most all metrics. First, why did the variances in Table 1 so small, smaller than one tenth of other methods in most lines, under Monte Carlo sampling? Why did STSW outperforms other baselines in terms of runtime? The theoretical complexity the authors provided cannot explain it. Why didn't the better runtime translate to a similar margin in reducing the training time in Table 2? Another point is that the authors attributes the better performances to \"the ability to capture topological information of integration\ndomain\" of STSW but they didn't show a direct connection in the paper. The results from the CIFAR dataset are not that impressive and the visualization didn't help either in explaining 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": "fqyl9uT5Rr", "reviewer": "Reviewer_8AUo", "summary": "This paper introduces the Spherical Tree-Sliced Wasserstein (STSW) distance, an efficient optimal transport (OT) metric for measures on a spheres. By leveraging a novel spherical Radon transform that integrates over spherical tree structures, it provides closed-form OT solutions and maintains computational efficiency. Theoretical analysis and experiments, including gradient flows and self-supervised learning, confirm STSW’s effectiveness and its performance against recent benchmarks.", "questions": "* I was wondering if incorporating a Markov chain over the distributions of the slices, instead of using a uniform distribution, could help in generating more informative tree-slices.\n\n* I am interested in understanding the topology of the tree corresponding to the most informative slice in spherical trees and compare its effectiveness to the most informative slice in SSW, and essentially comparing them to the most informative slice in SWD. (By the most informative slice I mean Max-slice). This can be done on a chosen benchmark.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* Paper is very well-written.\n* Through extensive experiments, the effectiveness of the STSW has been investigated.\n* Paper introduces a novel Radon transform for the measures on the spherical trees.", "weaknesses": "* While STSW aims to be efficient, the spherical Radon transform and tree-slicing require considerable computation, especially as the number of edges or the dimension of the hypersphere increases. This could limit scalability for very high-dimensional or densely-sampled spherical data, impacting runtime in large-scale applications. Although I understand that the authors have provided extensive runtime comparisons, I would like to see the scalability of the method on higher-dimensional tasks beyond CIFAR, MNIST, and similar datasets.\n\n* The effectiveness of STSW relies on the quality of sampled spherical trees, which introduces variability in metric accuracy. If the sampling process fails to capture diverse spherical structures adequately, STSW’s results might be inconsistent, especially in complex distributions where more refined tree structures are necessary. I would like to see which strategies (e.g., Markov Chains, Random Paths, etc.) could be applied here to sample more informative slices.", "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": "jfnHLBOg4x", "reviewer": "Reviewer_k7Wt", "summary": "This paper introduces the Spherical Tree-Sliced Wasserstein (STSW) distance, a novel metric designed for optimal transport on spherical domains. The key innovation lies in the integration over spherical trees as the domain, rather than traditional one-dimensional lines or great semicircles used in existing spherical Sliced Wasserstein approaches. This change allows STSW to better capture the underlying topology of spherical data, offering closed-form solutions that enhance both performance and computational efficiency.\n\nThe authors introduce a variant of the spherical Radon Transform tailored for spherical trees and prove its injectivity. Defining the STSW in terms of this transform is essential for establishing the metric properties of the distance, including its invariance under the action of the orthogonal group.", "questions": "Besides the experimental comparisons of the new STSW with SW, SSW, and S3W variants, are there any analytic comparisons among them? Which are the differents in the topologies defined by these different approach?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well-structured, presenting clear objectives and a comprehensive review of related work. \nThe efficiency of the new metric is well presented in the experiments. \n\nLeveraging on the ideas presented in Tran et al. (2024b), as said before, this reviewer things that the key innovation of this article (over articles as Bonet et al. (2022) and Tran et al. (2024a)) lies in the integration over spherical trees for defining the new metric between spherical probability measures.", "weaknesses": "The approach builds on previous work by Bonet et al. (2022) and Tran et al. (2024a), in the sense that uses the same hight-level ideas. \nHowever, while the research incrementally follows the line of previous studies by Bonet et al. (2022), Tran et al. (2024a), and Tran et al. (2024b), it offers meaningful advancements by developing a metric specifically adapted for spherical data analysis.\nAlso, the experiments closely follows experiments previously presnted in papers as Bonet et al. (2022).", "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": "5Rgz6si1du", "reviewer": "Reviewer_mQXD", "summary": "The paper proposes a variant of the Wasserstein distance for distributions defined on the hypersphere. The proposed metric, called the Spherical Tree-Sliced Wasserstein (STSW) distance, adapts the Tree-Sliced Wasserstein distance to be applicable to the hypersphere by defining a novel spherical Radon transform. The proposed metric is invariant to orthogonal transformations, and can be computed efficiently.", "questions": "- **Impact uniformity loss**: How should we understand $STSW(z^A,\\nu)$ in eq. 20?  $z^A$ is a single point, so I assume we are considering a Dirac distribution at $z^A$. Theoretically, the distance from a Dirac distribution to a uniform distribution in the sphere is constant regardless of where the Dirac distribution is placed, due to invariance to rotations, right?  Why does this term favour uniformity if it is a constant term? Or am I misunderstanding something?