Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions
Authors: Khai Nguyen
Nhat Ho
Venue: NeurIPS 2024 poster (2024)
Keywords: Sliced Wasserstein
Heterogeneous Joint Distributions
Optimal Transport
3D shapes as points.
Primary area: machine_vision
TL;DR: We propose a scalable metric for comparing heterogeneous joint distributions i.e., distributions have marginal supports on different domains.

Abstract: Sliced Wasserstein (SW) and Generalized Sliced Wasserstein (GSW) have been widely used in applications due to their computational and statistical scalability. However, the SW and the GSW are only defined between distributions supported on a homogeneous domain. This limitation prevents their usage in applications with heterogeneous joint distributions with marginal distributions supported on multiple different domains. Using SW and GSW directly on the joint domains cannot make a meaningful comparison since their homogeneous slicing operator, i.e., Radon Transform (RT) and Generalized Radon Transform (GRT) are not expressive enough to capture the structure of the joint supports set. To address the issue, we propose two new slicing operators, i.e., Partial Generalized Radon Transform (PGRT) and Hierarchical Hybrid Radon Transform (HHRT). In greater detail, PGRT is the generalization of Partial Radon Transform (PRT), which transforms a subset of function arguments non-linearly while HHRT is the composition of PRT and multiple domain-specific PGRT on marginal domain arguments. By using HHRT, we extend the SW into Hierarchical Hybrid Sliced Wasserstein (H2SW) distance which is designed specifically for comparing heterogeneous joint distributions. We then discuss the topological, statistical, and computational properties of H2SW. Finally, we demonstrate the favorable performance of H2SW in 3D mesh deformation, deep 3D mesh autoencoders, and datasets comparison.

Hierarchical Hybrid Sliced Wasserstein: A Scalable Metric for Heterogeneous Joint Distributions
Khai Nguyen
Nhat Ho
Department of Statistics and Data Sciences The University of Texas Austin Austin, TX 78712
Department of Statistics and Data Sciences The University of Texas at Austin Austin, TX 78712
khainb@utexas.c edu
minhnhat@utexas.c edu
Abstract
Sliced Wasserstein (SW) and Generalized Sliced Wasserstein (GSW) have been widely used in applications due to their computational and statistical scalability. However, the SW and the GSW are only defined between distributions supported on a homogeneous domain. This limitation prevents their usage applications with heterogeneous joint distributions with marginal distributions supported on multiple different domains. Using SW and GSW directly on the joint domains cannot make meaningful comparison since their homogeneous slicing operator, ie.. Radon Transform (RT) and Generalized Radon Transform (GRT) are not expressive enough to capture the structure of the joint supports set. To address the issue, we propose two new slicing operators, i.e., Partial Generalized Radon Transform (PGRT) and Hierarchical Hybrid Radon Transform (HHRT). In greater detail, PGRT is the generalization of Partial Radon Transform (PRT), which transforms subset of function arguments non-linearly while HHRT is the composition of PRT and multiple domain-specific PGRT on marginal domain arguments. By using HHRT, we extend the SW into Hierarchical Hybrid Sliced Wasserstein (H2SW) distance which is designed specifically for comparing heterogeneous joint distributions. We then discuss the topological, statistical, and computational properties of H2SW. Finally, we demonstrate the favorable performance of H2SW in 3D mesh deformation, deep 3D mesh autoencoders, and datasets comparisor
Introduction
Optimal transport 55 47 is powerful mathematical tool for machine learning, statistics, and data sciences. As an example, Wasserstein distance [47]. defined as the optimal transportation cost between two distributions, has been used successfully in many areas of machine learning and statistics, such as generative modeling on images 53). representation learning [55, vocabulary learning [57]. and so on. Despite being accepted as effective distance, Wasserstein distance has been widely known as a computationally expensive distance. In particular, when comparing two distributions that have most supports, the time complexity and the memory complexity of the Wasserstein distance scale with the order of (n log n) [45] and Onn²) respectively. In addition, samples approximate continuous distribution with its
the Wasserstein distance requires empirical distribution in high dimension since its sample complexity is of the order of O(n -I/d) 20) where n is sample size and dl the number of dimensions. Therefore, Wasserstein distance not statistically and computationally scalable, especially in high dimensions.
Along with entropic regularization [17] which can reduce the time complexity and memory complex- ity of computing optimal transport to (n3) and (n²)) in turn, sliced Wasserstein (SW) distance [I]
Code for this paper is published at
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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is one alternative approach for the original Wasserstein distance. The key benefit of the SW distance that it scales the order (n log n) and (n)) in terms of time and memory respectively. The reason behind that fast computation is the closed-form solution of optimal transport in one dimension. To
leverage that closed-form, sliced Wasserstein utilizes Radon Transform [33] (RT) to transform a high-dimensional distribution to its one-dimensional projected distributions, then the final distance is calculated as the average of all one-dimensional Wasserstein distance. By doing that, the SW
distance has very fast sample complexity i.e., O(n O/22) 1/2). which makes it computationally and statistically scalable in any dimension. Therefore, the SW distance has been applied successfully in
various domains of applications including generative models [19], domain adaptation (32 cluster-
ing (27). 3D shapes [30 29], gradient flows 34 g
Bayesian inference computation 37 58 texture
synthesis [22] and many other tasks.
Despite being useful, the SW distance is not as flexible as the Wasserstein distance in terms of choos- ing the ground metric. In greater detail, the number of ground metrics in one dimension is limited, especially ground metrics that lead to the closed-form solution. As result, the role of capturing the structure of distributions belongs to the slicing/projecting operators. To generalize RT to non-linear projection, generalized Radon Transform (GRT) is introduced in [3 with circular projection (28). polynomial projection [5T], and so on. With GRT, Generalized Sliced Wasserstein (GSW) distance is proposed in (26). In addition, there is line of works on developing sliced Wasserstein variants on
different manifolds such as hyper-sphere 6 54 40 50 hyperbolic manifolds 77, the manifold of
symmetric positive definite matrices [10) general manifolds and graphs (52] In those works, special variants of GRT are proposed.
Although the SW has become more effective on multiple domains, no SW variant is designed specifically for heterogeneous joint distributions joint distributions that have marginals supported on different domains, except for the product of Hadamard manifolds [9]. It is worth noting that marginal domains of heterogeneous joint distributions can be any metric space and are not necessary manifolds. Heterogeneous joint distributions appear in many applications, e.g., domain adaptation domains 15 4
comparing datasets with labels [I) 3D shape deformation [29], and so on. this case, Wasserstein distance can be adapted by using mixed ground metric, i.e., a weighted sum of metrics on domains [15] 1 In contrast to the Wasserstein distance, the adaptation of SW has not been well- -nvestigaggg.dd Using GSW directly with one type defining function for all marginals cannot separate the information within and among groups of arguments.
Contribution: In this work, we tackle the challenge of designing a sliced Wasserstein variant for heterogeneous joint distributions. In summary, our main contributions are three-fold:
1. We first extend the partial Radon Transform to the partial generalized Radon Transform (PGRT) to inject -ineariy into local transformation. We discuss the injectivity of PGRT for some choices of defining functions. We then propose a novel slicing operator for heterogeneous joint distributions, named Hierarchical Hybrid Radon Transform (HHRT). In partieular, HHRT is hierarchical transformation that first applies partial generalized Radon Transform with different defining functions on arguments of each marginal to gather marginal information, then applies partial Radon Transform on the joint transformed arguments to gather information among marginals. We show that HHRT is injective long the partial generalized Radon Transform is injective.
2.
We propose Hierarchical Hybrid Sliced Wasserstein (H2SW) which is novel metric for comparing heterogeneous joint distributions by utilizing the HHRT. Moreover, we investigate the topological properties, statistical properties, and computational properties of H2SW. In particular, we show that H2SW is a valid metric on the space of distribution over the joint space, H2SW does not suffer from the curse of dimensionality and enjoys the same computational scalability as SW distance.
3. A 3D mesh can be effectively represented by a point-cloud and corresponding surface normal vectors. Therefore, it can be seen as an empirical heterogeneous joint distribution. We conduct experiments on optimization-ban based 3D mesh deformation and deep 3D mesh autoencoder to show the favorable performance of H2SW compared to SW and GSW. Moreover, we also illustrate that H2SW can also provide meaningful comparison for probability distributions on the product of Hadamard manifolds by conducting experiments on dataset comparison.
2

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Organization. We first provide some preliminaries on SW distance, GSW distance, and joint Wasserstein distance in Section 2 We then define the hierarchical hybrid Radon transform and hierarchical hybrid sliced Wasserstein distance in Section 3 Section 4 contains experiments on 3D) mesh deformation, deep 31) mesh autoencoder, and datasets comparison. We conclude the paper in Section 5 Finally, we defer the proofs of key results, and additional materials in the Appendices.
2 Preliminaries
Wasserstein distance. For P 2 1, the Wasserstein-P distance [55 47 between two distributions J E (x)) and E P()), where and y are subsets of Eº and they share ground metric e: X x + R+, defined as:
/ dz, ()
w)(n := inf xEll(a, Xxy
(1)
where I(,,)) = { E x VH Sy dr(z, ((x), dr v(v) vv))))
When and are discrete with at most n supports, the time complexity and the space complexity of the Wasserstein distance is On log n) and (nn)) ) in turn which are very expensive. Therefore, sliced Wasserstein is proposed as an alternative solution. We first review the definition Radon Transform.
Radon Transform
23 The Radon Transform R (RR)) ) + IL, (R x 8d-1) is defined as:
(R)()))) = J [(x)((1 (x,, 0))de.
Radon Transform defines a linear bijection (23]. Given projecting direction 8, (+ 0) is an one-dimensional function. With Radon Transform, we can now define the sliced Wasserstein distance.
Sliced Wasserstein distance. For 2 1, the Sliced Wasserstein (SW) distance [ of p-th order between distributions μr P(X) and P()) with an one-dimensional ground metric R x R R is defined as follow:
SW c) = Es- (g Rotv;:)]
where Radp and Rotv are the one-dimensional push-forward distributions created by applying Radon Transform (RT) [23] on the pdf of and with the projecting direction 6. The computational benefit of SW distance comes from the closed-fore solution when the one-dimensional ground metric c(z, v) h(x y) for h is strictly convex function:
Wg Rotu; c) f (FRrdau()) ( (z) dz,
where Red and Raiv inverse CDF of Rejj and Rotv respectively, When μ and are discrete with at most supports, the time complexity and the space complexity of the closed-form is On logn) n) and (n) respectively.
Generalized Radon Transform and Generalized Sliced Wasserstein distance. To generalize RT non-linear operator, the Generalized Radon Transform (GRT) was proposed [3. Given defining function [26] g Rd Ω + I, the Generalized Radon Transform GR L, (Rd) + LL (R x ??)
defined
f f(x))(t g(x, 0))dr.
(GRf)(t,
For example, we can have the circular function 228 i.e., g(z, || r012 for r E R+ and Ω gd-1 homogeneous polynomials with odd degree (51) (mn) i.e., g(x, Eja =m 0,2 - x; Ω Sd., and so on. Using GRT, the Generalized Sliced Wasserstein (GSW) distance is introduced in (26). which is formally defined as follow:
with (an,...,,aa, Od1, Nd., |al| do Oliy σ
GSW (oo--gs-1//WgGGGR GR
(4)
It is worth noting that the injectivity of GRT is required to have the identity of indiscernible GSW.

