Gaussian-Smoothed Sliced Probability Divergences
Authors: Mokhtar Z. Alaya
Alain Rakotomamonjy
Maxime Berar
Gilles Gasso
Venue: Accepted by TMLR ()

Abstract: Gaussian smoothed sliced Wasserstein distance has been recently introduced for comparing probability distributions, while preserving privacy on the data. It has been shown that it provides performances similar to its non-smoothed (non-private) counterpart. However, the computational and statistical properties of such a metric have not yet been well-established. This work investigates the theoretical properties of this distance as well as those of generalized versions denoted as Gaussian-smoothed sliced divergences ${\text{G}}_{\sigma}\text{SW}_p$. We first show that smoothing and slicing preserve the metric property and the weak topology. To study the sample complexity of such divergences, we then introduce $\hat{\hat\mu}_{n}$ the { double empirical distribution} for the smoothed-projected $\mu$. The distribution $\hat{\hat\mu}_{n}$ is a result of a double sampling process: one from sampling according to the origin distribution $\mu$ and the second according to the convolution of the projection of $\mu$ on the unit sphere and the Gaussian smoothing. We particularly focus on the Gaussian smoothed sliced Wasserstein distance ${\text{G}}_{\sigma}\text{SW}_p$ and prove that it converges with a rate $O(n^{-1/{2p}})$. We also derive other properties, including continuity, of different divergences with respect to the smoothing parameter. We support our theoretical findings with empirical studies in the context of privacy-preserving domain adaptation.

Published in Transactions on Machine Learning Research (11/2024)
Gaussian-Smoother Sliced Probability Divergences
Mokhtar Z. Alaya
alayaetm@uto.fr
Université de Technologic de Compiègne,
LMAC (Laboratoire de Mathématiques Appliquées de Compiègne); CS 60 319 60 203 Compiègne Ceder
Alain Rakotomamonij
arahotonamoriiy@eettt.com
Criteo AI Lab, Paris, France,
Maxime Berar
maxime. mazime. aaazinee.erar@vnniivruueenn
Unio Rouen Normandie, INSA Roven Normondie, Universile Le Havre Normandie Normandie Univ, LITIS UR4108, Rouen, France
Gilles Gasso
gilles.gosoo@hnss-oooeee
INSA Rouen Normandie, Unio Rouen Normandie, Universite Le Houre Normandie, Normandie Univ, LITIS UR4108, Rouen, France
Reviewed on OpenReview:
https: // openreveew. net/ forum? wd=weuALLNUV2
Abstract
Gaussian smoothed sliced Wasserstein distance has been recently introduced for comparing probability distributions, while preserving privacy the data. It has been shown that it provides performances similar to its non-smoothed (non-private) counterpart. However, the computational and statistical properties of such a metric have not yet been well-establiseed. This work investigates the theoretical properties of this distance as well as those of generalized versions denoted as smoothing and slicing preserve the metric property and the weak topology. To study the sample complexity of such divergences, we then introduce f the double empirical distribution for the smnoothed-rrojetted μ.. The distribution fn, a result of double sampling process: one from sampling according to the origin distribution μ and the second according to the convolution of the projection of μ on the unit sphere and the Gaussian smoothing. We particularly focus on the Gaussian smoothed sliced Wasserstein distance S,SW, and prove that it converges with a rate Onn-1/2p). We also derive other properties, including continuity, of different divergences with respect to the smoothing parameter. We support our thcoretical findings with empirical studies the context of prreervvir domain adaptation.
sliced divergences GoSD,. g. We first show that
1
Introduction
Divergences for comparing distributions have been shown be important for achieving good performance
in the contexts of generative modeling (Arjovsky et al.) 2017 Salimans et al. 2018), domain adaptation (Long
et al.
2015
Courty et al., 2016 Lee et al. 2019),, and in computer vision (Bonneel et al. 2011
Solomon
ct al.
2015)
among many more applications Kolouri et al. 2017 Peyré & Cuturi 2019 Nguyen et al. 2023)
Examples of divergences that have proved useful for these tasks are the Maximum Mean Discrepancy (Gretton
ct al.
2012
Long et al. 2015 Sutherland et al. 2017), the Wasserstein distance (Monge
1781
Kantorovich
1942
Villani 2009 or its variant the sliced Wasserstein distance (SW) (Kolouri et al.
2016
Bouneel
Coeurjolyy
2019
Kolouri al. 2019b Nguyen al.] 2021 2022 2024).
