Slicing the Gaussian Mixture Wasserstein Distance
Authors: Moritz Piening
Robert Beinert
Venue: Accepted by TMLR ()

Abstract: Gaussian mixture models (GMMs) are widely used in machine learning for tasks such as clustering, classification, image reconstruction, and generative modeling. A key challenge in working with GMMs is defining a computationally efficient and geometrically meaningful metric. The mixture Wasserstein (MW) distance adapts the Wasserstein metric to GMMs and has been applied in various domains, including domain adaptation, dataset comparison, and reinforcement learning. However, its high computational cost—arising from repeated Wasserstein distance computations involving matrix square root estimations and an expensive linear program—limits its scalability to high-dimensional and large-scale problems. To address this, we propose multiple novel slicing-based approximations to the MW distance that significantly reduce computational complexity while preserving key optimal transport properties. From a theoretical viewpoint, we establish several weak and strong equivalences between the introduced metrics, and show the relations to the original MW distance and the well-established sliced Wasserstein distance. Furthermore, we validate the effectiveness of our approach through numerical experiments, demonstrating computational efficiency and applications in clustering, perceptual image comparison, and GMM minimization

Published in Transactions on Machine Learning Research (08/2025)
Slicing the Gaussian Mixture Wasserstein Distance
Moritz Piening
picroing bu-berlin
Institut für Mathematik
Technische Universität Berlin
Robert Beinert
beinert@male tu tu-berlin,ce de
Institut für Mathematik
Technische Universität Berlin
Reviewed on OpenReview: https: // openreview. net/ pet/forue forum? id- yPBt.4JPuu
Abstract
Gaussian mixture models (GMMs) are widely used in machine learning for tasks such as clustering, classification, image reconstruction, and generative modelling. A key challenge in working with GMMs is defining a computationally efficient and geometrically meaningful metric. The mixture Wasserstein (MW) distance adapts the Wasserstein metric to GMMs and has been applied in various domains, including domain adaptation, dataset comparison, and reinforcement learning. However, its high computational cost- from repeated Wasserstein distance computations involving matrix square root estimations and an expen- sive linear program mrogram-limits. -imits its scalability to high-dimensional and large-scale problems.. To address this, we propose multiple novel slicing- based approximations to the MW distance that significantly reduce computational complexity while preserving key optimal transport properties. From a theoretical viewpoint, we establish several weak and strong equivalences between the introduced metrics, and show the relations to the original MW distance and the well-established sliced Wasserstein distance. Furthermore, our approach through numerical experiments, demonstrating computational efficiency and applications in clustering, perceptual image comparison, and GMM minimization.
validate the effectiveness of
1 Introduction
Gaussian mixture models (GMMs) are a fundamental tool in machine lcarning and statistics; widely used in applications such as classification (Wan el al., 2019), clustering (Zhang al. 2021 Adipoetra & Marting
2025), image reconstruction Zooann & Weiss 2011 Nguyen et al. 2023) Piening et al. 2024), 3d surface representation (Zou & Sester 2024), and generative modeling (Hagemann Neumayer 2021 Alberti al. 2024). That is why defining computationally efficient and geometrically meaningful metric on the space of GMMs is valuable for many data-driven applications The most prominent candidate for such tasks is
the mixture Wasserstein (MW) distance (Chen et al.) 2016 2018 Delon & Desolneux 2020), which can be interpreted as restriction of the classical Wasserstein distance to GMM transport plans. The main limitation of the MW distance its high computational cost. consists of an 'inner" and an 'outer' optimal transport problem. The inner problem requires the computation of all pairwise Wasserstein distances between the components of the given GMMs. Despite having closed-form solution, this step involves matrix square root calculations, which becomes highly costly for high-dimensional data. The outer problem consists in solving optimal transport problem, whose cost matrix is determined by the inner problem. If both GMMs have K Gaussian components, the minimization of the outer problem has a complexity of O(K3) for exact solutions and OKK2 log(K)) for approximate solutions (Peyré & Cuturi 2019). For this reason, the MW distance becomes prohibitive for large-scale applications with high component numbers. To address these challenges, we propose an accoleration approach leveraging projection techniques. Slicing the inner and outer optimal transport problems, i.e., applying a double slicing, we significantly reduce the computational burden while preserving the geometric properties of the MW distance. Additionally, our novel sliced MW

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approach facilitates fast pradient-based optimization in the space of GMMs; making the resulting metric particularly useful for machine learning applications.
1.1 Related work
Sliced probability metrics A fundamental task in machine learning and computer vision is the mean- ingful comparison of high-dimensional empirical data distributions, Geometrically meaningful metrics such as the Wasserstein distance (Santambrogio) 2015 Peyré & Cuturi 2019) are often computationally costly; therefore, sliced probability divergences are increasingly popular. In case of the classical Wasserstein dis- tance, the underlying high-dimensional transport problem is replaced by multiple 1d transport problems. At the core, the 1d problems reduce to efficiently solvable assignment formulations (Bonneel et al.) 2015 Nad- jahi et al.) 2020), which surmount the original, expensive linear program. The reduced computational cost makes the obtained sliced Wasserstein (SW) distance a valuable tool for image restoration (Tartavel et al. 2016), generative modeling (Deshpande et al. 2018 Liutkus et al. 2019). GMM estimation (Kolouri ct al. 2018). J, and small-data classification tasks (Aldroubi et al. 2021 Moosmüller & Cloninger 2023 Beckmann et al. 2025). Beyond the classical Wasserstein distances on Euclidean spaces, the slicing idea can be applied to more general divergences (Kolouri et al. 2022), the Stein discrepancy (Gong et al.) (021). the maximum
mean discrepancy Herrtiic et al. 2024 Hagemann et al. 2024), and the Wasserstein distance on the sphere
Quellmalz al. 2023 2024).
Metrics between probability mixtures The most prominent mixtures of parametrized probability mea- sures are GMMs. Since the Wasserstein distance between GMMs with more than one component cannot be computed in closed form, an alternative metric-the MW distance- has been introduced (Delon & Desol- neux 2020: Chen et al. 2018: 2016). The MW distance has been successfully applied in various domains, including the quality assessment of GMMs (Farnia al.] 2023), domain adaptation (Montesuma et al. 2024
perceptual image evaluation (Luzi al. 2023). singlo-cell data analysis (L.nn et al. 2023), and reinforcement
learning [Ziesche & Rozo 2023). Beyond GMMs, the MW distance has been extended to mixtures of non-
Gaussian probability measures (Alvarez-Melis & Fusi 2020 Bing et al. 2022 Dusson et al. 2023 Wilson
et al. 2024). The extended versions have, in particular, applications in the comparison of labeled datasets
(Alvarez-Melis & Fusi 2020). Upon completing our work, we bocame aware of two closely rolated preprints
on sliced optimal transport distances between mixtures of probability measures (Nguyen Mueller 2025
Nguyen al., 2025), which were developed independently and in parallel to our study. Notably, the proposed metrics for GMMs differ from our sliced MW variants. Moreover, we provide the first theoretical comparison of sliced MW versions with the original MW and SW distance. In particular, we establish forms of metric equivalence.