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Heterogeneous joint distributions comparison. We are given two joint distributions pz 222) E P(R, X2) and v(y1, E P()) x)2 where X, are Y share ground metric ei X x x3i + R-
and X2 are Y2 share a ground metric X2 x 3/2 - R with (c1 # c2). In this case, previous works utilize the joint distribution Wasserstein distance [5 to compare μ and
/ (c1(21, y1) + 02 (2 V2) (d) V1, V2, (5)
inf Wi rEll(u.) XxXXXXX1XX2
where ((, = E x Xe x D x D2) xyx du dz( T2, y1, 92) = (1 12),
Sx,xA x,, dr (z 22, V1, 92) v 12) We can easily extend the definition to joint distributions with more than two marginals (see Appendix B). contrast to the Wasserstein distance, there is no variant of SW that is designed specifically for this case. SW variants can still be used by treating Xi X.22 and 1 x V22 as homogeneous spaces X' and y which share the same Radon Transform variant and one-dimensional ground metric However, that approach cannot differentiate the difference between X1 and X2, and leverage the hierarchical structure, i.e., inside and among marginals.
Hierarchical Hybrid Sliced Wasserstein Distance
In this section, propose the Hierarchical Hybrid Radon Transform (HHRT) which first applies P(G)RT on each marginal argument to gather each marginal information, then applies PRT on the joint transformed arguments from all marginals to gather information among marginals. After that, we introduce Hierarchical Hybrid Sliced Wasserstein distance by using HHRT as the slicing operator.
3.1 Hierarchical Hybrid Radon Transform
We first introduce the first building block in HHRT, i..., Partial Generalized Radon Transform (PGRT). Definition 1 (Partial Generalized Radon Transform). Given defining function g: Red Ω - IR, Purtial Generalized Radon Transform PGR: I1 (TR.. x (Rdd)) + I, ( x Q x Rd2) is defined as:
(PGRf)(t, = Sr f(z, v)8(t - g(z, 0))
(6)
When g(5, 0)= (xr, O). PGRT reverts into Partial Radon Transform (PRT) 33
Proposition 1. For some defining function g such linear, circular, and homogeneous polynomials with an odd degree: the Partial Generalized Radon Transform is injective, i.e., for any functions fI, f2 E L' (R*). (PGR.f)(t, 0,
f2) (t,t. 0 y) Vt, 0 y implies f1 fz-
The proof of Proposition Tis ii given in Appendix A. The main idea to prove the injectivity of PGRT is show that given fixed y, the PGRT is the GRT of f(:y).
Definition (Hierarchical Hybrid Radon Transform). Given defining functions 91 Rd 22 - R and 92 dRda x S2 - TR Hierarchical Hybrid Radon Transform HHR LT (Reh x RRd2) - x G2 8) defined as:
L [R x 22
[ X 221, (1 19igg 01) (22 02)) day dx2;
(HHRJ)) 01 02 12)
(7)
where 3i (V/, 42) E S.
The reason for using PRT for the final transform is that the previous PGRT's are assumed to be able to transform the non-linear structure to a linear line. However, PGRT can still be used as a replacement for PRT. Definition 2 can be extended to more than two marginals (see Appendix B).
Proposition 2. For some defining functions 91 92 such as linear, circular, and homogeneous polyno- mials with an odd degree; Hierarchical Hybrid Radon Transform injective, i.e., for any functions fi, f2 E Tr (R.), (HHR..):(tt 01, 02, v (HHRf2)(C 01, 02, 1/)) Vt. 01, Oz, vi implies fi = f2-
The proof of Proposition is given in Appendix A.2 The main idea to prove the injectivity of HHRT is to show that HHRT is the composition of PRT and multiple PGRTs.

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We are given f(x) (21 (n 2 1, 0; 2 Vi). The HHRT of f(z) is (HHRf) 01,02,0) 2i a; (t- --11g (z,1,)
HHRT of discrete distributions.
, 0,5((=11,22)
(2,2, 02)). For 9I and g2 that are the linear function and (or) the circular function, the time complexity of the transform (d1 + d2) which the same as the complexity of using RT and GRT directly. However, HHRT has an additional constant complexity scaling linearly with the number of marginals, i.c., two marginals in Definition 2
Example 1. In this paper, we focus on 3D shape data (mesh) with points and normals representation, i.e., shapes as points representation [46]. In particular, we can transform a 3D shape into a set of points and normals by sampling from the surface of the mesh. In addition, we can convert back to the 3D shape from points and normals with Poisson surface reconstruction (25) algorithm. In this setup, a shape is represented by a 6-dimensional vector (x1, x2) where x1 Xi E R3 and x22 E X22 E $2. For the set X, E R33, we can use directly the linear defining function 91 (2) 01) (F1, 01) with 01 32. For the set X2 E S2 with r E + and 02 E 8 As alternative options for X2, we can use other defining functionn from special cases of GRT including Vertical Slice Transform [49], Parallel Slice Transform (50), Spherical Radon Transform (6), and Stereographic Spherical Radon Transform (54).
utilize the circular defining function 92 (2 (0222022))0) 02) = 1:22110212
Inversion. In Proposition 2 we show that HHRT is the composition PRT and multiple PGRT's Therefore, the inversion of HHRT is the composition of the inversion of multiple PGRT (invertibility of PGRT depends on the choice defining functions 3 28
and the inversion of PRT 23
3.2 Hierarchical Hybrid Sliced Wasserstein Distance
By using HHRT, we obtain a novel variant of SW which is specifically designed for comparing heterogeneous joint distributions.
Definitions. We now define the Hierarchical Hybrid Sliced Wasserstein (H2SW) distance.
Definition 3. For p 2 1, defining functions 91, 92, the hierarchical hybrid sliced Wasserstein-p (H2SW) distance between two distributions μ E P(X, X2) and P(), V2) with an one- dimensional ground meiric
IR x IR + R+ is defined as:
H2SW (H,, V;C, C, $1. 92) = (,, (2,00) x0xx8) x8) (a) (HHHR 91.2 $1, HHRg, 0, (v; (8)
where HHRe, uu and HHRRa w are the one-dimensional push-forward distributions cre- ated by applying HHRT.
Definition 3 can be easily extended to more than two marginals (see Appendix B
Topological Properties. We first show that H2SW is valid metric the space of distributions any sets X' x ) E dR x d2 (d1, 2 1).
Theorem 1. For any p 2 1, ground metric and functions functions 9, which lead to the injectivity of GRT, the hierarchical hybrid sliced Wasserstein H2SW, 91, 92) is a metric on P(R x Reda)
i.e. it satisfies the symmetry, non-negativity, triangle inequality, and identity of indiscernible.
5

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The proof of Theorem is given in Appendix A.3 It is worth noting that the identity of indiscemible property proved by the injectivity of HHRT (Proposition 2]. We now discuss the connection of H2SW to GSW and Wasserstein distance in some specific cases.
Proposition 3. For any 2 I, c(z, y) = Jz y), and μ.v. E P(R x IR we have: (i) H2SS (14, V/; C, 01, 92) _ (un, V1; c, 91) +SW (112, 0, where (x( }u( ( and p2(Y) = (R x Y) (similar with 1/1 and 122).
(ii) If 91 92 are linear defining functions, H2SW,(p, 1, 92) (1;c) W2 (iii) If 91, 92 are linear defining functions, (1,
The proof of Proposition 3 is given in Appendix A.4
Sample Complexity. We now discuss the sample complexity of H2SW.
p. c(x, v) = =|x- yl. 91 are linear defining functions or circular defining functions, and μ. V E P (124 x (2d) with the corresponding empirical distributions pn and Vn (n Z 1), there exists constant Cp.q depending on p, q such that:
Proposition 4. For any p 2 1, dimension d1, d2,d2 2
E. H22S ((1nn Mai 91,92) H2SW (3
$
1/2p if > 2p, n 1/2p log(n)² if = 2p,
9 Σ
Cpp.
q C (M)()) Mi(v))
(9)
n no if E (p. 2p),
where M1(1)) and My( are the q-th moments of μ and v, Cgi.g is constant depends on 91, 92-
The proof of Proposition 4 is given in Appendix A.5 The rate in Proposition is as good as the rate of SW in (38. however, it is slightly worse than the rate of SW in 44 due to the usage of the circular defining functions and simpler assumptions. To the best of our knowledge, the sample complexity of GSW has not been investigated.
Monte Carlo Estimation. Since the expectation in H2SW (Equation 8) is intractable, Monte Carlo estimation and Quasi-Monte Carlo approximation (39] be used to form practical eval- uation of H2SW. Here, we utilize Monte Carlo estimation for simplicity. In particular, we sample i,idd (S). After that, we form iidd U(12)), and th,,..., Dr. .a......21.
011,-- OI.. i.id (S,)) O2...... O21.
the following estimation of H2SW:
1 T,
H2SW
W p 91 Pa tu, HHR
Bar #tze).
(10)
Proposition 5. For any 2 1, dimension d1, d2 2 1, and v E P(Id x 1Rda), we have:
E H2SW p (n, 91. L) (4,
Var W WHH .2 Sp HHR 91 s
(11)
where the variance is with respect U(4 S22 8).
The proof of Proposition given in Appendix A.6 From the proposition, we see that the estimation error of H2SW is the same as SW which O (1.
Computational Complexities. The time complexity and memory complexity of H2SW with linear and circular defining functions are O Ln log n + L(d, + d2 + k)m) and O + (d1 + d2 + k)n) with A is the number of marginals i.c., 2. We can see that the complexities of H2SW are the same as those of SW in terms of the number of supports n and the number of dimensions d. We demonstrate discrete distribution with L realization of 01, 02 v
the process of HHRT compared to GRT in Figure [ Overall, the complexities of defining functions are often different in the number of dimensions, hence, H2SW always scaled the same SW in the number of supports i.e., O(n log n).
Gradient Estimation.
applications, it is desirable to estimate the gradient We can move the gradient operator to inside the expectation
V VoH2SW3( 92)