The SW distance has the advantage of being computationally efficient, since it uses a closed-form. solution for distributions with support on IR, by computing the expectation of one-dimensional (1D) random projections of distributions in Rdd. Owing this efficiency and the resulting scalability, this distance has been successfully
applied in several applications ranging from generative models to domain adaptation (Kolouri et al. 2019a

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Published in Transactions on Machine Learning Research (11/2024)
Deshpande et al. 2019 Wu et al. 2019 Lee et al. 2019 and its statistical properties have been well-studied in Nadjahi et al. (2020)
Recently, Gaussian smoothed variants of the Wasserstein distance and the sliced Wasserstein distance have been introduced respectively in (Neetert et al. 2021) and in Rakotomamonjj & Ralaivola (2021). One main motivation behind these variants is to provide a privacy guarantee for the distribution comparison task as Gaussian smoothing is known to be a mechanism for achieving differential privacy (Dwork et al. 2014). While the properties of the Gaussian smoothed Wasserstein distance have been extensively studied by Nictert et al. (2021), the properties of the Gaussian smoothed sliced Wasserstein distance have not been fully investigated yet although they are known to be more computationally efficient.
In this work, we focus on the slicing of Gaussian smoothed Gaussian-smoothed measure discrepancies by providing theoretical propertics of more general divergenocs inducod by some base distances or divergences for distributions defined in Rd. These base distances/ cestances/divergennee /ivergences encompass Wasserstein, maximum mean discrepancy, Sinkhorn divergence. As for main contribution, we first establish the topological properties of these divergences in term of metrization of the weak topology and semi-lower continuous property. Then we focus on the sample complexity of such divergences by introducing the double empirical distribution A. for the smoothed- smoothed-projected origin distribution μ.. The new empirical distribution is a result of double sampling process: one from sampling according to the origin distribution and the second according to the convolution of the projection of μ on the unit sphere and the Gaussian smnothing. The introducing of fün inspired from the implementation part: we sample XI...., X, from the raw distribution μ to define An then project it on the unit sphere and smooth this projection with a Gaussian distribution. This smoothing is a continuous measure that needs to be sampled. For that reason, we add double sampling and then provide Å.. We particularly focus the Gaussian smoothed sliced Wasserstein distance.
Given the importance of the noise level in the privacy/utility trade-off achieved by the divergence, we investigate order relation and contimuity result with respect to the noise level. These properties are of high impact as it supports a computationally cheap warm-start/ tann-sttrr/fi--tuunngg procedure when looking for a privacy/ution /utility compromise of the divergence. Our theoretical study is backed by some numerical experiments on toy problems and on domain adaptation illustrating how owing to the topology induced by our metric and its continuity, differential privacy comes almost for free (without loss of performance) and multiple models with different level of privacy can be cheaply computed.
Comparison with previous works. Here we highlight the position of this work compared to the most linked previous ones, in particular Nadjahi et al. (2020) and Rakotomamonij & Ralaivola (2021). The work of Nadjahi et al. (2020) is focusod on sliced Wasserstein distance and its statistical propertics, however our work is based on the properties of the Gaussian smoothed with general divergences (e.g. Wasserstein, MMD, Sinkhorn divergence). We argue that the properties cannot be directly derived from (Nadjahi et al. 2020 especially the sample complexity result. In Rakotomamonjy & Ralaivola (2021), the authors invostigated the smoothed Wasserstein distance and their theoretical finding was principally on proving the metric property, whereas we further investigate sample and projection complexities and the continuity properties w.r.t. the smoothing noise level. We emphasize that the novelty of the present paper consists in the theoretical properties dcrived from the definition of the empirical mcasure P. The smoothing of the raw mcasures, from a theoretical point view, is a continuous measure (see Lemma 3.5p that needs to be sampled. This entails to define the second sampling step and construct fa, an empirical version for the smoothing projection of μ.. To the best of our knowledge, this work is the first introducing the double randomness in the case of smoothing optimal transport discrepancies. Recent works (Goldfeld et al.) 2020 Nietert et al. 2021 addressed the smoothing Wasserstein an their theoretical results relied only on An.
Layout of the paper. The paper is organized follows: after introducing the notation and some background in Scction 2 we detail the topological propertics of Gaussian-mmoothed sliced divergence in Section 3.1 while the double sampling process and its statistical properties are established in Section 3.2 The noise analyses are provided in Section 3.3] Experimental analyses for supporting the theory and showcasing the relevance of our divergences in domain adaptation are depicted in Section 4 Discussions on the perspoctives and limitations are in Section 5 All the proofs of the theoretical results and some additional experiments are postponed to the appendices in the supplementary.