1.2 Contribution
The main contributions of this paper, which are also schematically visualized in Figure are as follows:
Starting from the original MW distance, we propose (i) to use the SW distance for the inner transport problem leading to the novel mixture sliced Wasserstein (MSW) distance and (ii) to slice the entire MW distance using 1d projections, which results in a sliced mixture Wasserstein (SMW) distance. For the efficient computation of the latter, we propose a further slicing based on the identification of 1d GMMs with 2d empirical measures. This procedure yields the new double-sliced mixture Wasserstein (DSMW) distance. The new distances are closely related and significantly reduce the computational complexity, while maintaining background in optimal transport. Moreover, they are well-defined metrics on the space of all GMMs; see Theorem 3.1
Providing a thcoretical analysis, we show that, under mild conditions, the MW, MSW, SMW, and DSMW distances are weakly equivalent, i.e., convergence of a sequence in one distance implies the convergence in all others; see Theorem 3.4 Under the same mild conditions, we show that the SMW and DSMW distances are actually strongly equivalent, i.c., the distances can be estimated by each other; see Theorem 3.2 Unfortunately, this equivalence cannot be extended to the MW and MSW
distance; see Theorem 3.3
2

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Inducing Weak Topology
Prop
3:6
Prop
5.7
Weak Equivalence"
SW
Cp SW
Thm
AI
AI
MW
MSW
SMW
> DSMW
A1
1-Strong Equivalence
G-Strong Equivalence"
Prop
3.3
SW +'COV'
Thm 3
compact paramettrr
Figure Relations between the considered GMM metrics; more precisely, between the proposed mixture sliced Wasserstein (MSW), slicod mixture Wasserstein (SMW), double sliced mixture Wasserstein (DSMW) distance and the original mixture Wasserstein (MW) and sliced Wasserstein (SW) distance. The SMW distance can be bounded from above by the SW distance and non-vanishing additive term 'COV' depending on the involved covariances. Under mild conditions, the DSMW distance also induces the SW distance with bounding constant Cp depending on the considered subset of GMMs.
We compare the introduced metrics with the original SW distance. More precisely, we show that our MSW and SMW distances are upper bounds for the SW distance. Consoquently, the MSW and SMW distance introduce the usual weak convergence of measures (restricted to the space of GMMs); Proposition 3.6 For suitable subsets of GMMs, this property also be established for the DSMW distance; see Proposition 3.6
The efficiency and scalability of our slicing approaches are demonstrated through comprchensive experiments that focus on a runtime comparison especially for high-dimensional GMMs with large numbers of components and possible practical applications of the new distances.
1.3 Outline
The remainder of this paper structured as follows: In Section 2 we review optimal transport distances, especially, the original Wasserstein distance and its acceleration through the sliced Wasserstein distance. Furthermore, review the original MW distance for GMMs. In Section g we introduce
novel slicing- based GMM metrics and provide theoretical results about the weak and strong equivalences. Moreover, we study the relations of our distances to the MW and SW distance. In Section 4 we discuss the mumerical implementation and computational performance of our slicing procedures. In Section 5 the practical poten- tial for clustering, perceptual evaluation, and gradient-baser optimization evaluated. Finally, Section f summarizes our findings and discusses future directions.
2 Preliminaries
2.1 Wasserstein and sliced Wasserstein distance
Optimal transport-based metrics like the Wasscrstein distance and its variants gauge the similarity between different probability measures living on a common measurable space. In the following, we restrict ourselves measures on Euclidean spaces. More precisely, for any X Rd the space of all Borel probability measures X with respect to the Euclidean metric is denoted by P(X). The subset of probability measures with finite pth moment is defined by
(:00=((ER0) (MEP(X)) |a| du(x) < x 15p<0. du(ao
Px(X) =
A measure μ E P(X) transferred to another domain X' Rd via a mapping T: X X! by it's push- forward: T3p 1a = o T-1
3

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A transport plan between No, M1 E (X(X) is probability measure y E P(X X) whose marginals coincide with the given probabilities. Mathematically, based on the canonical projections onto the ith component given by Ti (z = Ií, the set of all transport plans between μ사 and reads as
T'uo, p1) {< P To. 140, 11477 m1).
Based on these plans, the p- Wasserstein distance also known as p-Kantorovich Rubbissteinn metric bet Po, M2 E (Rd))) is given by
W Wy(uo:M)
inf rEr(uo))))
1 x (/
(1)00
(1)
The Wasserstein distance defines metric; so (Pp Wp) becomes metric space
Villani
2003
Santam-
brogio 2015
For dimension > 1, the computation of the Wasserstein distance is, in general, numerically challenging, In the special case d 1, the Wasserstein distance can, however, be calculated analytically (Villani 2003 On
the basis of the curullativ distribution function:
FØ(x) p((-00,x1), ZER,
and its generalized inverse:
in Pa(a)> LE (0,1),
the Wasserstein distance has the closed-form solution:
Wp(no: m1) -F
(2)
For empirical measure, the generalized inverse may be calculated using efficient sorting algorithms Bonneel 2015 Nadjahi et al. 2020).
et al.
In order exploit the computational benefits of the one-dimensional Wasserstein distance, the so-called sliced Wasserstein distance is introduced in (Kolouri et al. 2019; Bonneel al. 2015; Kolouri al. 2016
which is closely related to the Radon transform. Exploiting the slicing operator:
Ne: Rd -R
e Dds
where gd-1 = Rd p- Wasserstein distonce between /4o. nn E (TRd) I is defined as
Rd |||||| = 1} denotes the sphere, and the Euclidean inner product, the sliced
SW u1) = (f WT no, To.s 11) d0)
The spherical integral can be approximated using Monte Carlo methods (Bonneel et al. 2015 Nadjahi et al.
2020) or Quasi-Monte Carlo methods (Nguyen et al. 2024 Hertrich et al. 2025
From a topological point of view, the sliced Wasserstein and Wasserstein distance are closely related. On the one side, SW, SWpppo.ir w1) S Wp(Ho: f1) for any Po, My E (R()). On the other side, for compact subsets X C Rd, both distances are p(d+1) (id++)-strongll equivalent (Bonnotte) 2013), i.e., there exists a constant Cx > 0 such that
SWp M1) & Wp(oo.44 & Cx SW FWaT
(1) Ypu, A1 E P(X).
Beyond compact sets, the sliced Wasserstein and Wasserstein distance weakly equivalent (Nadjahi et al.
2019 2020), i.e., SWp(Mi, 14) - 0 is equivalent to Wp('n,))) + O n + oo for any sequence (Un))eN C P R and any measure E (Rd). Moreover, the convergence under W and SW imply weak convergence
Pn μ in P(Rd) (Nadjahi et al. 2019 2020). i.c., for every continuous function Φ: Rd + R vanishing at infinity, holds
lim
((x) dum(x)) Jgd ((x) dp(x))
n00 gd
4

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2.2 Wasserstein distance between Gaussian mixtures
Computing the Wasserstein distance between arbitrary probability distributions is numerically challenging, but, in some specific cases, closed-form solutions are available (Santambrogio 2015 One such instance is ~ A (m,, Eo) and ~ (m1,,, E1) on R with means m E Rd and covariance matrices Σ, E S2; which are symmetric and positive semi-defiitte in Raxdd For these measures, we have the explicit. formula (Delon Desolneux 20200:
the 2- e-sssserstein distance between two Gaussian distributions
W p1) = |lmo mill + tr + E1 2(2) En 23 ))
(4)
On the real line with μr (mm, of), this simplifies to
Wz(uo, m1) (mg m1)² + (oo -1) [1 σo) (m² 01)l2
(5)
Unfortunately, there exists no explicit formula for Gaussian mixtures. Restricting the transport plans y in ① Gaussian mixtures as well, Delon and Desolneux (Delon & Desolneux 2020) propose an adapted
Wasserstein metric that explicitly exploit the closed forms 14 and 5/. More preciscly, finite Gaussian misture model (GMM) has the form
Σ
whr n² ~ ",mm,,
(6)
with weights = (W in the probability simpler AK = tw E RRK w 1x = 1}, where 1x E R denotes the all-ones vector. The sot of Gaussian mixtures on Rdd with at most K components denoted by GMMa(K). Obviously, GMMa(K) GMM,(K)) for K < K'. The collection of all finite Gaussian mixtures is defined by
GMM4(00) = U GM)
K=0
Using the discrete transport plans between wp E Ak, and w1 E Ax, that are defined by
T(w,, w1) = (r R KaxKi 1k, Wo, 1x, wi).
the mirture Wassesssrn (MW) distance (Delon & Desolneux 2020 no E GMM{(Ko) and μı GMM,(K,) with components in 6 reads as
Chen et al.