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Table 1: Summary of joint Wasserstein distances across time steps from deformation from the sphere mesh to the Armadillo mesh.
Destunees
Step 100 (W. 12... 4)
Step 300 (W....
Step 500 (W.,r
Step 1500 (W.... J)
Step 4000 (W.
5000
(W.,.a
1=100
1852.510: 1,236
1436 686--3.00 3,056
1071 2222-222.444
452-42.55
6.19:00.07
2.726-0
GSW L-10
1893.438.3.205
1192.5223.274
143.51811.00
8.73100353
H2SW L-10
1840.73-1.282
1422.66747..813
165%.171+5.362
95.672=4.3766
6.326+00115
2.602=0.201
SW L=100
18:7,572.0,300 303
1226.125....8 5252
16991127.1.100
89.693100,998 0,73
4.453.10.22
0.050
GSW 1-100
1889 989--312-00885
1525 269++1 Loms
1179.1 100.1+22.00 052
122.618=1.172 1.75
7:905+00.573
3,226=0. 388
H2SW 10000
18.99,47::1.990
1417.1
1048.99554..08
86.078:0.625
4.61=0443
1.096=0.117
Table 2: Summary of joint Wasserstein distances (multiplied by 100) across time steps from deformation from the sphere mesh to the Stanford Bunny mesh.
Distances
Step 100 (W... em 3)
Step (W...s rs
Step 500 (W....
Step (W..>> 3)
Step 4000 (W...s
Step (W.,.o J)
SWI=10
06.86800.579
4.46.00.999
1.52:10.08]
0.023:0.024
0,14:00.188
GSW 1=10
26.837-0.490
4.378--0.12 128
1.548+00.02
0.653+6.01
0.173-0.018
6.146-0.013
H25W L-10
0.119
2.22110.124
1.45210.005
0.636.00.055
0.177:10:009
0,089_0..022
SW 1-100
26.678-0. 168
4.109+0.138
1.458+00 142
0.362-0.028
0.072+0.017
0.849-0.006
GSW L=100
26,795_.0,202
4.08411,1109
1.375.000019
0.372.10.026
10,00
0.012_00:11
H2SW L-000
23.772=0.19
2.288=0.009
1.358=00051
0.458:00.026
0.064+00.01
0.08=000007
and then apply Monte Carlo estimation. The gradient V. WWCHHHR .2 can be computed easily since the functions 91, 92 are usually differentiable.
8,.. 2, HHR (2. c)]
Beyond uniform slicing distribution. H2SW is defined with the uniform slicing distribution in Definition 3 however, it is possible to extend it to other slicing distributions such as the maximal projecting direction [18) distributional slicing distribution [2),, and energy-based slicing distribu- tion 41 Since the choice of slicing distribution is independent of the main contribution i.e., the slicing operator, we leave this investigation to future work.
H2SW for distributions on the product of Hadamard manifolds. A recent work 9 extends sliced Wasserstein on hyperbolic manifolds 7 and on the manifold of symmetric positive definite matrices (T0) to Hadamard manifolds i.e., manifold non-positive curvature. The work discusses the extension of SW to the product of Hadamard manifolds. For the geodesic projection, the closed-form for the projection is intractable. For the Busemann projection, the Busemann projection on the product manifolds is the weighted sum of the Busemann projection with the weights belonging to the unit-sphere. In the work, the weights are a fixed hyperparameter i.e., Cartan-Hadamard Sliced-Waad Wasserstein (CHSW) utilizes only one Busemann function to project the joint distribution. In contrast, H2SW utilizes the Radon Transform on the joint spaces of projections i.e., considering all distributed weighted combinations which is equivalent to considering all Busemann functions under a probability law. As result, the H2SW is a valid metric as long as the Busemann projections can be proven be injective (the injectivity of the Busemann projection has not been known the moment) while Cartan-Hadamanl fixed weighted combination not trivial show. Moreover. H2SW does not only focus on the product of Hadamard manifolds i.c., H2SW is a generic distance for heterogeneous joint distributions in which marginal domains are not necessary manifolds e.g., images [40], functions [2T]. and so on. In the later experiments, we conduct experiments on comparing 3D shapes which are represented by a distribution on the product of the Euclidean space and the 2D sphere (not Hadamard manifold).
Sliced- Wasserstein is only pseudo metric since the injectivity of
4 Experiments
In this section, we first compare the performance of the proposed H2SW with SW and GSW in the 3D mesh deformation application. After that, we further evaluate the performance of H2SW in training a deep 3D mesh autoencoder compared to SW and GSW. Finally, we compare H2SW with SW and Cartan-Hadamard Sliced-Vo -asseestess (CHSW) in datasets comparison on the product of Hadamard manifolds. In the experiments, we use c(x, y) = |xr yl and p 2 for all SW variants.
4.1 3D Mesh Deformation
In this task, we would like to move from a source mesh a target mesh. To represent those meshes, we sample 10000 points by Poisson disk sampling and their corresponding normal vectors of the mesh surface those points. Let the source mesh be denoted as X(0) = (x1 (O). ., xn, (0) } and the target mesh be denoted as Y = {y1..... }. We deform (0) to Y by integrating the ordinary differential
equation (t) -n Xx(o [S ( En 8(x xt),)), 1 Σ Ei= 8(y -y))], where S denotes a SW
7

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GPW
Source
Step 500
Step 1600
Step
Stop
Step
Figure 2: Visualization deformation from the sphere mesh to the Armadillo mesh with L 10.
SW
cow
HZON
Stop 500
Step 1500
Step 4000
Step 5000
Target
Figure 3:
Visualization of deformation from the sphere mesh to the Stanford Bunny mesh with L
10.
variant. We utilize the Euler discretization scheme with step size 0,01 and 5000 steps. The normal vectors projected back the sphere after taking an Euler step. For evaluation, we the joint Wasserstein distance in Equation 5 with the mixed distance from the Euclidean distance and the great circle distance. We use the circular defining function for GSW, and use the linear defining function and the circular defining function for H2SW. We vary the number of projections L E {10, 100) } for all variants. For H2SW and GSW, we select the best hyperparameter the circular defining function E (0.5, 0.7, 0.8, 0.9, 1, 5, 10, 50, 100).
Results. We compare H2SW with GSW and SW by deforming the sphere mesh to the Armadillo mesh [59 We report the quantitative results in Table ] Jfter 3 independent runs and the qualitative result for L 10 Figure 2 and 100 in Figure 6 Appendix From Table
observe that H2SW helps the deformation convergence faster at the beginning and better at the end in terms of the joint Wasserstein distance, especially for small value of the number of projections i.c., = 10. The result for L = 100 is better than L = 10 which is consistent with Proposition 5 The qualitative results in Figure and Figure also reinforce the favorable performance of H2SW since they are visually consistent with quantitative scores. We also conduct deformation to the Stanford Bunny mesh f16 59 Table Figure 3 and Figure Appendix Dnd we observe the same phenomenon that H2SW is the best variant for 3D meshes. From those experiments, H2SW has shown the benefit of the HHRT in transforming joint distribution over the product of the Euclidean space and the 2D sphere compared to the conventional RT of SW and the conventional GRT of GSW.
4.2 Training deep 3D mesh autoencoder
We utilize the processed ShapeNet dataset [2] from [46], then sample 2048 points and the cor- responding normal vectors from each shape in the dataset. Formally, would like train an autoencoder that contains an encoder fo that maps a mesh X E 2024886 to latent code E R 10224 and decoder gy that maps the latent code back to the reconstructed mesh X 2041866 We adopt

---

Table 3: Joint Wasserstein distance reconstruction emros (multiplied by 100) from three different runs of autoencoders trained by SW, GSW, and H2SW with the number of projections L = 100 and L 1000.
Distance
Epuch 500 ce,oz())
Epoch 1000 We.ee(() (+)
Epoch 2000 W es
SW L=100
136.87 i 1.18
133.24 + 0.50
131.10 0.34
GSW L=100
136.51 + 0.20
133.36 + 0.24
130.80 0.46
H2SW L=100
135.54 + 0.72
132:24 + 0.36
130 24 = 0 0.47
SW L0
135.85 0.92
132.88 0.27
130.93 0.11
GSW L=1000
136.40 0.10
133.02 0.98
130.76
H2SW L=1000
135.47 + 0.64
132.17 + 0.10
129.87 0.44
SW
H2SW
Figure 4: Visualization some randomly selected reconstruction meshes from autocncoders trained by SW, GSW, and H2SW in turn with the number of projections = 100 at epoch 2000.
architecture to construct the autoencoder. We want to train the encoder fo and the decoder 9p such that X (x)(( se X for all shapes X in the dataset. To do that, we solve the following optimization problem:
Point-Net 48
min Xx>m(x)(S(P) Po.
6.7
where S is sliced Wasserstein variant, and Px - - 8(x xx) denotes the empirical distribu- tion over the point cloud X (Ii,..... ). We train the autoencoder for 2000 epochs on the training set of the ShapeNet dataset using an SGD optimizer with learning rate of 1e 3. and a batch size of 128. For evaluation, we also use the joint Wasserstein distance in Equation with the mixed distance from the Euclidean distance and the great circle distance measure the average reconstruction loss on the testing set of the ShapeNet dataset. We use the circular defining function for GSW, and use the linear defining function and the circular defining function for H2SW. For H2SW and GSW, we select the best hyperparameter of the circular defining function E {0.5. 0.7, 0.8, 0.9, 1, 5, 10). For more details such as the neural network architectures, we refer the reader to Appendix B
Results. We report the joint Wasserstein reconstruction errors (measured in three independent times) the testing set in Table with trained autoencoder epoch 500, 1000, and 2000 from SW, GSW, and H2SW with the number of projections I. = 100 and L, = 1000. In addition, we show some randomly reconstructed meshes for epoch 2000in Figure 4 and for epoch 500 in Figure in Appendix D From Table 3 we observe that H2SW yields the lowest reconstruction errors for both L 100 and L 1000. Moreover, we see that the reconstruction errors are lower with L = 1000 than ones with L = 100 for all SW variants. The qualitative reconstructed meshes in Figure Figure &reffect the same relative comparison. and the qualitative performance of autoencoders can be improved by using more powerful neural networks. Since we focus on comparing SW, GSW, and H2SW, we only use light neural network i.e. Point-Net [88 architecture. The trained autoencoders can be further used to reduce the size of 3D meshes for data compression and for dimension reduction, however, such downstream applications not focus in the current investigation of the paper.
worth noting that both the qualitative