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2 Preliminaries
For the reader's convenience, we provide a brief summary standard notations and definitions used throughout the paper.
Notation. For d E N let P(RR) be the set of Borel probability measures R' and Pp(R*) C. P(R"), those with finite moment of order p, i.e., (IR))) 2 {u E P:: [==][/dp(x)) oo), where || is the Euclidean norm. We denote Mp(H)) J& [x))))))((r)) For two probability distributions μ and P, we denote their convolution as 2*v (( namely (μ * (A) fz Jy 1a(x + y)dd(z)dv(()), where 1a(+) is the indicator function over A. Given two independent random variables X ~~H and Y ~ V, we remind that X ~ u The d-dimensional unit-sphere is noted gd-1 & {0 e R²: |0||| = 1}. We denote by ud the uniform distribution on gd-1 and we use 5(-) to denote the Kronecker delta function. We note as E the expectation of the function f with respect to μ..
Let P: F: - R be the Gamma function expressed as [(u) 10-etree-' for 0. For h E EN,, (+), denoted the Pochhammer symbol, also known in the literature as rising factorial, namely 1, a, and [(a+k) a(a + +) 1)...)).a-- k 1), for k 2 1. We denote by 1Fi m)) the Kummers confluent. (a)x T(E)
=
hypergeometric function (Olver 2010) and defined by (a, m2) = Σx= (a)s
(1)x ATT
Sliced Wasserstein distance. We remind in this paragraph several measures of similarity between two distributions. The Wasserstein distance of order p E [1, oo) between two measures in Pp( ) given by the relaxation of the optimal transport problem, and it is defined as
wp(,v) (
1/p
inf (Ell(uu))
J || x' 1l" (x, Jdrde'
where 11(,,)) A (T E P(Rd Rd) |T = μ4 72#7 = v} and Ii π2 are the marginal projectors of y on each of its coordinates. When d 1, the Wasserstein distance can be calculated in closed-form owing to the cumulative distributions of μ and RRacevv & Rüschendorf 1998). In practice for empirical distributions, the closed-form solution requires only the sorting of the samples, which makes it very efficient. Because of this efficiency, efforts have been devoted to derive a metric for high-dimensional distributions based on 1D Wasserstein distance. The main idea to project high-dimensional probability distributions onto a random one-dimensional space and then to compute the Wasserstein distance. This operation can be theoretically formalized through the use of the Radon transform, leading the so-called sliced Wasserstein distance Kolouri et al. 2016 Bonneel & Coeurjolly 2019 Kolouri et al. 2019b Nguyen et al. 2021
Definition 2.1. For any p E (1)) and two measures μ1 E (,(())), the sliced Wasserstein distance (SW) reads
swp(u,)) & (1 Wi(RRal, Ruv) Ruu)ua(u)dd)
1/P
where Ru is the Radon transform of probability distribution, namely Rup(-) = San p(s)o(- u)ds. In practice, the integral is approximated through a Monto-Carlo simulation leading to a sum of 1D Wasscrstoin distances over a lixed number of random directions u.
Gaussian-smoothed sliced Wassorstein distance. Based on this definition of SW, replacing the Radon projected measures with their Gaussian-smoothed counterpart leads to the following definition:
Definition 2.2. The o-Gaussian-smoode p-Siiced Wasserstein distance between probability distributions p and v in (Rd) writes
(1, G,SW,H,, & (1 Wo(Rall * No, Ray * No No)
1/p
G,SW
ga-1
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where No =2) is the zero-mean o'-variance Gaussian mcasure. It is important to note here that the smoothing (convolution) operation occurs after projection onto the one-dimensional space. Hence, assuming X X~ J, Y ~ V, for a given direction u, we compute in the integral the one-dimensional Wasserstein distance between the probability laws u + Z and u y Z' where Z, Z' ~ No are independent random variables. The metric properties of G,SW, for p 2 1 have been discussed in a recent. work (Rakolomamonjy Ralaivola, 2021). This latter work has also shown, the context of differential privacy, the importance of convolving the Radon projected distribution with Gaussian instead of computing the SW distance of the original distribution smoothed with a d-dimensional Gaussian μ Nol.. where La denotes the d x d identity matrix.