2016
2018
between
Kn
Ki
ke Wa W3( )²
MW p1) = min ((
Amg
The MW distance is geodesic metric GMMa(co) that is neither strongly nor weakly equivalent to the 2- Wasserstein distance (Delon & Desolneux 2020 However, (Delon & Desolneux 2020 Prop. 6) establishes the lower and upper bounds
W2(Ho, S MW(Ho. MI)
Kg M
Ki w tr(I)o) + 2 Σ w/ tr(E ). k1=l
S W2(Ho,
On the real line, the MW distance may be interpreted as (classical) Wasserstein distance the parameter space IR x R2 of the Gaussian distributions. Mapping the parameters of GMM(K) to the point measure
K
v(n) = wk 8(m (ot))
Σ
5

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Published in Transactions on Machine Learning Research (08/2025)
where 8 denotes the Dirac measure, and using 5 we obtain
Ko
(i =
min (eT(w))))))
:00) -mm . )1²)
k
= =Wa(u(mo): v(1))
for all No GMM1 (Ko) and μ1 (M,).
3 Sliced distances for Gaussian mixtures
From a computational point of view, there are two main disadvantages of the MW distance: 1. The com- putation of the Wasserstein distance between two Gaussians for d > 1 requires the calculation of matrix square roots and, therefore, of costly cigenvalue dccompositions. 2. The outer optimal transport between the Gaussian components becomns expensive for large numbers of components, even when using entro regularization (Cuturi 2013). To overcome both issues, we propose different slicings of the MW distance.
3.1 Mixture sliced and sliced mixture distances
As remedy to the first disadvantage- the define the miature sliced Wasserstein (MSW) distance:
the computation of Wasserstein distances between Gaussians we
Ka
K2
MSW ASS (1) min
Ke SW2(40 No p!t)
Ag =1=1
where No E (M,((K) and μr E GMM(K)) have form 6 The MSW distance is espocially useful for GMMs with few high-dimensional components, where the slicing yields a significant speed-up, and where we are interested in recovering a transport plan between Gaussian components. However, still requires solving an optimal transport problem, which becomes problematic for high numbers of components, and therefore aim for 'fully sliced' distance.
As the first step in this direction, we instead consider the sliced misture Wasserstein (SMW) dislance:
SMW (Mo.M) m1)
MW2 (w. (o. p1) do W2(vo(Ho)
Jga-1
Jgu-1
where (04s = v( 14). Since No, E GM,) the SMW distance well-defined. However, this does not offer computational advantages as it requires solving a non-trivial 2d optimal transport problem for each projection. To speed up the computation, we switch the sliced Wasserstein distance yielding the double-sticed mirture Wasserstein (DSMW) distance:
DSMW2 (Ho. p1) = J -g (o) de.
This distance solves both computational issues since we have neither to deal with Wasserstein distance between Gaussians nor to minimize an optimal transport problem. Instead, we can exploit the closed-form
solution 22) to compute the required Wasserstein distances between point measures on the line.
Theorem 3.1. MSW, SMW, and DSMW are metrics on GMMa(oc) salisfying
DSMW m1) < SMW (1o, < (po, mi) S MW(Ho,
Po:141 GMM(2o).
(9)

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Proof. For any ba E GMM( i == 0, there exists K, N such that μ4 GMM(K)). The first and last inequalities directly follow from SW2(No: mi) s W2(uo, (1). The remaining inequality can be established by
SMW? u1)
{
min yET
Σ
A W p =0.4 p1 do
J8-
Ki
min (
Wa(To Po Nv T0.1 10 ) = MSW
=1=1 Sa-1
Ro=
The positivity and symmetry of all distances are clear. The triangle incqualities for SMW and DSMW directly follow from the triangle inequalities for W2 and SW2 together with the triangle inequality for the 2-norm on gd-1, To show the triangle inequality for MSW, we consider 14 E GMM )) = 0,1,2. Tet 2011+ and y1.2.* realize (Po. and MSW(H u2) respectively, and choose E R xKxxx such that 20
K2
Ko
0,1,* and ka
1,2,* K2i,k2
11 =
K1o =
Ro=1
The existence of such 1) is guaranteed by the so-called gluing lemma given by 0.2 ka,k2 = Eki=1 ks satisfies 70 E I(wo, W2), we obtain
Villani 2003
Lem 7.6). Since 70.2
Ko
K₂
Ka
K
K2
MSW (2) S
7al
0.2
( No 1122
=
Ks K2
SW .12 k22
Ag
No
R1
lgg
Ko
K1
Ki
K2
0.1 SW m1
1,2.
SW ,
MSW(HO, 11) MSW MSW(P1, (p1,p2).
where we exploited the triangle incquality of SW₂ and of the weighted 2-norm.
It remains show the definiteness. II ps fh, then MW ui) 0, which is inherited to the other distances. Contrary, DSMW m1) 0 implies SW2 (Wo(()())) (o()) for a.e. 0 e S Since SW₂ is metric, we have vo(Ha) = vo(1) and (1)) (0,((14) for a.c. f. Hence
0 = f
W2( Po. To p1) do = SW2(Ho,
ga-1
implying Ao = -11 Due to
99 this is transferred to SMW and MSW.
3.2 Strong and weak equivalences
We next study the topologies induced by the proposed sliced metrics. More precisely, we are interested in reverting the inequalities in Theorem or, if this is not possible, in weak equivalences. For the majority of the results this section, we assume that the means and covariances of the involved Gaussians are contained in compact set. For a parameter set P C Rd x St. we define
GMMa,P(() = (ur GMM4(K) | (m" ( P}.
This expression again extended to GMM,,P(0o) by taking the union over all K E N. From the view of practical machine learning tasks, the compactness assumption is unproblenatic as long as we consider GMMs whose variances and means are bounded for instance. In this case, can be chosen as sufficiently large closed ball in Rd x St. For compact P, we can transfer the strong equivalence 3 to SMW and DSMW.
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Theorem 3.2. SMW and DSMW are 6-strongly equivalent GMM4,p(>0) with P C R Sj compact, i.e., there emists Cp > O such that
DSMW(HO;; S (MW(HO; S Cp (Ho Aya, E GMMap(00))
Proof. For 0 E gd-1, and n == n² E (GMMa(K) with μ² ~ N(mm, Ex), we observe Te, / E GMMM (K) where the means and standard deviations of the projected components wa,; n¹ are given by m = 0 m transformed by To Σ) (0 m, (0TE))² Due to the
and of (0T (0T )) ie., the parameters compactness of P, we find X R 120 compact, such that T&(P) C X for all E gd-1, Applying (3) pointwiscly and Jensen's incquality, we have
[ Wa(vo(Ho)
(po, (w)(o)))) s Cx DSMW (1).
SCx f 130-1
for all E GMM)(
Generally, Wasserstein and sliced Wasserstein distances are not 1-strongly equivalent (Park & Slepčev 2025 Similarly, no 1-strong equivalence 33) can be encountered for MW and MSW. Morcover, the countcrexample in the following proof indicates that 1-strong equivalence may not even hold when restricting to Gaussians whose parameters are contained in compact sets.
Proposition 3.3. MW and MSW are not 1-strongly equivalent on GMM,(K) with d >
Proof. We restrict ourselves to the case (Barrkkarr & Guo 2021 More preciscly, we consider
and follow a construction first presented in the errata of
~ ((()). and (~N((())(18))
Since MW W2 and MSW SW2 on GMM,(1), the calculations in (Bayraktar & Guo 2021 imply
MW(He.
W2(ue:
MSW(He,
SW2(He,H1)
0o.
(2 +2+00 c)x 10 cos (o)l(cos?()) csin dø
Using L'Hôpital's rule and Lebesgue's dominated convergence theorem, we observe that the right-hand side diverges for e c- 0; so MW and MSW cannot be 1-strongly equivalent. The construction can be generalized by extending the diagonals by and respectively.
For GMMs with parameters in a compact set, for instance, GMMs with bounded means and variances, we can establish the weak equivalence of the remaining distances. Intriguingly, the weak equivalence holds true although SMW is not immediately related MSW and MW. While we conjecture that the weak equivalence holds true for all GMMs, the employed proof technique does not allow to skip the compactness assumption.