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Table 4: Relative error to the joint Wasserstein distance of SW. CHSW, and H2SW.
Distances
100
500
1000
2000
SW
4.618 + 0.744
4.253 + 0.398
4.233 + 0.310
4.198 + 0.238
CHSW
4.449 0.497
4.063 0.254
4.059 0.167
4.035 0.145
H2SW
4.381 0.695
4.001 0.267
4.048 0.182
3.998
0.142
SW
CHSW
H2SW
Joint Wasserstein
Figure 5: Cost matrices between datasets from SW, CHSW, and H2SW with I. 2000.
4.3 Comparing Datasets on The Product of Hadamard Manifolds
We follow the same experimental setting from 9 Here, we have datasets as sets of feature-lahel pairs which are embedded in the space of Rd 42 where 4 denotes Lorentz model of d2 dimension (a hyperbolic space). We uses MNIST [31 dataset, EMNIST dataset [14], Fashion MNIST [24]. For CHSW, we use Busemann projection on the product space of Euclidean and the Lorentz model. For H2SW, we use the linear defining function and the Busemann function on the Lorentz model. We refer the reader to Appendix B for greater detail on Busemann functions and experimental setups. We compare SW, CHSW, and H2SW by varying L E {100, 500, 1000, 2000]. For evaluation, we use the joint Wasserstein distance as the ground truth. In particular, let Cw be the cost matrix from the joint Wasserstein distance and C' be a given cost matrix, we use IC/ max - Cw/ maxxaaaa (Cw) as the relative error.
dataset [56]. KMNIST dataset [3] and USPS dataset
Results. We report the relative errors from SW, CHSW, and H2SW in Table @bbleaafie Jaterr 100 independent runs. In addition, we show the cost matrices from SW, CHSW, H2SW. and joint Wasserstein distance with L = 2000 in Figure 5 Cost matrices for L 100, L 500, and L = 1000 are given in Figure 9 in Appendix D From Table 4 we see that H2SW gives lower relative error than CHSW and SW. Therefore, using H2SW for comparing datasets is the most equivalent to the joint Wasserstein distance in terms of the relative error. We also observe that increasing the value of the number of projections also reduces the relative errors for all SW variants. Again, we would like to recall that H2SW can be used for heterogeneous joint distributions beyond the product of Hadamard manifolds shown in previous experiments.
5 Conclusion
We have presented Hierarchical Hybrid Sliced Wasserstein (H2SW) distance, a novel sliced probability metric for heterogeneous joint distributions i.c., joint distributions have marginals different domains. The key component of H2SW is the proposed hierarchical hybrid Radon Transform (HHRT) which is the composition of partial Radon Transform and multiples proposed partial generalized Radon Transform. We then discuss the injectivity of the proposed transforms and theoretical properties of H2SW including topological properties, statistical properties, and computational properties. On the experimental side, we show that H2SW has favorable performance in applications of 3D mesh deformation, training deep 3D mesh autoencoder, and datasets comparison. In those applications, heterogeneous joint distributions appear in the form of joint distributions on the product of Euclidean space and 2D sphere, and the product of Hadamard manifolds. In the future, we will extend the application of H2SW to more complicated heterogeneous joint distributions.
10

---

Acknowledgements
NH acknowledges support from the NSF IFML 2019844 and the NSF AI Institute for Foundations of Machine Learning.
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Supplement to "Hierarchical Hybrid Sliced Wasserstein; A Scalable Metric for Heterogeneous Joint Distributions"
We first provide skipped proofs in the main paper in Appendix ^ We then provide some additional materials including additional background and extended definitions in Appendix] B After that, we discuss some related works in Appendix C Finally, we report computational infrastructure Appendix E
We report additional experimental results in Appendix D
A Proofs
A.1 Proof of Proposition !
For any f,0,u y, we are given (PGRfi)(t, (P)R/2)),, 0. y). By Definition 1 we have:
J (i(,, g(x, 0))dr = [ f2 ()( g(x, 0))dx.
For any RR we have:
J 2d1
fi(2,))o(t g( 0))c (2.() dxdy 1 | fe(2, ()),
(22 -22 drdy.
Applying the Fubini's theorem, have:
J fi(2,D) SSe.
e dyō(1 g(2, 0f)do Jeen Jacn Ja(ur, w)e (2 dyo(l g 0) d),
Rd1
Junz
which is:
SR / y)e (2nte,n) dy) GR
((z,8))-2nieen))
/ f2(x,
fa(x,v)e -)
By the injectivity of GRT, we have:
S f1 v) 12x (s.0)
/ f2 u)e -o)e--2xtee dy.
Then, for any E TRd,, we have
Jee [ Jeez fi (2m(=.u) e (e,) dydr = Jun Jxd, fe(2, D) 12 -127le,,) dydr.
which is (Ffi(z,)) (Ff2(z, v))) with F denotes the Fourier transform. By the injectivity of the Fourier Transform, we have Si (2. = [2(2, y) for any x, y, which concludes the proof.
A.2 Proof of Proposition
2
We first show that HHRT is the composition of PGRT and PRT. We have
(PR(PGR(PGRf))(C 01 02 v))
1 I
Rd Ja f(a. - (1)o02))((tz g2(v.02))0f (2) - thtt - 22t2 dxdyddddydt dtz
[ [ ((x / 8(11 gi 01) 0 - 92 (v. 02)) vidh idd2ddd
JRL J (12,8)0 (i 01) - V2 02)) drdy (HHR))(L 01, 02, v).
For any t, ,, 02 2b, we are given (HHHRf))(t (t, ( v), which is equiva-
lent to:
(R
(P 02, v/)
By the injectivity of the PRT and the PGRT, we obtain fi(2,y) f2(x, v) for any x, y which completes the proof.
14

---

A.3 Proof of Theorem 1
To prove that the hierarchical hybrid sliced Wasserstein H2SW p C, 91, (2) is a metric on the space of distributions on P(R4 x (d)) ) for any p 2 1, ground metric and defining functions gl.92, we need to show that it satisfies non-negativity, symmetry, triangle inequality, and identity of indiscernible.
Non-Negativity. Since W b. Fu, HHR
02.. F c) 2 0 47 for any f1, 82, 10, we have:
(6, (0) x (HHHR 92 un, HHR o2 uu c) 2 0,
which means that H2SW (M, C, 92) 2 0 for any μ and
Symmetry.
Since we have the symmetry of the Wasserstein distance WP (HHH s HHRS 02.0, (V;c) = any 01, 02 1. we have:
az Wg(HHR Dr 02 tv HHR
0, o
oz
47 for
W (HHRR 8 82
S HHR 9 92 tt;;c)
= 00) x W (HHHR 8,8, 0 tv, HHR a
8, cp;
which means that H2SW y (a, 9,, 92) H2SW p (v, 15 c, 91 92) any μ and
Triangle Inequality. Given e to a valid metric on R. we can use the triangle inequality of the Wasserstein distance. For any distributions μl, M2, #s E P(R x Rd)), we have:
H2SW (15,p(11,44 C, 92. (0,0, 02.0) x0x
8..2. tu HHR uiee e)l)
02 x (HH
#u Hug HHR 02 #
d2 + +W, 82.0 Fus. HHR 2 9. #2 8
uuz ()))))))
E(,,, (2.0000,) x0xx
02 tun HHR 02 tuo;
(20 x
91 gs Eus, HHR de 1 Aw2;c)))
= H2SW (P1, P3i 91, 92) + H2SW (U3, 42; 91.92),
where the final incquality is due to Minkowski's ' inequality. Therefore, we complete the proof for the triangle inequality of the hierarchical hybrid sliced Wasserstein.
Identity of indiscerníble. For any p 2 1, ground metric c, and 91.92. when V, we have HHR S2 02, Ff = (HHR)) 8, ge 02. tV.
Therefore, we have
WB(HHRR 02. $2. & A/1, HHHR2. (112;c) Now, assume that H2SW,
0 which leads H2SW C, H,W. (H, V;C, 92. =. then WH(HHR 92 ab 2p,, HHR 51 .82 =P c)
0.
for almost everywhere 0, E Ω1 E Q2 E S. By applying the identity property of the Wasserstein distance, we have HHRg) P2 Ej = (HHRg)) 0,,0, e for almost everywhere
0 E Ωr Og Ω2 v E S. Since the HHRT is injective (proved in Proposition 2.
we obtain v.
A.4 Proof of Proposition 3
(i) For any 2 1, c(z, v) = J - y/. and u. P(Rc x R2), we have:
H2SW (14, V: 91,
(2xx)) ( 20 H 91.02 22 tu; tv;cc] 1
/ (e1 91 (1),))) + 9n (02, 12)) 22, 22, 122 1) 1 =(B (rein
inf
15

---

By applying the
inequality, we have:
H2SW p(1, V; 91, 92)
inf ( / (Vvi+ (V
(g2(02, g2 (02, 92)) )2))) (x 22 81 882)
E
inf (Ell((nN) / - 91) (92(02,,
E
g2 v2) pd (err(((0 (21 22, 21. 822)
/ - - (0 yr. v1.v2)
E
inf xEll(p,))
inf FEEl(ee) / (922(22.2)) 22) 92(02, 22) 12, 31. 122)
+ E
=
=(B
( / (0r,0 - 2)
inf
inf pEll( / 192((0).2)) 92 g2 y22) F (22, 82)
+
E
GSW Miggi, e) +
M2: 92, c),
where the last inequality is due the Minkowski's 's inequality.
(ii) From (i), we have H2SW (/,, v; 91, 92) I GSW GSWp(M1, vi; 91, c) + GSW GSW,(M2, 92, c). When,
01. 92, and c(x, y) = z yl are linear defining funetions, we have:
inf FEll( / /(10T0: 0T -Ty2 (d
GSW (SWp(M1, =
E
{ (110112114 yn 11 (x11 v1)
inf rel(()m))
inf [Fll(pp..))) / (11m 1lllddd (211 21))
E
inf (ET /
Similarly, we have GSW (12, C) S u2(pz, Therefore, we obtain the proof of H2SW (14, v; G, I1, 92) S W, Up(pi, + W UiC.
(iii) When 91 92 are linear defining functions, have: H2SW (Hi, vi c, 91. 92)
inf / (11T 21 -0, v1 + 10] x2 02 (dl 22, B1, y2) 1 =ETl(L.))
inf [ / (10T21 - 0, v1)1 )l + 10] x2 0g vel)² do (11, x2, 81, (sn
E
/ (101--11 (:hi-y1)1 81)l + 222 dd (21, 22, 11 82)
E
inf x
inf rEl((..))
(101 y1)1 |02 (21,
When = 1, we obtain:
[(es [(1o1-w)1 ((e1-v))1= vz)l l222 122)dd((11 22 Ma, 222
(1,
inf (
= -,,,,,s,,
16