Gaussian-smoothed sliced divergence. The idea of slicing high-dimensional distributions before feeding them to a divergence between probability distributions can be extended to distances other than the Wasserstein distance. These sliced divergences have been studied by Nadjahi et al. (2020) Similarly, can define Gaussian-smoothel sliced divergence, given a divergence Dg (TR)) x (R() IR+ for d 2 1 as:
Definition 2.3. The (Gaussian-smoothlee p-Sliced Divergence between probability distributions and in P (IR") associated to the base divergence D DR, p p 1 is
GoSDp(H, (/Sz/1 D'(RuHt No, Ruv * No (oae)aa(u (ud(uuuu)
1/D
D"(uut
Typical relevant divergences are the maximum mean discrepancy (MMD) Gretton et al. 2012) or the Sinkhorn divergence (Genevay et al. 2018 Peyré & Cuturi 2019). In Section
we report empirical findings
based on these divergences as well on the Wasserstein distance.
3
Theoretical properties
In this scction, analyze the propertics of the Gaussian-smoohhel sliced divergence, in terms of topological and statistical properties and the influence of the Gaussian smoothing parameter on the distance.
3.1 Topology
It has already been shown in Rakotomamoniy & Ralaivola (2021) that the Gaussian-smoothed sliced Wasscrstein is a metric on P(R"). In the next, we extend these results to any divergence D(.,-) under certain assumptions.
Theorem 3.1. For any o > 0,P 2 1, the following properties hold:
1. if D(;))) is non-negative (or symmetric), then G,SDp(:)) is non-negative (or symmetric)) if D(:-) satisfies the identity of indiscernibles, i.e. for p¹, P(R). v') if and only if n¹ v., then this identily also holds for G,SDp(:.)) for any u1 U (R()))
if D(.-) salisfies the triangle inequalily then G, satisfies the triangle inequality.
The above theorem shows that under mild hypotheses over the base divergence D, as being a metric for instance, the metric property of its Gaussian-smoothed sliced version naturally derives. As exposed in the appendix, the more involved property to prove is the identity of indiscernibles.
We further postponed to the appendix the proofs of the two other topological properties: (i) GD metrizes the weak topology on P,, (TR.) and (ii) G,SD is lower semi-continuous with respect to the weak topology in Pp (IR.).
Now, we establish under which conditions on the divergence D, the convergence of sequence in SD implies weak convergence in Pp (Rd). We say that {μk heeN converges weakly to and write, 14 = n., if f f())duu(z)) + f(x) as ko + co, for every f in the space of all bounded continuous real functions. Theorem 3.2. Let σ > 0,p 2 1, e (R,(), and {ux E (R((R a sequence of distributions. Assume that the divergence D is bounded and metrizes the weak topology on P(R). Then, lim (P /) if and only is pa = p.
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Note that Theorem 3.2
extends the results of Nadjahi et al. (0200 to Caustian-mmotheed distributions, as retrieve them special case for = O. In addition, based on Theorem 3.2 by Lin et al. (0021) and the above, we can also claim that the Gaussian-smoothed SWD metrizes the weak convergence.
Proposition 3.3. Let 0 > O,p 2 1 and assume that the base divergence D is lower semi-continuous w.r.t. the weak topology in P(R). Then, GoSD, is lower semi-continuous with respect to the weak topology in Pp (Rd).
When the base divergence D is equal to the Wasserstein distance Wp, that is lower semi-continuous (Villani 2009), then Proposition 3.3 shows that the smoothed sliced Wasserstein distance is semi-lower continuous too.
3.2 Statistical properties
The next theoretical question we are interested in is about the incurred error when the true distribution y is approximated by its empirical distribution A.. Such case is common in practical applications where only (high- dimensional) empirical samples are al disposal. Specifically, we are interested in quantifying two key properties of empirical Caussian-smoothed divergence: (i) the convergence the double empirical G,SD, (An, P%)
(see Definition 3.6 to G)SDp(H,)) (ii the convergence of GoSDp(H, (see approximating the expectation over the random projection with sample mcan.
(1) ((,DD( when
Let A. Σin 5x. and in + Σim 9y, be the empirical probability measures of independent obscrvations.
The smoothed Gaussian sliced divergence between An, and ùn is given by
(1s D2 (Rufin No, Rub No (6))uu()dd))
I/D
GoSDD(PPns 0n) 1:::(()) ga-1
Remark 3.4. Remark that for a fixed u E $ the distributions Rufr * No and Rud, No are continuous, in particular they are a mixture of Gaussian distributions centered on the projected samples with variance a².