Theorem 3.4. DSMW, SMW, MSW, and MW are weakly equivalent on GMM,, with P C Rd x S compact, i.e.,
DSMW ()) + O
SMW(Un, (4) +0 +
MSW 0
MW p1) 0
for all (HN)NNN GMMM, and μ GMMap(K).
To show this statement, we require the following lemma ensuring that almost all (AA) projections of two Gaussians do not collide.
Lemma 3.5. For non-equal Po, 11 E GMM2(1), holds
ma.1 14 n0, VVD E $6-1
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Proof. We consider non-equal μ4 N (m, Σ,). If mo mı, then 8 (m m1) # 0 for a.c. 8 gd-1, yiclding the assertion. Otherwise, if mo and En * En, then 0 0 (Eo - Σ1) = 0 implies E ker (Eo E1). Since dimm(er((o - 21)) < d, 0 & ker(>i- - 21) n almost surely, which finishes the proof.
Proof of Theorem 3.4
2k whi n in GMMa,r(K) satisfy Let pu E = who no and -
K
K
SMW (Hm: (4) = Jg--
n.O.* Kk
Mx Te. ( d0 - 0 W2
No
where nm.0,* denotes the optimal MW plan for fixed 0 sd-1. Due to the L2((9=1) convergence, for any subsequence of Pn: we find further subsequence (Knz)JN such that
K
(2 Mny Ten x0 VV0 E S R, ko
(10)
Rig mr
pointwisely. Since P is compact, (Pn, )EN can be chosen such that uny converges to some t E GMMa and why some ho E R2o.
Assume that there exists ji with Who > 0 such that jtto + ur for all kn with who > 0. Due to Lemma 3.5
we find / gd-1 such that [0] converges and
Ta.i
_o + Tap
Vki E (k > 0}.
Since W2 33 Te. p111) is thus bounded away from zero for ny large enough, the pointwise convergence for all hı {k w > 0}, i... an entire row in n,. which sums up to non-zero weight vanishes. This, however, contradicts that y0.0,- is discrete transport plan. Consequently, #oo converges to a non-zero component of
in 10 implics .ko
For any E gd-1 such that 10 hold true and Fo pro + no, pr for the components with non-zero weights, we observe
if t + nh and wh w > 0,
8.1 k1
+ +
if to = =0,
and hence
0y Tho.k ko
if f = r and ao w > O.
+
Exploiting that the Wasscrstein distances betweon all involved Gaussians in GMMa,p(1) are bounded, we finally have
Z MW (un pa)
S
0,.0. Thuiks W
pk1 +
j +0.00 00.
R
Because this holds true for any subsequence of (px (uN),EN. the assertion is established. The remaining impli- cations follow from the inequalities Theorem 3.1 and 3.2
3.3 Weak topology
Beyond computational speed-up, the sliced Wasserstein distance is closely related to the original Wasserstein distance. Specifically, both imply weak convergence and thus induce the weak topology on P2(R)), see (Nadjahi al.] 2020). In this section, we aim explore this property for our proposed sliced distances. In first. step, establish analogue of (8) for SMW.
Proposition 3.6. For μ4 E GM,) MSW satisfies
SW2(Ho.14) & (MNWo..11)
Ko 2
Ki
< SW2(Ho: +
wp
ko
TT
0
rok 0 +
2 > whi /
o
ToT 000.
#o=1
gd-1
1==l 8a-2
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Proof. The lower bound follows from the definition of MSW and W2(Uo, μ1) I MW p1). The upper bound follows from applying in the definition of MSW together with the triangle inequality on (2
Since SW₂ induces the weak topology, also MSW yields the weak convergence. Delon and Desolneux (Delon & Desolneux 2020) give example that MW and W2 are weakly equivalent and that the upper bound in is tight. This example carries over to SW2 and SMW. Owing to the additional slicing in DSMW, which decreases SMW, Proposition 3.6 cannot be exploited to study the relation of DSMW to the weak topology. For GMMs whose parameters are contained in a compact set, the following statement yields a first approach in this direction.
S compact, there erists Cp > O such that Proposition 3.7. For PC
PCR Rd
SW2(Po, S Cp DSMWW(HO Ypo, pu E GMMa,((o)).
Proof. The statement immodiately follows from combining the strong equivalence in Theorem 3.2 with the lower bound Proposition 3.6.
As SW2 metricizes weak convergence, lim,-+oo DSMW (pn, 1)) results in Pn - μ P(O). Proposition 3.7 does not imply the strong equivalence of SW₂ and DSMW on GMMa,p(0o) since the second bound is not established. Beyond compact parameter sets, SW₂ and DSMW are not even weakly equivalent. Henceforth, 0₂ E Rd denotes the all-zero vector, Ia E Raxdd the identity matrix, and the gamma function.
Proposition 3.8. For m Rdd and 0, let nr Σ Ko I 8mzo and μn ~ NN IaI)) be given. Then
ko=1 Ko
2x 242
DSMW2 (MoW
r
Proof- Due to the uniqueness of the transport plan between one-point measure and any other measure, direct calculation yields
(Ho:
(W3(vo(p)) vo(u1)) do
Jga-1
Ko
(( m..) cos(o) + a sin(o)) dode
3ga-1
Kg
kg
2m
Ko
[00 mkg )2 )2 (0) + 0² sin?( død0.
gd
Ko
Here, the mixed terms in the extensions of the squares vanish since they are integrals over odd functions. Dropping the cosine terms, we obtain
20 2x [(2) (on 20 d+3
/
DSMW2 (Na, (1)) 2
I(g)
For sequence of point measures Pa approximating a Gaussian NO o² I2), i.., lim SWa 11) 0. Proposition 3.8 implies lim DSMW ()) 2 20 du2 oquivalent.
/T(g) + 0; so SW₂ and DSMW cannot be weakly
In particular, this means that samples from Gaussian do not yield (weak) approximation.
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Algorithm Implementation of MSW
Require: GMMs == > wi pa
hs with u ~ N(m )
> bi=00 0,
Require: number of projections L
1: for ky = -.....: K,, = 0, do
1 Chp.k1 L
02 m 0. m )²
Exo 0z))) 0,) 100
-(0, M 10, 01)))
b 02 + 1(
end for return minyeT(ww.wi)
Ko Ki Σ
ko=1 Eki=1 dy
D optimal transport solver
4 Numerical analysis
In this section, we present the computational implementation of MSW and DSMW, a runtime comparison with MW, and numerical experiments showcasing the potential of our novel sliced metrics. Experiments were conducted on system equipped with a 13th Gen Intel Core 15-00K CPU and an NVIDIA GeForce RTX 3060 GPU with 12 GB of memory. Our implementation is based on Python 3.12 and employs the Python optimal transport library (POT) (Flamary et al, 2021 for optimal transport solvers and PyTorrh Paszke et al. 2019) for automatic differentiation.
4.1 Implementation details
A common, popular approach implement the sliced Wasserstein distance relies on the so-called Monte Carlo integration (Nadjahi al. 2020). For this, we independently draw L random samples Og E gd-1 with respect to the uniform distribution ((S-11) on the sphere (in pseudocode: Or + U($d-1)) and approximate the sliced p- Wassersttnn distance by
(1
Σ W (To Mo, Mu, To..s 3
SW SW. (Pu,
(11)
where the Wasserstein distances on the line can, in general, be caleulated via 22) Independent of the dimension the Monte Carlo approximation converges with rate ()) More precisely, denoting the uniform measure by u, have the following estimate.