---

which completes the proof.
A.5 Proof of Proposition
Let p 2 1, c(2; y) |x - yl. ja E ((R) with the corresponding empirical distribution Maz, assume that there exists q > p such that the q th order moment of i.e My = Ja [x)d)(()), is bounded by B < co. From Theorem in (20), there exists constant Cy.9 such that:
n 1/2 if >>97>0 2p, 1/2 log(n)) à if 2p, n (9-p1// if E (p. 2p).
E [WP(un, H:c] & C.p.
We show that HHR .02 81. 12.02,1 Fpi has finite bounded moments. In particular, we have:
[ 111t ( (t)
M& (HHRR) 92
h/))
02.02,
[
24 xRd2 1119 x1) + |
f Jd xRd2
(07 +
(0), 11 (9)
(2 /
$ f Ju x (0,1 x
وo (02, x2)) du(xx,,
where the first inequality is due to the Sauchy-Schwaad inequality and the second inequality is due
to the fact that | < |a/. For the linear defining functions g(0, = 0 a, we have 1g(0, x> =
je 2| S || For the circular defining functions g(f,z) 2 r0112 I || rf11, S ||>||| +
||ol]]
|0:]| + r. Therefore, we have:
f
Ma (HHR) 02 01,02 (p) S
(121) + [22 Co du([11 22)
R xRd2
k
( J xR
Σ
ks (1(aal + |12)) 91.92 d2( 222)
i=0
k' Oh-
(111 + du(z1,
JRc x
1=0
b k K'Ok-i Mi(H), g1.g2
where Cg..02 = 0 if g1 92 are linear, Co..o = if 91 and 92 are linear and circular respectively (exchangeable), and C9..92 2r if both 91 and g2 are circular.
Now, using the triangle inequality of H2SW (Theorem we have:
E H2SW, (Hm, P,ic,91.92) - H2SW , (1, 91))))
< E H2SW (14, Ph; c, 91, 02) + H2SW un; (n 92)l
< E H2SW, H2SWp(1s, (13, 91. 91 + E [H2SW (M, (,.9,99922)
EE H2. c, 91 92) P (E (v, 92)))
where the last inequality is due to Holder's inequality. Combining with previous results, we obtain:
E H2SW p (Mn, Pni 92) - H2SW (M,, P3 pF, 91)
71 1/2p if > 2p,
S Cr.
Σ
+ M((1))
n 11/2p log(n) > if q = 2p.
n (0-1p/ if q E (p. 2p),
which completes the proof.
17

---

A.6 Proof of Proposition
For any p 2 1, and p. E ((R RR, using the Holder's inequality, we have:
H2SSW " (1, v.C, 91. 92, L) H2SW, 91.
E H2SW (1, V;,, H2s.:
NHHHR 01.92 on.0. ti, HHR 22.
0,162 #V; W ( uu, 02
L
t=1
81 bb, tp, HHR 0n, Bu # AUS
Var
W WHHH
L
Var W
bh tp, 92
01 b, te; c) c)}²
VL
which completes the proof.
B Additional Materials
HHRT with more than two marginals. We now extend the definition of HHRT to K > mariginals. Definition 4 (Hierarchical Hybrid Radon Transform). Given K 2, given defining functions {gu dRde x 24 - IR. |R the Hierarchical Hybrid Radon Transform HHR Tu (R24 x x
Rdx) + L1 (R x .t... Qx x -1) is defined as:
(HHR))(L .......0.. (0.)),,,........................
K Xxxxx..xxxx (x. IK)6 (1 Σo than 0s)
(1-
0n) de ... da
(12)
H2SW with more than two marginals. From the new definition of HRRT K > 2 mariginals, now can define H2SW between joint distributions with K mariginals.
Definition 5. For p 2 1,K > 2, defining functions gl.... IK,, the hierarchical hybrid sliced Wasserstein-p (I22WW distance between two distributions E P(X x... X Xx) and v E P(Y x ... x Vx) with an one-dimensional ground metric e IR x R R+ defined as:
H2SW? (,,,,,
........... -E ---0 x0x XBK 18 [ (WCHRR 0..... H1, HHROL.0. 9x tu; e)l,
(13)
where HHR OK tu and H.R....9 N created by applying HHRT.
tu are the one-dimensional push-forward distributions
Lorentz Model and Busemann function. The Lorentz model T_d RR++1 of d-dimensional hyperbolic space is
7
Ld { (x1, (21...., IT) Rd+11 2030 T.yi - >0
Given a direction E T. nsd , x E Ld, the Busemann function is:
B f) = log( (x. 2o + 0))-
Busemann function on product Hadamard manifolds. For distributions supports on the product of
K 2 2 Hadamard manifolds with the corresponding Busemann functions B,...., Bx, we have Busemann function of the product manifolds is:
B(a TK. ., 0k) =
Be
Rm
18

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GPW
Source
Step 500
Step
Step
Stop
Step
Figure
Visualization of deformation from the sphere mesh to the Armadillo mesh with L = 100.
for (A1,- .A,.... Ax) ESK-1 The Cartan-Hyperbolli Sliced- Wasserstein distance fixed value of (AI..... Ag) e.g., (AI..... Ax) (1/VK, //KK) (see In our proposed H2SW, we treat (A1..... Ax) random variable follows ((( 1) and the value of H2SW is defined as the mean of such random variable.
C Related Works
HHRT and Generalized Radon Transform, HHRT can be also seen as a special case of GRT with the defining function g(2, 0) Vig1((1 01) + v222 (y. 02) with (x1,42) and 0 (01,0, 16) (i2 = 9 x l2 x S). However, without approaching via the hierarchical construction, the injectivity of the transform might be challenge obtain.
HHRT and Hierarchical Radon Transform. Hierarchical Radon Transform (HRT) is the composition of Partial Radon Transform and Overparameterizee Radon Transform, which is designed specifically for reducing projection complexity when using Monte Carlo estimation. Morcover, HRT introduced with linear projection and does not focus the problem of comparing heterogeneous joint distributions. In contrast to HRT, the proposed HHRT the composition of multiple partial Generalized Radon Transform and Partial Random Transform, which is suitable for comparing heterogeneous joint distributions.
HHRT and convolution slicers. Convolution slicers 40 are introduced project an image into a scalar. It can be viewed as a Hierarchical Partial Radon Transform i.e., small parts of the image transformed first, then be aggregated later. Although convolution slicers separate global and local information as HHRT, they focus on the domain of images only and have not been proven to be injective. Again, HHRT is designed to compare heterogencous joint distributions and is proven to be injective in Proposition
2 As a result, H2SW valid metric while convolution sliced Wasserstein 40 is only pseudo metric. Moreover, H2SW also convolution slicers when having marginal domains as images.
D Additional Experiments
3D Mesh Deformation. As mentioned in the main text, we present the deformation visualization to the Armadillo mesh with L = 100 in Figure and the deformation visualization to the Stanford in turn. The quantitative result for the
Bunny mesh with L 10 and L 100 in Figure Armadillo mesh is given in Table 2 Here, we set the step size to O.I. From these results, we see that the proposed H2SW gives the best flow deformation flow in general. The performance gap is especially larger when 10 i.e., having small number of projections.
and _no_eaaaa__daoaFFFF
Cartan- Htw.aaaaaaaaaaaaaaaaaa xp sp_otdd/OTDD. ipynb
/
19

---

GPW
Source
Stop 308
Step 500
Step
Step
Figure 7: Visualization deformation from the sphere mesh to the Stanford Bunny mesh with L 100.
Data
GSW
Figure 8: Visualization randomly selected reconstruction meshes from autoencoders trained SW.
GSW, and H2SW turn number of projections at epoch
SW
CHSW
H2SW
W
Figure 9: Cost matrices between datasets from SW, CHSW, and H2SW with L = 100.
SW
CHSW
H2SW
W
Figure 10: Cost matrices between datasets from SW, CHSW, and H2SW with 500.
20