\n- **Radon Transform measure preservation**: Why does the proposed Radon transformation in eq. 8 transform a probability distribution $\\mu$ defined on $\\mathbb{S}^d$ into a probability distribution defined on $\\mathcal{T}$? This is mentioned in lines 332-333, but in line 268 it says that $||\\mathcal{R}_{\\mathcal{T}}^{\\alpha}f||\\leq||f||_1$.  So it does not immediately follow that the Radon transform preserves the measure.\n- **STSW Computation on continuous measures**: In section 5 you explain how to compute STSW in practice, but it is assumed that the probability distributions are discrete.Is it possible to get a closed form analogous to that in eq. 19 for non-discrete distributions? \n- **Injectivity of the radon transform**: In Theorem 4.3 it is proved that if the splitting map is $\\mathcal{O}(d+1)$-invariant, then the spherical Radon transform is invariant. What would be the consequences of using a non-injective spherical Radon transform? What structure might be missed?\n\n**Minor & Typos:**  \n- l. 170 \"be **a** positive\"\n- l. 372 $\\nu(x)=\\sum_{j=1}^n$ \n- In l. 322 change notation of $\\delta$ to other letter, in order to avoid possible confusion with the Dirac delta function.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* The paper is well-written. \n* The paper proposes an efficient extension of the Sliced Wasserstein distances applicable to hyperspheres and the construction of the Radon transform is well-motivated.\n* The authors also show desirable properties of their proposal, such as invariance to orthogonal transforms.\n* On the practical side, the authors propose an efficient way to compute such a metric and show its effectiveness and superior performance in various experiments.", "weaknesses": "* The proposed metric is limited to hyperspheres. The paper's contributions appear to be incremental, primarily combining previously established concepts (tree systems and Sliced Wasserstein distances) and extending them to the sphere setting.\n* One of the claimed contributions is a bit misleading. Specifically, the authors claim to derive a closed-form expression; however, its computation still relies on approximations: first, due to the need for sampling trees, and second, by considering only discrete distributions in the explanation of how it is computed. This should be clarified, as the current claim gives the impression that the distance can be computed exactly due to the closed-form expression.\n* Line 65 states that the use of tree systems enhances the capture of topological information. However, it is not so clear to me why is this the case. An experiment demonstrating this advantage would be useful.", "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": "hDmk4lxWCS", "reviewer": "Reviewer_Gf54", "summary": "The paper introduces a novel way to measure distances between probability distributions on hyperspheres, coined Spherical Tree-Sliced Wasserstein (STSW). The core technical contribution lies in adapting tree-based structures from [1] to work on spheres (spherical trees) and defining a new type of Radon transform for these structures. The authors prove this approach leads to closed-form solutions for optimal transport problems and show that STSW is a valid distance metric on $\\mathcal{P}(\\mathbb{S}^d)$. Through various experiments including gradient flows, density estimation, and self-supervised learning, they show that STSW can perform competitively with/better than the baselines while having faster runtime.\n\n---\n\n[1] Tran, Viet-Hoang, et al. \"Tree-Sliced Wasserstein Distance on a System of Lines.\" arXiv preprint arXiv:2406.13725 (2024).", "questions": "- What makes STSW run faster than S3W [3], and in some cases, SW?\n\n- The tree structure is supposed to capture 'richer' topological information per the authors' claim. How does that translate to practical results? Have the authors explored different hyperparameters to confirm that the superior performance in these setups is due to the tree component of the method? Traditional trees in euclidean spaces often have a hierarchical structure; here, it appears that our design choice is not hierarchical? If so, then what are the concrete benefits?\n\n- Are there relationships to the OT distance on the spheres?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "S1. This method addresses an important problem of comparing distributions on the sphere, which has applications in many fields.\n\nS2. Extending the tree-sliced concept to spherical domains is novel and non-trivial.\n\nS3. The writing in the main paper is rigorous, coherent, and easy to follow.\n\nS4. STSW is proved to be a proper metric (compared to SSW [2] which is only known to be pseudometric) with a novel splitting maps that preserve orthogonal invariance.\n\nS5. STSW appears to outperforms various baselines in terms of runtime and other quantitative metrics.\n\nS6. The algorithm is straightforward to implement.\n\n---\n\n[2] Bonet, Clément, et al. \"Spherical sliced-wasserstein.\" arXiv preprint arXiv:2206.08780 (2022).", "weaknesses": "W1. Sampling: In the algorithm, the authors propose sampling uniformly from R^{d+1} and then normalizing to get points on S^d, which does not produce a uniform distribution on the sphere. Would this induce a bias (and implications)? \n\nW2. Ablation: The paper would be strengthened if there are more insights provided via ablations on different design choices (i.e., rays, trees). How does the current tree structure help capture the data better than existing methods? Are there limitations, theoretical issues, or numerical instability associated with different components of the method (i.e., S3W [3] has the north pole issue). Can the splitting maps be learned? etc.\n\nW3. Runtime and Complexity: It would be nice to have an explicit discussion of the computational/memory complexity (this is aside from the information provided in Appendix B).\n\nW4. Experiments: This may be a minor point, but setup and hyperparameters could be better documented for all methods. In addition, there is no comparison with Vertical SW in appropriate setups. For generative experiments, there are no samples, of quantitative measures of the quality of images (i.e. the FID score).\n\n---\n\n[3] Tran, Huy, et al. \"Stereographic Spherical Sliced Wasserstein Distances.\" Forty-first International Conference on Machine Learning. 2024.", "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)