Thcorem 4.1 (Nadjahi et al. 2020 Thm 6)). For No,Hi E Pp(RR), and independent samples 0r, 1, 1,..., L, with respect to U(Sa-1 the expected absolule ervor is bounded by
Eg,..... er ISWP p. (Mo. M1) SWP(Mo- (1)]
VL (f W (mo.s No. m1) (1) du(o))
ge-
The term on the right-hand side may be interpreted the standard deviation of the Wasserstein distance W (TT, Ho. To, f1) the line with respect to US$d-1)) Incorporating the slicing of Gaussians and the Id Wasserstein distance (5) we obtain simple closed-form expression for the outer costs of MSW. The corresponding transport problem may then be solved using POT FFlamary et al. 2021). The details of the resulting MSW implementation are presented in Algorithm [ The Monte Carlo integration may also be
replaced by quasi-Monte Carlo methods (Nguyen et al. 2024 Hertrich et al. 2025).
For the double slicing behind DSMW, we have implement the following two major steps: 1. Determine the parameters of the GMMs Ta, μt, i.e., calculate v0((4) E P2(R R20). This step can be realized by transforming the parameters of the Gaussian components via (m, (0 m. (0 ()))). 2. Slice the resulting point measure in (R x R2o). Using the standard parametrization of S1 we realize this step by
https:///gichub..com/eeet
Wa__fo__GGMMM_ GMMs
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Algorithm 2 Implementation of DSMW
Require: GMMs == -Ei= with ht with u ~ (mmm,,
> i= 0,
Require: number of projections L
1: for == I,....
......L L do
US)
de -- u((0, 2x))
Ki
Eite = Σ
8 8.p
pit = (0)- m cos(0r) + (0) 0g) sin(
1==1
i = 0,
4: wy- = (1,
b. 1d optimal transport via 2
5: end for
return 1 Et wr.
calculating the inner product with (cos Φ5 sin Φ) where & E [0, 2T). Concatenation of both steps yields the parameter transformation Eo.s: GMM((o) + P2((R) that is, for fixed E gd and Φ E [0, 2r), given by
K
(p) = Σw wh oper p² = (0 m.) cos( + (0 Σ 0) sin(o).
k=1
Drawing L independent samples (0e, (z) with respect to the uniform measure U($ [0, 2x)), we propose to approximate DSMW by
DSMW (1),
( Σ (r( Ee (o)))
(12)
The details of the implementation are given in Algorithm where we again rely POT (Flamary et al. 2021) to compute the ld optimal transports. Similarly SWp, ps the simultaneous Monte Carlo integration with respect to both slicing directions yields convergence rate of O(L 2).
Theorem 4.2. For Po: Mi E GMMg(oc), and independent. samples (0e, (e) l 1,..., L, with respect to U($-11x x [0, (m),, the expected absolute error is bounded by
(1)..d)).....0....... (DSMW DSMW2 (No, H1)
1 s VT.
/./ (W2(E, (R.)))) DSMW3 u1)) du(( o)) 9d x(002m
o))* &
Proof. Applying Hölder's incquality, and exploiting that the samples (0e, (e) drawn independently, obtain
E(0,:0))...(00..... DSMW (Mo, m1) DSMW3 (wo, #1)l
DSMW? (MD,
(L)
(3a 20
(/
DSMW? DSMMW,(Ho.HL) v1) - DSMW3 (Ho, du(01, (84 2
du(Ol, du)
[2 (Mo)), Eee (p1)) - DSMW (mo. ui)l du(Oe, (z))
E=1 gd 2m
(/ (W3(5,, (H,( DSMW3 (o p1)l du(0, (d) gd-2x|0027]
Similarly to the convergence rate for SW p: the integral on the right-hand side corresponds to the variance of Eo with respect to (S (0)2)))
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To study the behavior of MSW and DSMW for GMMs with higher numbers of components, we perform Moreover, report the averaged number of Published in Transactions on Machine Learning Research (08/2025)
Table I: Average CPU computation time in seconds (mean plus-minus standard deviation) for GMMs with varying dimensions and component number K for MW, MSW, and DSMW. Additionally, we approximate the outer transport behind MW using efficient Sinkhorn iterations with regularization parameter e 0.1 (entropic MW). The time for MW quickly surpasses 60 seconds, whereas MSW takes only seconds and DSMW is calculated within single second. The Monte Carlo iterations are stopped when reaching precision of 10-3. Note that GPU usage achieves further acceleration in practice.
components K
components K
dim
10
100
500
dim d
10
100
500
10
0.05
1.85
46.4100.2
10
0.02+10.00
1.67:00.02
41.57:10.27
100
2.59=0.19
>60
>60
100
0.74+0.35
>60
>60
500
>60
>0
>60
500
>60
>60
>60
MW
Entropic MW
components K
components K
dim
10
100
500
dim
10
100
500
10
1.21=0.15
1.40:t0.19
1.28:10.09
10
0.15:0..05
0.02+00001
0.01#00011
100
0.89+0.44
3.11+0.15
10.2+0092
100
0.13+0008
0.05+00.03
0.08+0.04
500
4.63+
44.7=1.95
>60
500
0.34+0.24
0.89+0.441
1.05
MSW
DSMW
4.2 Runtime comparison
To evaluate the computational speed-up of Algorithm[and22 present side- side-by-side CPU time comparison in Table 1 For each dimension d and component number K we calculate MW, MSW, and DSMW for 10 pairs of random GMMs. The GMMs are generated with uniformly distributed weights in AK and uniform means in [0, 10 Based on the Cholesky decomposition, the covariance matrices are generated as QQ based on lower triangular matrices Q with uniform entries in [0, 1]. The Monte Carlo estimates SW2 (II for the MSW costs Algorithm and of DSMW (12) (Algorithm 2 are calculated up to convergence. More precisely, we slop the Monte Carlo ileralion soon as the mean precision atlains
Ko
SWa,r pt) SWa,L-1014 .
S 10-3
(13)
K0Kı
RD
=1=1
and
JDSW 2(Ho, - DSMW, L-10 p1)l < 10-3;
(14)
so we stop as soon as the estimate nearly stagnates over 10 iterations. The number of 10 iterations is here chosen manually to reduce the stochastic impact of the random process on subsequent iterates. Table [ that MW, unlike MSW and DSMW, does not scale well beyond toy GMMs. To reduce the numerical burden of the outer transport problem behind MW, we additionally solve this via efficient Sinkhorn iterations (Peyré & Cuturi, 2019) that rely on an entropy regularization. This adaption only yiclds minor acceleration showing that the analytical solution of the inner transports are the major bottleneck. We observed an analogous slight acceleration for regularized version of MSW-not recorded in Table ! The usage of the original MW distance is thus limited in image reconstruction methods typically employing GMMs with 100- 300 components on dimension of size 50 100 ZZornn & Weiss 2011). In such cases, MSW and DSMW are computed in a second or less, whereas MW takes minutes. Interestingly, we consistently observed accelerated convergence of DSMW for higher dimensions for < 500.
additional experiments in dimension d 50, Table 2
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components K
Table Average CPU computa- tion time in seconds (mean plus- minus standard deviation) and av- >60
1000
2000
5000
10000
time
8.9+1.03
47.1342.81
>60
erage number of projections until convergence for GMMs with varying component numbers K in dimension = 50.
MSW
proj-
8967
8816
time
0.02++0.01
0.04+0.01 15
0.09
0.16+0.00
DSMW
proj.
15
12
10
MSW
DSMW
Average CPU computation time in seconds (mean plus-minus standard deviation) and average number of projections until convergence for GMMs in varying dimensions d and K = 10 components.
Table
dim d
time
proj.
time
proj.
1000
25.40
6932
1.00
225
2000
53.09+7.82
6928
3.43+11.72
228
3000
>60
6.90+3.22
215
projections required to reach the convergence criteria 13 and 14). While the outer transport behind MSW leads to exploding computation times, DSMW is still computed under one second. The averaged number of required projections slightly deceases for higher component numbers. The reason for this behaviour might be that more and more components in the generated GMMs become alike. For very high component numbers, the entire generated GMMs are closely related such that the variance in the upper bound from Theorem 4.2 becomes small; so the absolute error for DSMW decreases extremely rapid. This effect is much less distinct for MSW since the majority of the single components still differ.