---

ww.. meam soun
SW
CHSW
H2SW
Figure 11: Cost matrices between datasets from SW, CHSW, and H2SW with 1000.
Deep 3D mesh autoencoder. We first report the neural network architectures that we use in the experiments.
The encoder: Convld(6,64,1)) + BatchNormid LeaeyyyeLLUUL - Convld(64, 128,
1) BatchNormid LeakyReLU(0.2) Convld 256, 1) BatchNormld + LeakyReLU(002)) - Conv1d(256, 512, 1) + BatchNormid + LeakyReLU(0.2)) + Conv 12512, 1024, 1) BatchNormld + LeakyReLU(0.2) Max-Pooling Lin- ear( (0044 1024).
The decoder: Linear( 1024, 1024) - BatchNormid - Leaky - Linear( 1024, 2048) - Batchhormid (e0.) - Linear(2048, 4096) - BatchNorm Id - LeakyReLUU(0.2) - Linear(2048, 2048*6). The output of the decoder is the concatenation of the location and normal vector. We normalize the normal vector to the unit-sphere.
As mentioned in the main text, we report the reconstruction of randomly selected meshes for 100 epoch 500 in Figure 8 We that the reconstructed meshes at epoch 500 visually worse than the reconstructed meshes at epoch 2000. Therefore, the joint Wasserstein distances Table are consistent with the qualitative results.
Dataset Comparison. We follow the same procedure in Section 6.2 in 9 We refer the reader to the reference for detailed description. Here, we show the cross-dataset cost matrices with the number of projections L 100 Figure L 500 in Figure 10 and L 1000 Figure]
E
Computational Infrastructure
For the nn-deee--eeamig experiments, HP Omen 25L desktop for conducting experiments. For 3D mesh autoencoder experiments, we use a single NVIDIA A100 GPU.
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Reviews:
[{"review_id": "3QGPRLpO73", "reviewer": "Reviewer_E3oU", "summary": "To address the issue that Sliced Wasserstein (SW) and Generalized Sliced Wasserstein (GSW) GSW are only defined between distributions supported on a homogeneous domain and using SW and GSW directly on the joint domains cannot make a meaningful comparison, this paper first extends the partial Radon Transform to the Partial Generalized Radon Transform (PGRT) to inject non-linearity into local transformation, and then proposes a novel slicing operator for heterogeneous joint distributions, named Hierarchical Hybrid Radon Transform (HHRT). The authors conduct experiments on optimization-based 3D mesh deformation and deep 3D mesh autoencoder to show the favorable performance of H2SW compared to SW and GSW. In general, this paper is well structured and presented.", "questions": "Please clarify the main challenges of designing a sliced Wasserstein variant for heterogeneous joint distributions in the Introduction section, and how the proposed H2SW addresses these challenges.", "limitations": "The authors have addressed the limitations of their work.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. This paper addressed the issue of that Sliced Wasserstein (SW) and Generalized Sliced Wasserstein (GSW) GSW are only defined between distributions supported on a homogeneous domain, and using SW and GSW directly on the joint domains cannot make a meaningful comparison.\n2. This paper is well structured.", "weaknesses": "1. The authors did not clarify the main challenges of designing a sliced Wasserstein variant for heterogeneous joint distributions in the Introduction section. Additionally, they failed to explain how the proposed H2SW addresses these challenges. I suggest that the authors elaborate on these points to provide a clearer understanding of the significance and innovation of this work.\n2. The paper involves many similar concepts, such as PRT, GRT, and PGRT. What are the relationships between them? Is the proposed PGRT a combination of PRT and GRT? These questions are confusing. I suggest that the authors provide a detailed explanation of the differences and connections between these concepts.\n3. The text in Figure 1 is too small to read clearly, and the main idea it aims to convey is not clear enough. I suggest that the authors add some elements to make the figure easier to understand.\n4. The definitions in the paper are sometimes too concise, lacking detailed explanations and context. For instance, when introducing new transforms and metrics, more background information and examples would help readers understand the practical applications of these definitions.\n5. Some descriptions in the paper are overly verbose and the language lacks precision, which affects readability and may cause confusion. I recommend that the authors refine the writing to make it more concise and precise.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for the time and the feedback.\n\n**Q1**. The authors did not clarify the main challenges of designing a sliced Wasserstein variant for heterogeneous joint distributions in the Introduction section. Additionally, they failed to explain how the proposed H2SW addresses these challenges. \n\n**A1**.  As discussed in the introduction, heterogeneous joint distributions are joint distributions that have marginals supported on different domains with different supported ground metrics. As a result, the joint ground metric between two points in the joint spaces is often a weighted combination of marginal ground metrics between corresponding marginal dimensions. For example, if the marginal domains are manifolds, the weighted combination of ground metrics is the geodesic ground metric on the joint space [1].\n\nTo design a SW variant for heterogeneous joint distributions, we need to map probability distributions on the joint space to one-dimensional distributions on the real line to apply the closed form of the one-dimensional Wasserstein distance with a strictly convex-based ground metric. A meaningful map should be injective, computationally efficient, and geometrically meaningful. Injectivity is essential for preserving the identity of the indiscernible property of the sliced metric; i.e., when all one-dimensional projections are equal, the original distributions are also equal. Moreover, the map requires fast computation to maintain the near-linear computation time of one-dimensional Wasserstein distances. The geometric meaning is necessary for preserving the topological structure of the original space of distributions under the Wasserstein distance with the geodesic ground metric to some extent. In more detail, the composition of the map and the chosen strictly convex-based ground metric in one dimension create a pull-back metric on the original joint space. The pull-back metric will not account for the difference between the two marginal domains if we use conventional maps from the Radon Transform and the Generalized Radon Transform. As a result, the geometry of the original joint space is lost.\n\nTo address these challenges, the proposed H2SW utilizes the hierarchical hybrid Radon Transform (HHRT), which is the composition of the Partial Radon Transform and multiple domain-specific Partial Generalized Radon Transforms. Intuitively, HHRT first projects marginal domains onto real lines and then projects the joint space of those lines onto the final real line. By doing so, HHRT can retain more of the geometric structure of the original joint space. When marginal domains are Hadamard Manifolds and marginal Busemann projections are used, the HHRT is equivalent to using Busemann projections (not unique) on the joint space [2]. We would like to emphasize that HHRT cannot entirely preserve the original geometry; however, it is the first attempt to tackle the challenge since it is injective and has fast computation with efficient choices of marginal generalized Radon Transforms.\n\n[1] Learning Mixed-Curvature Representations in Product Spaces, Gu et al.\n\n[2] Metric spaces of non-positive curvature, Bridson et al.\n\n**Q2**. The paper involves many similar concepts, such as PRT, GRT, and PGRT. What are the relationships between them? Is the proposed PGRT a combination of PRT and GRT? \n\n**A2**. We would like to clarify the relationships of PRT, GRT, and PGRT as follows: \n\nFirst, the RT maps a function over $\\mathbb{R}^d$ to a function over $\\mathbb{S}^{d-1} \\times \\mathbb{R}$ by integrating over hyperplanes (Equation 2). Given a projecting direction $\\theta  \\in \\mathbb{S}^{d-1}$, the RT provide a  linear map  from $\\mathbb{R}^d$ to $\\mathbb{R}$. However, a linear map is not geometrically correct for some domains such as manifolds in $\\mathbb{R}^d$. Therefore, the generalized Radon Transform is introduced by changing from integrating over hyperplanes to integrating over hypersurfaces (controlled by the defining function - line 116). PRT  is also an extension of RT which transforms only a subset of arguments via integrating over hyperplane on the corresponding sub-space of the arguments. PRT is useful in signal-processing problems where we are particularly interested in only a subset of arguments due to some prior knowledge. As a natural extension, the proposed  Partial Generalized Radon Transform (PGRT) generalizes PRT by changing from integrating over the hyperplanes on the corresponding sub-space to integrating over the hypersurfaces. Overall, PGRT serves as a building block of HHRT since it is used to transform marginal arguments of the joint spaces. Since the marginal spaces are heterogeneous, different types of PRGT are used (with different defining functions).\n\n**Q3**. The text in Figure 1 is too small to read clearly, and the main idea it aims to convey is not clear enough.\n\n**A3**. We will increase the font sizes and revise the figures accordingly in the revision.\n\n**Q4**. The definitions in the paper are sometimes too concise, lacking detailed explanations and context. For instance, when introducing new transforms and metrics, more background information and examples would help readers understand the practical applications of these definitions.\n\n**A4**. We will add the background information of the previous transforms to the revision as discussed in **Q2**. In the paper, we also present Example 1, which discusses the usage of HHRT in the application of a 3D mesh with shape as points representation. We will add more detail on how HHRT is applied step by step and the computation of H2SW in this example in the revision. Additionally, we will include a new example comparing datasets using Busemann projections for a more comprehensive discussion.\n\n**Q5**. Some descriptions in the paper are overly verbose and the language lacks precision, which affects readability and may cause confusion.\n\n**A5**. We will adjust the writing in the revision based on your suggestions.", "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": "HPp0gMjsta", "reviewer": "Reviewer_deuA", "summary": "Wasserstein distance measures the distance between two distributions and is particularly important in machine learning, especially in generative models. However, computing the Wasserstein distance is computationally intensive. For homogeneous distributions, Sliced Wasserstein (SW) and Generalized Sliced Wasserstein (GSW) significantly reduce time and memory complexity. However, while they are computationally friendly for heterogeneous distributions, they fail to capture the underlying heterogeneity of the spaces. This paper addresses these issues with the following contributions:\n1.Novel Slicing Operator: The paper proposes a novel slicing operator for heterogeneous joint distributions, called the Hierarchical Hybrid Radon Transform (HHRT).\n\n2. Hierarchical Hybrid Sliced Wasserstein (H2SW): Using HHRT, the authors develop the Hierarchical Hybrid Sliced Wasserstein (H2SW) distance tailored for heterogeneous distributions.\n\n3.Empirical Validation: The superiority of HHRT is demonstrated on heterogeneous distribution datasets.\n\n4.Extension of Radon Transform: The partial Radon Transform is extended to the partial generalized Radon Transform (PGRT), and the monotonicity of PGRT is proven, thereby confirming the monotonicity of HHRT.\n\n5. Comprehensive Analysis: The paper investigates the topological, statistical, and computational properties of H2SW.", "questions": "see the weakness.", "limitations": "yes", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1.1Pioneering Metric for Heterogeneous Joint Distributions: The paper introduces Hierarchical Hybrid Sliced Wasserstein (H2SW), a novel metric specifically designed for comparing heterogeneous joint distributions, a task that conventional methods struggle with.\n\n1.2A Novel Slicing Operator for Heterogeneous Data: To address the challenges of comparing heterogeneous data, the paper proposes a novel slicing operator for heterogeneous joint distributions, named Hierarchical Hybrid Radon Transform (HHRT).\n\n1.3 Extending Partial Radon Transform: The paper extends the concept of Partial Radon Transform (PRT) to the more general Partial Generalized Radon Transform (PGRT), expanding its applicability.\n\n1.4. Unveiling Properties and Connections:\n\na.  Monotonicity of HHRT and PGRT: The paper delves into the properties of HHRT and PGRT, establishing their monotonicity, a crucial aspect for ensuring meaningful comparisons.\n\nb. Unveiling the Link between HHRT and PGRT: The paper explores the intricate connection between HHRT and PGRT, shedding light on their underlying relationship.\n\nc. A comprehensive analysis of H2SW is conducted, encompassing its topological properties, statistical properties, and computational properties.", "weaknesses": "The article lacks attention to detail in several areas.\nThe paper needs to clearly explain heterogeneous joint distributions,\nsuch as by detailing why 3D shape deformation is considered a heterogeneous joint distribution.\n• P22-849: The explicit expression for GSWp p(u, v; g, c) is not immediately obvious. A detailed derivation from the definition should be provided.\n• The proof of Proposition 1 seems to be simplified. By fixing y, Let\nfy\n(x) = f(x, y), then PGRf degenerates to GRf. So f1,y(x) = f2,y(x) ⇒\nf1(x, y) = f2(x, y).\n• Page-5: In Proposition 4, what is the specific meaning of the subscripts\nn for µn, vn? How do they differ from and relate to µ1 in Proposition 3?\nWhat is the connection between the definition of P\nq and the parameter\nq?\n• Here are some writing issues or areas that need further elaboration:\n– Page-3,100: L1 should be clarified as the Lebesgue integrable function space.\n– Page-3,110:The full name of CDF :Cumulative Distribution Function.\n– Page-4,157: The functions f1, f2 should belong to the function\nspace L1(Rd1 × Rd1).\n– Pqge-4,162: The variable x should be subscripted as x1,i, x2,i.\n– Page-20,812-: The equation is not balanced on both sides. The\nleft side should be rewritten in the form of the right side.\n– Page-21,840/844: redundant left parenthesis\n– Page-22,851-: there is an extra ’-’ sign in the first inequality.\n– Page-22,854-: The θ1, θ2 symbol should be moved outside the absolute value symbols.\n– Page-22,855-: for p = 1, 1\np\nshould be removed", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to express our gratitude for the time and feedback from the reviewer. We would like to answer questions from the reviewers as follows:\n\n**Q6**. The paper needs to clearly explain heterogeneous joint distributions, such as by detailing why 3D shape deformation is considered a heterogeneous joint distribution\n\n**A6**. Thank you for your comments. It is worth noting that the notion of heterogeneous joint distributions arises only when considering the metric spaces of marginals, which are not usually accounted for in conventional probabilistic models. We define heterogeneous joint distributions as joint distributions that have marginals supported on different domains with different supported marginal ground metrics. For example, $\\mu(x,y) \\in \\mathcal{P}(\\mathbb{R}^d \\times \\mathbb{S}^{d-1})$ is a  heterogeneous joint distribution. The reason is that $\\mathbb{R}^d$ supports the Euclidean metric $c_1(x_1,x_2)=\\|x_1-x_2\\|$ as the geodesic while $\\mathbb{R}^d$ supports the great circle metric $c_2(y_1,y_2)=arccos(\\langle y_1,y_2\\rangle)$. The heterogeneity comes from the difference in geodesic distances for each subset of arguments.\n\n3D shapes are often presented as triangle meshes, which contain vertices and faces. Vertices are location vectors in 3-dimensional real space ($\\mathbb{R}^3$) while faces are sets of 3 vertices that are pairwise connected. In their original form, 3D shapes are expensive to store and difficult to manipulate. As a result, implicit representations have been proposed, one of which is the shapes-as-points representation [3]. In this representation, a 3D shape is represented by a set of points that lie on the surface of the 3D shape (constructed manually or by sampling). Additionally, for each point, the associated normal vector—the unit directional orthogonal vector to the surface at the point—is stored. As a result, a 3D shape is represented as a set of points on the product space $\\mathbb{R}^3 \\times \\mathbb{S}^2$. This representation is lightweight, suitable for neural network processing on 3D meshes, and can be converted back to the triangle mesh representation using Poisson surface reconstruction.\n\n\nTo convert a set of points to a probability distribution, we can assign a weight to each point (the summation of all weights is 1) to turn the set of points into a discrete distribution. Since points are on the product space $\\mathbb{R}^3 \\times \\mathbb{S}^2$, the resulting discrete distribution is a heterogeneous joint distribution as discussed above. In the experiments, we assign uniform weight to points i.e., turning the set of points into an empirical distribution.\n\n[3] Shape As Points: A Differentiable Poisson Solver, Peng et al.\n\n**Q7**. P22-849: The explicit expression for $GSW_ p(\\mu, \\nu; g, c)$ is not immediately obvious. A detailed derivation from the definition should be provided. \n\n**A7**. Thank you for your comments. Since the measures $\\mu$ and $\\nu$ are supported on an open set and the defining function $g$ is continuous to the support of $\\mu$, the defining function $g$ is measurable. By the definition of the push-forward measure and change of variable formula [4], $\\int   |g(\\theta,x)-g(\\theta,y)|^{p}d\\pi (x,y) = \\int   |x’-y’|^{p}d\\pi’ (x’ ,y’) = $ where $\\pi’$ is a joint distribution of $\\mathcal{GR}^g_\\theta \\sharp \\mu$ and $\\mathcal{GR}^g_\\theta \\sharp \\nu$ by pushing forward $\\pi$ through $g(\\theta,x)$ and $g(\\theta,y)$.  