Lemma 3.5. Conditionally on the samples (X:)i=1..... and (Yijim one has: Rufin No Y2,0)).
1 NE_,N((" (u"X,, X2, and RuD, No - N(u
Note that we further need to sample with respect to the continuous mixture Gaussian measures Lemma 3.5 in order to get a fully empirical measure version of G,SD(p, To this end, we next define the double empirical divergence of G,SD.
3.2.1 Double empirical divergence of G,SD
Let .i...., T and TV, ,....., Tw be i.i.d. obscrvations of Rufn NN and Rudn+NG respectively. Sampling i...d. J is given by the following scheme: for i 1,..., n, we first choose the component Mu (V from the mixture NEnNN" x2,0²) then we generate T.E u X; 2i, where Z ~ No. Hence, we set, for given
A
Au TTx X and in 1
87
n
6 TY+
71
The measure Å. E P(R) defines an empirical version of the continuous Rud * No denoted as Rud, * No (similarly é, Ruō * No). Using the aforementioned notation, we define.
Definition 3.6. The double empirical smoothed Gaussian sliced divergence reads as
G,, Dn) A (1 (R 8n @n)ua(u)du)
1/p
Remark 3.7. (i) It is worth to comment the double randomnessos showing in the definition of p,)): the first comes from sampling according to the original probability measure (μ or v) whereas the second lakes place from sampling according to the mixture "E M(u X x2.
(ii) The empirical measure of the convolution Ruji * No could be written Er doz+0z allowing to
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sample in one shot ii.d. samples U + Of such that U ~ Rull and Of ~ No. From an empirical view, sampling according to Ruju * NG is intractable. For that reason, our theoretical results and numerical experiments are based on ing in and hence with respect to (Rn:,,
3.2.2 Sample complexity of G,SW, p
Herein, our goal is to quantify the error made when approximating G,SWp(r u) with GoSWp Dn)- More preciscly, we are interested in establishing an order of the convergence rate of G,SD, Pa) towards G,SD,(M, v), according lo the sample size n. This rale slands for the so-called sample complerity.
The convergence results in the sequel are given in expectation. Recall that the empirical distributions are derived from a double sampling process, which leads to consider a double expectations, wrt the origin distribution Ene, and wrt the sampling from the Gaussian smoothing E where μ and N/ are the n-fold product extensions of f and No. respectively. We first consider the conditional expectation given the samples X1.-.., Xn, Le. E N (X Xn], and then apply Eys.. We denote by
n IN [ EE N [| ................................ Xnl]
Next, we focus on the sample complexity for the special case of Gaussian-smoothed sliced Wasserstein distance.
Proposition 3.8. Pix a > 0,P 2 and > 22. For X ~ μ, assume that S0 e e ||IP| > E) < co. Then,
(log logn))/p
non W Won
(GSSW ((ns 4]] S Ep,oo //ppp
+ T..
n/pp
where E =..,0
-
2
(T(p 2)(
4To #2-2
20 + 45500
-0
PIIXI
<ldE)) and Y Pp
11/2p
2 $ 1/ 0² (T(p
Ex
(pp) (-1*
1/v with Cp is positive constant depending only on p. M2)(1))
(2)x (204)
It is worth to note that for p E N e.g. p 2 (standard choice for numerical experiments), the (psendo) confluent hypergeometric function Ei_ (-ple (-1)*
(3 (2o2]aR M2s(11) only depending on the 2k-th moments of p for k= 1, P. since (-p) 0 for k 2 p+1. Now, let sketch the proof of Proposition 3.8 we first insert the proxy term of mixture Gaussian distribution 1 "E N(TTx, .²), then by an application of the triangle inequality on the Wasserstein distance are faced to control two terms (i) WP W 1 Ei= ((,Tx,,o")) and (ii) W E (NTT x², 11). For (i) we get a standard order of O ((ok))), which comes from a by- by-product of Fournier & Guillin (2015). For (ii), through a coupling via the maximal coupling using the total
x,
n
variation distance (Thcorem 6.15 in Villani 2009)), obtain the order O(n O(n-1/2). The control technique for
(ii) was inspired from Goldfeld el al. (2020)) and Nietert el al. (2021).