Finally, we repeat the experiment for GMMs in higher dimensions, where the number of components is fixed by K = 10, see Table 3 As before, we observe a significant impact of the double slicing on the computation time. While MSW and DSMW take longer for increasing dimensions, the average projection number remains stable; the increasing computation times mainly trace back to the increased numerical effort for each projection. Again, we observe that DSMW requires much less directions than MSW for convergence, which leads to an additional speed-up.
5
Experiments and applications
5.1 Detection of cluster number
When clustering classifying data, a common approach is to fit likelihood- maxihoozimmmimmiiingggggg GMM using the EM algorithm (Wan et al. 2019). For this, however, the number of Gaussian components has to be specified, which requires additional data analysis. In practice, we may rely on a wide range of classification indices and criteria to find suitable number (Xu et al., 2016). As an alternative, we propose to use GMM-based metrics by tracking the distance between the fitted GMMs μk GMM{()) and μ4+1 GMM + 1). Figuratively, try find the smallest N such that adding more components does not change the fitted model. Tn other words, we increase & until the distance between μ4 and μ4411 vanishes. To illustrate this approach, we sample 1000 data points from four 2d distributions, each with four or six clearly separated modes. Then, GMMs with k = ,, Gaussian components are fitted using the EM algorithm, and the distances MW (1k: 11)1), , MSW (1k, 1k+1) and DSMW(Hk, Hk+1) are computed. The evolutions of the fitted models are presented in Figure 2 For each distribution, we observe convergence in all three metrics at = 4 or the true number of components. The experiment suggests that the GMM-based matrices can indeed be used to specify the number of clusters. A broader study and comparison with other methods, especially when the classes not well separated, is left for future research.
k 6, which corresponds
5.2 Perceptual metric
The Fréchet Inception Distance (FID) Heusel et al., 2017) widely used metric for evaluating generative models in computer vision. Tt estimates the distance between real and generated image distributions by
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dislance between pls Ma-11
MWIJA, P)
MWV Ma
MSWOL Ma
MSWIU.
MSWON. DSMWIph.Peil
DEMWO..He12)
DSMWipe.Pe121
2
6
8 2
4
Components 8
Components
Components
Components
Detection of cluster numbers by tracking the distances between μ4 and 1k+1, where 14k is fitted GMM with k components. For the input data (top), the plots (bottom, y-axis from 0 to maximum) show the distance between μa and Mk+1 with respect to MW (blue), MSW (orange), and DSMW (green). The distances become zero after k =4 (left) and & = 6 (right) indicating that higher component numbers only yield reparametrizations of the former models. The first zero hence corresponds to the correct cluster nunber. Notice that MW seems to be slightly less stable than MSW and DSMW.
Figure 2:
assuming that their deep feature representations follow multivariate Gaussian distributions. In more detail, the features of the given empirical image distributions are extracted using pre-trained neural network. Both fealure sets are then modeled as normal distributions, whose Wasserstein distance is estimated by 4 Despite good performance, the Gaussian assumption does not align with empirical feature distributions [Luzi et al. )023). As an alternative, the Wasserstein on Mixtures (WaM) metric fits two GMMs to the feature distribution using the EM algorithm and estimates the mixture Wasserstein distance 7 to capture more visual information (Tuzi et al. 2023). Both perceptual metrics FID and WaM-have been shown align with human perception.
In this experiment, adapt the WaM metric by replacing MW with MSW and DSMW. As feature extrac- tor for FTD and WaM, we employ pre-trained Inception-v3 network (Szegedy et al. 2016) with latent representation of 2048 dimensions. The obtained feature set is then fitted by a multi-variate normal dis- tribution or by a five-component GMM respectively. For the real image set, we employ a subset of 1000 CIFAR1O images (Krizhevsky) 2009). For the artificial set, we disturb these images with Gaussian noise (standard deviation between 0.01 and 0.2), Salt&Pepper noise (corruption ratio between 5% and 30%), and Gaussian blur (slandard deviation between 0.1 and 1.5). The resulting FID and WaM between the real and artificial image sets are visualized in Figure 3 All perceptual metrics react very similarly to the different distortions. Stronger distortions lead to increased distances, and the FID and the (adapted) WaM curves display comparable shapes for the same distortion types. Given the fitted GMMs, MW calculation took around 11 minutes, MSW calculation took around minute and DSMW calculation took around 20 seconds.
5.3 Quantization of Gaussian mixtures
The focus of the next experiment is to demonstrate that our DSMW distance may be used in the context of gradient-based optimization. Exemplarily, we aim to reduce the component mumber of a given GMM while prescrving key statistical properties, aiding in model compression. Earlier work on this topic has explored optimal component selections and divergence- based approximations to minimize the information loss (Runnalls] 2007 Crouse al. 2011 Assa. & Plataniotis 2018). To quantize given μ E GMM2(K), we
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WaM/MWW /MW
250
Wald WaM/MSW /MSW
WaM /DSMW
100
10
0.36
0.5
0.30
6,20
0.36
standard deviation
standard deviation
standard deviation
standard deviation
FID
Gaussion noise
WaM/Mi /MW Gaussian noise
WaM/MSW MSW Gaussion noise
WaM FUMM/DMWW Gaussian noise
100
200
240
WaM/MMN /WW
300
Wabd/ /SWW
12
Waa MESSEES
200
260
10
corruption ratio
corruption ratio
corruption ratio
corruption ratio
FID Salt&Pepderr
WaM/MW Salt&Peover
WaM/MSW Salt&Peeoer
WaM/DSMW Salt&&Peueer
130
Wal /DSMWW
2.3
WWW/
WaM Wa.MMMSW /MSWW
FID
116
100
1.0
20
0.5
0.6
0.8
10 1.2
1,4
0.8
2.
1.2
0.8
0.8
3.0
3.2
0.6
0.8
1.0
1.2
14
standard deviation
standard deviation
standard deviation
standard deviation
FID
Gaussian blur
WaM/MW Gauseian blur
WaM/MSW Gaussian blur
WaM/DSMW Gaussian blur
Figure 3: FID (left), WaM with MSW (center left), WaM with MW (center right), and WaM with DSMW (right) between real and artificial CIFAR10 image sets based on different types and levels of image distortions (standard deviation of Gaussian noise, corruption ratio of Salt&Pepper noise, standard deviation of Gaussian blur). All resulting curves of the evaluated perceptual metrics show comparable behaviors.
instead propose to solve
min (KG
DSMW2 (4",)4) with K < K.
In favor of gradient descent minimization, we parametrize μ² according to (Gepperth & Pfülb 2021 For this, denote the set of lower triangular matrices with non-negative diagonal by Trizold). Given parameter vector (w* m Q with w E R, m² Rd, and Q* E Trizo(d), consider the parametrized GMM:
exp(w** K- 2t=1 exp(we)
Mp Σ
~ Q.Q + La))),
(15)
where we add the identity Ia for numerical stability. Using fixed number of random directions 02 in 12 we employ the Adam scheme (Kingma & Ba 2014 in combination with automatic differontiation to minimize
min (EMMW,(pppp)
The updated diagonal entrics of Qh are here projected back to Rzo. As target GMM we use two- dimensional Gaussian mixtures with 100 components in the form of 28 x 28 MNIST digits (LeCun el al., 1998), which are calculated by employing the EM algorithm (Dempster et al. 1977 to the pixel intensities, see Figure 4 Applying Adam with step size 0.03 for 200 itcrations and 20 random initializations, and choosing L 100 and a (pixel), we quantize the inputs by 50-component GMMs. Each gradient descent cycle requires less than second GPU time. The resulting densities on [0, 28)2 shown in Figure 4
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2
4
S
6
7
Tnput
Input
Input
Input
Input
Input
Input
Input
Input
Input
Output
Output
Output
Output
Output
Output
Output
Output
Output
Output
Figure 4: Quantization of input GMMs with 100 components (top) GMMs with 50 components (bottom).