As a result, we obtain $W_p^p(\\mathcal{GR}^g_\\theta  \\mu,\\mathcal{GR}^g_\\theta  \\sharp \\nu;c)$ as defined in Equation 4 in the paper. We will add a more detailed derivation to the revision of the paper.\n\n[4] https://en.wikipedia.org/wiki/Pushforward_measure\n\n**Q8**. The proof of Proposition 1 seems to be simplified. By fixing y, Let $f_y (x) = f(x, y)$, then PGRf degenerates to GRf. So $f_{1,y}(x) = f_{2,y}(x) \\Rightarrow f_1(x, y) = f_2(x, y)$.\n\n**A8.** Thank you for your detailed comments. We will include this derivation in the revision of the paper. \n\n**Q9**. Page-5: In Proposition 4, what is the specific meaning of the subscripts n for $\\mu_n$, $\\nu_n$? How do they differ from and relate to µ1 in Proposition 3? What is the connection between the definition of $\\mathcal{P}_q$ and the parameter $q$? \n\n**A9**. Thank you for pointing this out. Here, $\\mu_n$ and $\\nu_n$ are empirical distributions over $n$ random samples from $\\mu$ and $\\nu$ respectively. In particular,  let $X_1,\\ldots,X_n \\overset{i.i.d}{\\sim} \\mu$, $\\mu_n = \\frac{1}{n} \\sum_{i=1}^n \\delta(x-X_i)$. It seems that this notation overlaps with the marginal distribution notation $\\mu_1$ in Proposition 3. We will change the notation from $\\mu_n$ to $\\hat{\\mu}_n$ in the revision.  For the definition of $\\mathcal{P}_q$, it is the set of all probability distributions with finite $q$-th moments ($q\\in \\{1,2,3,\\ldots,...\\}$). We will add a paragraph to explain notations in the revision of the paper.\n\n**Q10**.  On writing issues or areas that need further elaboration:\n\n**A10**. Thank you for pointing out. We will fix all writing issues in the revision of the paper.", "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": "25WeebhNK3", "reviewer": "Reviewer_p3fW", "summary": "The paper proposes a new metric for comparing heterogeneous joint distributions, called Hierarchical Hybrid Sliced Wasserstein (H2SW). The proposed metric is tested for shape deformation, for training shape autoencoder and comparing datasets.\nThe proposed metric is based on a new slicing operator (Hierarchical Hybrid Radon Transform, HHRT).", "questions": "Can authors clarify the application 4.3 ? i.e. what is the data and it is encoded ($\\mathbb{L}^{d_2}$ ?)?", "limitations": "The work presented focuses on contribution to defining an  OT metric however it may be that HHRT has applications in other areas (?)", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper's contributions are in defining a novel slicing operator (HHRT) and its associated metric (H2SW).\n\nThe paper is clear, well written and interesting to read.", "weaknesses": "The third contribution listed (lines 81-82) is confusing (and in my opinion should be rephrased/tone down): the paper ignores  other attempt(s) for deforming shapes using heterogenous joint distributions using likewise point clouds combined with normals - e.g. see https://doi.org/10.1016/j.patcog.2018.02.021 for an example with 2D and 3D shapes using L2 metric between heterogenous distributions. \n\nStandard error (e.g. tab 3) are too large to rank/compare tested methods with confidence.\n\nExperiments in section 4.3 are not easy to grasp ( rely on reader to read recent [9])\n\nMinor remarks:\n\n- reference [56] == [57] ?\n\n- [50] published 10.1088/1361-6420/acf156 but arxiv cited\n\n- line 49: \"interested distributions\"  ?\n\n- line 85: \"Hadamand\" \n\n- line 89: \"distance s\"\n\n- Fig 1 too small fonts\n\n- line 248 \"the injectivity of the Buseman projection has not been known at the moment\" : do you mean no math proof available yet?\n\n- line 251-252: \"H2SW is a generic distance for heterogeneous joint distributions in  which marginal domains do not necessary manifolds\":  \"do not necessary manifolds\"  meaning??\n\n- line 259-60 \"we compare H2SW with SW and  Cartan-Hadamard Sliced-Wasserstein (CHSW) in comparing datasets on the product of Hadamard  manifolds\" : repetition compare/comparing ... sentence unclear\n\n- Fig 4 too small fonts\n\n- line 909 \"Hierarchial\"\n\n- line 918 \"Armadill mesh\"\n\n- Fig 9/10/11 too small font\n\n- Buseman projection -> Busemann projection ?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for insightful comments and constructive feedback. We would like to answer questions from the reviewers as follow:\n\n**Q11**. The third contribution listed (lines 81-82) is confusing (and in my opinion should be rephrased/tone down): the paper ignores other attempt(s) for deforming shapes using heterogenous joint distributions using likewise point clouds combined with normals - e.g. see https://doi.org/10.1016/j.patcog.2018.02.021 for an example with 2D and 3D shapes using L2 metric between heterogenous distributions.\n\n**A11**.  Thank you for your insightful comment. We will adjust the writing to tone down our claim. We would like to clarify that we do not claim that we are the first that utilize point clouds combined with normals for representing 3D shapes.  The main contribution of the paper is the proposal of a hierarchical hybrid sliced Wasserstein (H2SW) which is a new variant scalable of sliced Wasserstein for heterogenous joint distributions. In greater detail, the time complexity of H2SW is $\\mathcal{O}(n \\log n)$ and the space complexity is $\\mathcal{O}(n)$ for $n$ is the number of supports (the number of points of the 3D shape). Moreover, H2SW leverages optimal transport which retains the geometric structure. In contrast, L2 distance requires evaluating density at all supports of two distributions which makes the time complexity and space complexity $\\mathcal{O}(n^2)$. In addition, to create smooth transitions, the L2 distance relies on KDE methods which are not necessary for H2SW. We will include the mentioned paper in the reference and add the discussion to the revision.\n\n**Q12**. Standard error (e.g. tab 3) are too large to rank/compare tested methods with confidence.\n\n**A12**. Thank you for the comments. Since SW variants are approximated with the Monte Carlo estimation with $L>0$ projecting directions. Increasing the number of projections $L$ can help to reduce the randomness (the standard error) as demonstrated in Table 3. Due to the limitation of the computational infrastructure, we could not increase $L$ further to obtain results in an acceptable time. Nevertheless, we believe H2SW can show its benefits when $L=2000$ since both the mean and the standard error are lower. Moreover, compared to SW, H2SW is better considerably which demonstrates the benefit of the hierarchical hybrid Radon Transform.\n\n**Q13**. Experiments in section 4.3 are not easy to grasp ( rely on reader to read recent [9])\n\n**A13**. Thank you for your comment. We refer the reader to **Q15** for a detailed discussion.  We will add this discussion to the revision of the paper.\n\n**Q14**. Minor remarks and typos.\n\n**A14**. Thank you for pointing out. We will fix typos and revise the paper based on your suggestions.\n\n**Q15**. Can authors clarify the application 4.3 ? i.e. what is the data and it is encoded ($\\mathbb{L}^{d_2}$?)?\n\n**A15**.  In  Section 4.3, we want to compare datasets as discussed in [5] and follow the approach in [6]. In particular, a dataset consists of samples with features and labels. For example, the MNIST dataset has images of digits with associated labels (the class index of the digit). For each class, we want to embed to $\\mathbb{L}^{d_2}$. To do that, we represent each class as an empirical distribution of images and compute the pair-wise Wasserstein distance between class distributions to obtain a pair-wise cost matrix between classes from two datasets. We then use multidimensional scaling to obtain the embedding of classes on $\\mathbb{L}^{d_2}$. Finally, we represent a dataset as a set of samples on $\\mathbb{R}^{d_1} \\times \\mathbb{L}^{d_2}$ with $\\mathbb{R}^{d_1}$ is for the image and $ \\mathbb{L}^{d_2}$ for the class embedding. We will add this detail to the revision of the paper.\n\n[5] Geometric Dataset Distances via Optimal Transport, Alvarez-Melis et al.\n\n[6] Sliced-Wasserstein Distances and Flows on Cartan-Hadamard Manifolds, Bonet et al.\n\n**Q16**. The work presented focuses on contribution to defining an OT metric however it may be that HHRT has applications in other areas (?)\n\n**A16**. Thank you for your insightful questions. From the perspective of signal processing, HHRT can be used as a feature extractor [7] for other tasks like Radon Transform e.g., image classification tasks [8]. In this case, the feature from HHRT can be used to classify 3D shapes or datasets as discussed in the paper. Moreover, we believe that HHRT can be also used for data denoising i.e., transforming noisy data and then reconstructing a less noisy version with inverse transforms.\n\n[7] Data representation with optimal transport, Martin et al\n\n[8] Radon Cumulative Distribution Transform Subspace Modeling for Image Classification, Shifat-E-Rabb et al.", "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": "90leLtTDjK", "reviewer": "Reviewer_XX3o", "summary": "This work focuses on introducing a Sliced-Wasserstein distance between heterogeneous joint distributions, i.e. distributions whose ambient space lies in a product space, with possibly different geometries for each space. This new distance relies on a new hierarchical hybrid Radon transform. The authors derive several theoretical properties of this new discrepancy, and apply it to three tasks: 3D mesh deformation and autoencoders whose underlying space can be seen as $\\mathbb{R}^3 \\times S^2$, and dataset comparisons, with datasets embedded on $\\mathbb{R}^{d_1}\\times \\mathbb{L}^{d_2}$ where $\\mathbb{L}$ is the Hyperbolic manifold.", "questions": "In the statements of Proposition 1 and 2, it is stated \"For some defining function g such as linear, circular, and homogeneous polynomials with an odd degree\". I believe it works for defining functions which are injective? If it is the case, it would be more clear in my opinion to state, let $g$ be injective, and write somewhere else examples of injective defining functions.\n\nUnder Definition 2, it is written \"The reason for using PRT for the final transform is that the previous PGRTs are assumed to be able to transform the non-linear structure to a linear line\". Without reading the appendix and understanding that HHRT can be seen as a composition of PRT and PGRT, which is stated after, this sentence is not clear. \n\nIn Proposition 3, I guess $W_1(\\mu,\\nu,c,c)=W_1(\\mu,\\nu,c)$.\n\nA missing reference on injective defining functions, which might give another example for which HHSW is a distance is [1].\n\nOnly 2 examples of heterogeneous data are given (3d meshes and datasets). Are there other examples?\n\nTypos:\n- Line 52: \"In addition, a line of work...\": misses verb\n- In Definition 2, I do not think $\\mathbb{S}$ is defined.\n- In the proof of Proposition 3, in the equation after line 851: $\\|x_1--y_1\\|_2^p$\n- Line 317: \"By varying $L\\in\\{\\dots\\} For\"\n- Refs [20] and [21] are the same.\n- Refs [56] and [57] are the same.\n\n\n\n[1] Chen, Xiongjie, Yongxin Yang, and Yunpeng Li. \"Augmented sliced Wasserstein distances.\" arXiv preprint arXiv:2006.08812 (2020).", "limitations": "Yes", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "This work introduces a new Sliced-Wasserstein distance to compare distributions on any product spaces. The relevance of this new distance is shown on different experiments, and many theoretical properties are derived.\n\n- New distance to compare probability distributions on product spaces\n- Theoretical properties such as conditions under which the proposed discrepancy is a distance, projection complexity, and connections with other distances\n- Nice applications on 3D meshes and on dataset comparisons", "weaknesses": "To my opinion, there are few weaknesses. First, the applications on 3D meshes are nice and large scale (10000 points) but only in few dimensions. Moreover, on this application, the only distance used for the sphere component is the circular defining function. It would have been nice to compare it with e.g. a spherical Radon transform.\n\nOn the theoretical side, there are some parts which could be more clear, notably around Proposition 1 and 2. I give more details into the Questions section.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to thank the reviewer for the time and constructive feedback.\n\n**Q17**. First, the applications on 3D meshes are nice and large scale (10000 points) but only in few dimensions ... It would have been nice to compare it with e.g. a spherical Radon transform.\n\n**A17**. For the low-dimensional comment, H2SW can still be used in any dimension,  however, we believe 3D data is still a very important data modality. As discussed in Example 1, we can use other types of transformation for the spherical domain including Funk-Radon Transform, Vertical Slice Transform, Parallel Slice Transform, Spherical Radon Transform, and Stereographic Spherical Radon Transform. We agree that using a spherical-type Radon Transform is more theoretically sounded than a circular Generalized Radon Transform. However, circular GRT is easier and more efficient in computation and implementation.\n\nWe have replaced the circular GRT with the Stereographic Spherical Radon transform (SSRT) which is a spherical Radon Transform type, and run an additional experiment on deformation with the Armadillo mesh. We obtain the following result :\n\n| Distances | Step 100 ($W_{c_1,c_2}\\downarrow$)  | Step 300 ($W_{c_1,c_2}\\downarrow$) | Step 500 ($W_{c_1,c_2}\\downarrow$) | \n Step 1500 ($W_{c_1,c_2}\\downarrow$) | Step 4000  ($W_{c_1,c_2}\\downarrow$) | Step  5000 ($W_{c_1,c_2}\\downarrow$) |\n\n|    H2SW  L=10|1840.73$\\pm$1.282 | 1422.667$\\pm$7.813| 1058.171$\\pm$5.362 | 95.672$\\pm$4.376 | 6.326$\\pm$0.151 | 2.602$\\pm$0.201 |\n\n|H2SWs L=10| 1840.574$\\pm$4.505 | 1423.448$\\pm$2.994 | 1061.584$\\pm$3.9 | 95.618$\\pm$3.044 | 6.365$\\pm$0.106 | 2.595$\\pm$0.038|\n\n  |  H2SW  L=100|1839.347$\\pm$1.986 | 1417.1$\\pm$3.677 | 1048.895$\\pm$4.008 | 86.078$\\pm$0.623 | 4.61$\\pm$0.431 | 1.086$\\pm$0.177|\n\n|H2SWs L=100| 1839.577$\\pm$0.949 | 1417.785$\\pm$1.502 | 1050.187$\\pm$4.421 | 86.431$\\pm$1.878 | 5.192$\\pm$0.449 | 1.085$\\pm$0.191|\n\nFrom the result, we can see that the result of SSRT is similar to using circular GRT. However, it is worth noting that SSRT does not have hyperparameters while circular GRT has one hyperparameter. Hence,  SSRT might be a good alternative for circular GRT when we want to avoid hyperparameter tuning.\n\n[9] Stereographic Spherical Sliced Wasserstein Distances, Tran et al.\n\n**Q18**. In the statements of Proposition 1 and 2, it is stated \"For some defining function g such as linear, circular, and homogeneous polynomials with an odd degree\". I believe it works for defining functions which are injective? If it is the case, it would be more clear in my opinion to state, let $g$ be injective, and write somewhere else examples of injective defining functions.\n\n**A18**.  Thank you for the insightful comment. For the defining function, it is not necessary to be injective in order to make the GRT injective. For example, the linear defining function $g(x,\\theta)=\\langle x,\\theta \\rangle$ is not injective i.e., if $\\theta$ are orthogonal to $x_1$ and $x_2$ with $x_1 \\neq x_2$, we still have $g(x_1,\\theta)=g(x_2,\\theta)=0$. However, by considering all possible values of $\\theta$, the GRT is still injective. \n\nWe assume that the reviewer is referring to spatial Radon Transform [7] where an injective function can be used to transform the original support $x$, then the linear projection is applied. In HHRT, we have not used such a function to transform the original support. However, it is possible to extend HHRT into spatial HHRT i.e., using different injective functions to transform marginal arguments first, then apply generalized one-dimensional projection after that. By doing so, we can make the generalized one-dimensional projection easier to separate two distributions as discussed in [10]. The only challenge is to design meaningful injective functions on manifolds and special domains. We will include this discussion in the revision of the paper and leave this promising investigation to future work.\n\n[10] Augmented Sliced Wasserstein Distances, Chen et al.\n\n**Q19**. Under Definition 2, it is written \"The reason for using PRT for the final transform is that the previous PGRTs are assumed to be able to transform the non-linear structure to a linear line\". Without reading the appendix and understanding that HHRT can be seen as a composition of PRT and PGRT, which is stated after, this sentence is not clear.\n\n**A19**. Thank you for your comments.  We will add an equation to the main text as a rigorous explanation along with the sense.\n\n**Q20**. On Typos\n\n**A20**. Thank you for pointing out the typos. We will fix them in the revision.\n\n**Q21**. A missing reference on injective defining functions, which might give another example for which HHSW is a distance is [1].\n\n**A21**. Thank you for your reference. We will include the paper in the revision of the paper. As discussed in **Q18**, we believe that the augmented approach is a promising extension for H2SW. We will leave this extension to future work.\n\n**Q22**. Only 2 examples of heterogeneous data are given (3d meshes and datasets). Are there other examples?\n\n**A22**. Thank you for your great questions. We would like to give a few other examples of heterogeneous distributions.  The first example is distributions over features and class labels in domain adaptation [7].  The second example is distributions over data variables and latent variables in joint contrastive inference [8]. The third example is point-clouds with additional attributes e.g., colors, opacity, and so on.  In addition, distributions over multimodal data (e.g., text and image) can also be seen as heterogeneous joint distributions since the geodesic metric on the text space and the geodesic metric on the image space are different. However, the challenge is that for such complex data, the geodesic metrics are unknown which requires additional effort for metric learning.   \n\n[7] Joint distribution optimal transportation for domain adaptation, Courty et al\n\n[8] Wasserstein Variational Inference, Ambrogioni et al", "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:
Dear chairs and reviewers,