5.4 Barycenter of Gaussian mixtures
Barycenters are generalized Fréchet means, which in the context of optimal transport are defined measure-valued midpoints between multiple inputs. For general measures, Wasserstein barycenters (Agueh) & Carlier 2011 and sliced Wasserstein barycenters (Bonneel et al. 2015) as well as, for GMMs, mixture Wasserstein barycenters (Delon & Desolneux 2020) are studied. The latter have applications in compu- tational biology (Lin et al.] (2023) and domain adaptation (Montesuma et al., 2024). In more detail, for Ply. Pr GMMa(K) and Ar, MW barycenter the solution to
inf J (G
X MW3 pi).
(16)
The infirumm of (16) is always attained, where the components 4 M GMM(11) of the minimizer p are themselves Wasserstein barycenters between i . ie.,
arg min p (PPP)
Σ X, w
(17)
which can be calculated analytically (Delon & Desolneux 2020 The weights of these closed-form Wasser- stein barycenters be computed by solving lincar program, where at most (I 1) 1 weights are non-zero.
In the style of the MW barycenter problem (16) and as prooe-of-concept, we initially consider (fixed- component) DSMW barycenters μ² E GMM KT) with components 1 Ai...ky from 17 solving
inf (KC)
Σ A LDSMW3(4' (44", 1=1
(18)
Referring to Section 5.3 we solve 18] using stochastic gradient descent with automatic differentiation, where we parametrize 4², whose components are fixed, by
exp (.....)) R Σi1=1
Pw Σ
exp(w " de..kkk R Kx
24=1
For comparison, we adapt the MW interpolation proximating 18 using DSMW2 with 100 random directions, stochastic gradient descent finds the optimal weights parametrized by after 10 ilerations with step size of 0.03. The results presented in Figure 5 show that MW and DSMW both enable meaningful smooth interpolations despite their distinct geometry. A direct comparison convey the impression that the DSMW barycenter is less related to the prominent features of the imputs. Especially, in the middle, we nearly obtain unimodal GMMs. Note that the aim of
tasl first presented in Delon & Desolneux 2020 Ap-
httpps//githhb. com/judelo/gmmoo
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Z
7
Z
Z
MW barycenter
Fixed-component DSMW barycenter
Figure 5: Bilinear interpolation between GMMs at the corners via MW barycenters and fixed-comi exed-comppnent DSMW barycenters based on the example from (Delon Desolneux 2020 Both methods show reasonable, smooth interpolations between the inputs.
this toy example only demonstrate that the DSMW distance allows efficient barycenter computations using stochastic gradient descent. The exploration of the quality, especially compared other optimal transport-based barycenters, is left for future research.
Despite the underlying sparse structure of the MW barycenters, computation becomes increasingly costly beyond toy examples due to the necessary minimization over the weights of K Gaussian components <Lin et al. 2023 Based on the parametrization Ho E GMM,(K*) in [1) with o = 0.3 (pixel) and an arbitrary component number K* E N, we may alternatively use the Adam scheme with automatic differentiation to approximate (froe-component) DSMW barycenter by minimizing
min d DSMW> (Hp:
(19)
19 between five MNIST samples LeCunn et al. 1998) per digit. The MNIST samples themselves are approximated by 10-component GMMs using the EM method. We refer Figure we choose K* 100, L 100, and À = 1 13.. Taking 200 Adam gradient descent steps with step size 0.03, where the descent. is started from 10 random initial GMMs, we obtain the free-component DSMW barycenters in Figure Along the results, further display SW₂ barycenters estimated by minimizing
In the second experiment, we aim to compute the barycenters
t for the density contour plots of the inputs. For the barycenter computation,
min Z.ER
SW2 2.L
Σ 8rn Bin.
where jú, are empirical measures obtained from 10000 samples from Pn Similarly to the DSMW barycenter, we choose N = 100, L = 100, x = 5 15 and apply 200 Adam gradient descent steps with step size 0.03 and 9 restarts. The results are also visualized in Figure where similar shapes emerge for both ansätze. The SW2 barycenters seem to be more noisy. For the described setting, 200 gradient descent steps require around is GPU time for DSMW and around 5s for SW,- Besides the computational speed-up, the main benefit of using our DSMW distance is that the DSMW barycenter is an actual undegenerated GMM, where the SW, barycenter only yields (degenerated) point cloud. Note that the shape of the DSMW and SWa barycenter
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5
5
6
6
6
6
6
2
2
2
2
2
7
7
7
3
3
3
3
3
8
8
8
8 9
8
4
4
4
4 4
4
9
Inputa
DSMW
SW2
Inputs
DSMW SW2
Figure 6: DSMW and 2W₂ barycenters. Given an input set of five two-dimensional Gaussian mixtures in the form of MNIST digits (left: 0-4, right: 5-9), we display DSMW (in the form of GMM density) and 2W₂ barycenters (in the form of points). The DSMW barycenters are actual undegenerated GMMs, where the SW2 barycenters are only (degenerated) point clouds.
are closely related, and that the DSMW barycenter can thus be interpreted as smoothed version of the SW₂ barycenter.
6 Conclusion
In this work, we introduce two novel sliced versions of the MW distance for GMMs, namely, the MSW and the DSMW distance. Both variants significantly reduce the computational burden while relying on the Euclidean geometry of the underlying domains. Our theoretical results established connections wetween MSW, DSMW, and MW as well as relations to SW2. The latter especially ensures meaningful convergence and topological properties. Through extensive numerical experiments, we demonstrated the efficiency of our approach in various applications, namely, perceptual data comparison, unsupervised clustering, GMM quantization, and GMM barycenter computation.
Additionally, our framework can be readily adapted to so-called max-sliced distances, where the integral over the sphere is replaced by taking the maximum over the sphere. The original max-sliced Wasserstein distance has been shown to provide a meaningful alternative to the sliced Wasserstein distance by focusing on the most discriminative direction [Deshpande et al. 2019). This might be beneficial for gradient- optimization. Moreover, the max-sliced distance is 1-strongly equivalent to the Wasserstein distance on G1) (Bayraktar & Guo 2021). Consequently, replacing the sliced distance in the MSW construction with the max-sliced variant, we can strengthen the shown weak equivalence to an actual strong equivalence. Furthermore, it might be interesting to investigate metric equivalences with respect to the slicing techniques in [guyen& & Mueller
2025: Nguyen et al. 2025
Beyond GMMs, the MW distance has been extended to mixtures of non-Gaussian probability measures AAlvarez-Melis Fusi 2020
Bing et al. 2022 Dusson et al. 2023). While our focus has been on Gaussian mixtures, an interesting direction for future research is the extension of our framework to other relevant non-Gaussian mixture models, such as Dirichlet mixtures (Pal & Heumann 2022. Martin et al. 2024). Since many real-world datasets exhibit non- non-Gaussian characteristics (Luzi et al. 2023). developing Wasserstein- type distances tailored to these distributions would further broaden the applicability of our approach. Beyond the MW distance, a sliced extension to
optimal transport between Gaussian mixtures (Sllmona et al. 2024 Beier el al. 2025) might be of interest.
Author Contributions
Both authors MP and RB have equally contributed to this paper.
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Acknowledgments
MP gratefully acknowledges the financial support by the German Research Foundation (DFG), GRK2260 BIOQIC project 289347353.
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24


Reviews:
[{"review_id": "ihTWIPKzMN", "reviewer": "Reviewer_XP7B", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. The slicing approach provides a simple yet theoretically grounded means of making MW computations feasible for high-dimensional GMM applications.\n2. The paper rigorously analyzes the relationships between WM, MSW, SMW, and DSMW, including their inequality structure, equivalence properties, and topological behaviors.\n3. Runtime, clustering, perceptual quality, and barycenter examples illustrate both efficiency and utility.\n4. The theoretical derivations and algorithmic descriptions are clear and well-organized.\n\nWeaknesses:\n1. The paper offers limited discussion of alternative approaches to reduce the computational cost of the Wasserstein distance. For instance, a comparative analysis with entropically regularized OT methods such as Sinkhorn distances would be informative.\n2. While Theorem 4.1 and 4.2 showed the expected absolute error of SW and DSWM, since the right hand side of the inequalities, which are interpreted as standard deviation terms, increase as dimension d and component K increase, the scalability of $L$ required to reach convergence of the algorithm is still not clear.\n3. In the runtime analysis, DSMW demonstrates a significant empirical speedup over MW; however, the scalability of the proposed methods remains unclear due to the limited range of parameter settings explored in the experiments.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a family of sliced approximations to the Mixture Wasserstein (MW) distance for Gaussian‑mixture models (GMMs).\nBy repeatedly projecting the mixtures onto one‑dimensional subspaces, the authors get two computable metrics, i.e., Mixture Sliced Wasserstein (MSW) and Double‑Sliced Mixture Wasserstein (DSMW).\nThese metrics (i) retain the distance properties of MW, (ii) come with Monte‑Carlo estimators whose cost scales linearly in dimension and number of components, and (iii) dramatically reduced the computational cost of OT‑based tasks such as clustering, model selection, and barycenter computation.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. Discuss the the trade-offs between sliced Wasserstein distance and entropically regularized Wasserstein distances, e.g. comparing MSW with mixtures of sinkhorn distances. If feasible, also extend the experimental section to include this comparison.\n2. To show the scalability of the proposed method, extend the synthetic benchmarks to higher dimension d and component K in the experiments.\n3. To show how the number of projections $L$ needed scales with dimension d and component K, in the runtime analysis, include the time used for each projection, and the number of projections $L$ used to reach the convergence criteria, together with the total time used for each method.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "gGbSYQVzY2", "reviewer": "Reviewer_Mo6r", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n- [S1] Clear Computational Advantage: MSW and DSMW drastically reduce runtime compared to MW, as demonstrated in Table 1.\n- [S2] Solid Theoretical Foundation: Theoretical results (Figure 1, Theorems 3.1–3.4) provide important insight into metric properties and equivalence relationships.\n- [S3] Paper is well-organized and easy to follow structurally.\n\n**Weaknesses**\n- [W1] Limited explanation of compact parameter assumptions: Many key theoretical results rely on the assumption of compactness, but its implications and relevance to real ML applications are not sufficiently discussed, related to [C1].\n- [W2] Applications are somewhat illustrative rather than practical: Section 4 showcases several applications, but they feel scattered, limited in diversity, and do not fully leverage high-dimensional real-world datasets, related to [C2].\n- [W3] Notational inconsistencies and unclear formulations: Some equations (e.g., (3), (7), (8)) lack clarity or proper references; inconsistent use of notations (e.g., $W_p$ and $\\mathrm{W}_p$) also hinders readability, related to [C3].", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses the computational limitations of the Mixture Wasserstein (MW) distance between Gaussian Mixture Models (GMMs) by introducing slicing-based approximations.\n\nThe proposed alternatives—Mixture Sliced Wasserstein (MSW), Sliced Mixture Wasserstein (SMW), and Double-Sliced Mixture Wasserstein (DSMW)—are theoretically analyzed and shown to be metrics.\n\nTheoretical results regarding their weak and strong equivalences with MW and Sliced Wasserstein (SW) distances are provided.\n\nEmpirical applications include clustering, perceptual similarity assessment, quantization, and barycenter computation.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "**[Critical to securing my recommendation for acceptance & Comments to strengthen the work in my view]**\n\n### [C1]\nPropositions except 3.6 utilized the `compact parameter sets', but the explanation on the importance of this property is insufficient. Is this assumption just for proofing theorems? In contrast, do ML tasks have some merits on this?\n\n### [C2]\n\nSection 4.1 and 4.2 seems to be the main contribution and Section 4.3 - 4.6 show application examples; thus some main message is now degraded.\n\nSome comment for this:\n- A possible way is dividing the section.\n- Do we real data examples on high-dimensional domains?\n- For clarity, give an explanation to stop MC computation by comparing L and L + 10 results (i.e., 10 is a kind of magic number for eaders).\n- In Eq. (15), $\\mathrm{W}$ means $W_p$?\n- Please justify using $k=4$ examples in Figure 2: the results are insightful, but readers may not follow the importance and reason to show this result here.\n- Please clarify the advantages of DSMW in Figure 5 and Figure 6; some readers may be difficult to follow this qualitative comparisons using figures (basically, this just shows some gaussian crowds and points, digits).\n\n### [C3]\nPlease carefully write equations to make the discussions clearer.\n\nThe below are examples:\n- $\\mathrm{W}_p$ and $W_p$ are mixed.\n- [above Eq. (3)] What is the meaning of $\\subset\\subset$?\n- [below Eq. (3)] $SW(\\mu_n,\\mu)$ and $W(m_n, mu)$ mean $SW_p$ and $W_p$ for any $p$? (This paper did not define SW or W without a subscript.)\n- [Eq. (7)] What is the exact definition of this sigma ($\\sum_{k_0, k_1=1}^{K_0, K_1}$).\n- [Eq. (8)] Where from is this equation? Delon & Desolneux 2020?\n- [Eq. (9)] What is the exact definition of $\\leq$ between metrics?\n- [Proof of Theorem 4.2] Should 1/L in line 3 be in the parentheses? (to extract 1/sqrt(L))", "broader_impact_concerns": "Not applicable to this paper in my opinion.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "NegzV5SaEF", "reviewer": "Reviewer_cDSz", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "__Strengths:__\n- Overall, the paper is easy to follow; the objectives are clear and the main results are presented nicely, aided by a summarising figure in Figure 1.\n- Claims are backed by rigorous proofs, which I have checked and seem correct. These theoretical results are also insightful about the newly proposed metrics.\n- Numerical experiments are extensive and show the benefits and properties of the proposed distances both computationally and in terms of their various applications.\n\n__Weaknesses:__\n- Clarity on some of the experiments can be improved (see more below).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The work proposes two new distances on Gaussian mixtures; the mixture sliced Wasserstein distance (MSW) and the double-sliced mixture Wasserstein distance (DSMW). This addresses the problem of expensive computation of the mixture Wasserstein (MW) distance, previously proposed to compute distances between Gaussian mixtures, which arises due to (i) poor scalability in space dimension $d$ and (ii) number of mixture components $K$. The MSW addresses the former problem, and the DSMW addresses both problems, at the cost of being a weaker metric. Theoretical results show the weak equivalence between MW, MSW, SMW and DSMW, strong equivalence of SMW and DSMW and the weak topology induced by SMW and DSMW. Experiments demonstrate the computational benefits of using the newly proposed distances with various applications.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- In the proofs of Theorem 3.1 and 3.4, I believe there is a misprint in the notation of the optimal transport plan, where sometimes the notation $\\pi^{n, \\theta}\\_{k\\_0, k\\_1}$ is used instead of $\\gamma^{n, \\theta}\\_{k\\_0, k\\_1}$.\n- There is a misprint in the proof of Theorem 4.2: in the second-to-last line of the equation, $1/L$ should be inside the parentheses, not outside.\n- Misprint in page 13: \"Cholseky decomposition\" -> \"Cholesky decomposition\".\n- I am not sure what the phrase \"fitted GMM with $k$ components and a one with $k+1$\" in the caption of Figure 2 means.\n- Please add a label for the y-axis in Figure 2, as it can be confusing what is being displayed at first glance. My understanding is that these are the {MW, MSW, DSMW} distances between the true Gaussian mixture and the predicted one using EM. I would recommend adding a y-axis label \"GMM Distance\" or something to that effect, and explain in the caption or main body how the distances are computed (e.g. the blue curve displays MW distance between the true model and the predicted model, etc...).\n- Similarly, please add labels for the x and y-axes in Figure 3.\n- In the conclusion, please be clearer about the sentence \"while preserving the geometric properties\". What geometric properties are the authors referring to?", "broader_impact_concerns": "The work is mostly theoretical, and there are no immediate ethical concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)