First, we would like to thank the reviewers for spending time reviewing our paper and providing constructive feedback. We would like to summarize some additional results and common questions from the reviewers:

* As suggested by Reviewer **XX3o**, we added an experiment where we replaced the circular generalized Radon Transform with a spherical-based Radon Transform, i.e., Stereographic Spherical Radon Transform (SSRT), for the hierarchical Hybrid Radon Transform. We observed that the SSRT gives comparable results while having no hyperparameters. Overall, we would like to emphasize that HHRT can utilize any type of domain-specific transforms, which have been active research areas recently.

* As suggested by Reviewer **E3oU** and **deuA**, we will add a more detailed discussion on heterogeneous joint distributions in the revised paper. In summary, the notion of heterogeneous joint distributions appears when we consider the metric space of supports. We would like to refer to a short description in the paper: “Heterogeneous joint distributions are joint distributions that have marginals supported on different domains with different supported ground metrics.” Overall,  we can treat a heterogeneous joint distribution as a special multivariate distribution, which has different structures for some blocked marginal distributions.

* We will revise the paper by fixing the typos, adding a detailed discussion, improving figures, and providing more detailed proofs as suggested by the reviewers.

Best regards,

Authors

Meta-review: There is agreement among the reviewers that this paper contributes novel ideas and new concepts to an important topic and addresses a real need with potentially high impact on the ML research community.   The paper needs a careful round of editing by the authors prior to the final submission.
Decision: Accept (poster)