<] < needs P |||X|| E] goes to 0 faster than e =n5² for K < 2/02202. This can be satisfied when ||||| is a w-sub-gausssian (w 2 0). Namely, Ele" (x (X-E(X)) S w|اا for all n Rd. If the parameter w verifies w < 00/2, then the latter condition holds. Remark 3.10. Note that the sample complexity depends on the amount of smoothing through the moment of the Gaussian noise: : the larger the amount of smoothing (and thus the privacy), the worse is the constant of the complexity. Hence, trade-off on privacy and statistical estimation appears here as a reasonable guarantec on the differential privacy usually requires a large Gaussian variance.
22 Remark 3.9. The condition fo e. P |||||
Proposition 3.11. Under the same conditions of Proposilion 3.8 we have
(log (lgg)///> (T + T Ip... n/p
E E fe E [G,SW, |G Pw)]] < 31 SGG + p,,. 1/2p +3 +3-
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107
10
divergence
divergence
10
2012
SWDD GS SWD
SMB. ditt di
MMD
10
GS SKD
5KD, 2000
100
nb of samples
nb of samples
Figure 1: Mcasuring the divergence between two sets of samples in R of increasing size, randomly drawn from I). We compare three sliced divergences and their Gaussian-mmoothed sliced versions with a a = 3: (top) dimension has been set d = 50; (bottom) sample complexity with different dimensions. This plot confirms that the complexity is dimension-indepeer ddimension-independeet..
and
(log ny) n/p
G)SWp(H,P) < 33'- E o W E u: No (G.SW Pn)] + p.,.o n//2pp + 3 paa + T T..,,u
relies on double application of triangle inequality satisfied by Wasserstein distance as follows: Wp(An D S Wp(An, Rusi N&) + Wp * No. Ruv NG) + Wp(R * No bn), combined
Proof of Proposition
with Proposition
3.8 This gives non sharp convergence result since get the constant 3 in front of G,SW (Rn, or G,SW,(u,u However, when the power p we obtain sharp
E p E ee B
convergence result with O(n-1/2) namely
E JA E pa (Wn, Pn) G.SW v) 35.00 Vn (Y1,0 T 1, log
Despite that our theoretical results hold only for Gaussian-smoothed sliced Wasserstein distance, our empirical results show that given other base divergences D, shows that the sample complexity G,SDP is proportional to the one dimensional sample complexity of D" (p = 2). Figure provides an empirical illustration of this statement.
3.2.3 Projection complexity
To compute the Caussian-smoothee sliced divergence, one may resort Monte Carlo scheme numerically approximate the integral in G,SDp((,, Towards this, let define the following sum:
G, = ( I Dp RRy + No, Rug v xN) 1/p
(1)
where u, is random vector uniformly drawn from Sd-1 for / = L,..., L. Theorem 3.12 shows that for fixed dimension d, the root mean square error of Monte Carlo (MC) approximation is of order O ( which corresponds to the projection complexity. We denote by u uniform measure da on the unit sphere.
@r. and the L-fold product extensions of the
Proposition 3.12. Let 0,p 1. Then the error related the MC-estimation of GD is bounded follows
E oz [GSSD,''' P (11,V) GOSD A(p,o)
VL
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where (p Jge-1 (DD(Rni No, Ruy No) - -p)) "ud(u)du, with Ty Sga-1 D2 (RRRii No, Ruy No No)uu(u)duu du.
The term A (p) corresponds to the variance of ("( No. & with respect u ~ ud. It is worth to note that the precision of the Monte Carlo scheme approximation depends the number of projections L and the variance of the evaluations of the divergence DP. The estimation error decreases at the rate L L-1/2 according to the number of projections used to compute the smoothed sliced divergence.
Given the above results, we provide finer analysis of G,SW, v)'s sample complexity. Towards this ends, for a fixed random projection uy (1 1 L) we define fin. = Ei=1 xn]x++2; x (similarly for Dn,)) and set
GDSpp(An: Pn)) = ( (z(dm,;
4
1/8
1=1
The overall complexity of G,SD) v) consists in its approximation by sampling and projection of the origin probability measures M, M, iee through G.SDp(An, Pn). By application of triangle inequality, one has
SGS (in, Pn) G,SWa b SG (im; Pn) C.sw", Pn)) G. Pa) (I,
The first term the right- hidt-- (RHS) the latter decomposition can be controlled by Proposition 3.12 in the following way:
A(p.o) (Vunua ( 1/2 VL
oL [I,,Ww, (An, Pn) Gosw Bn)1]
VL
However we don't have a proper control for p 2 2 of the second term in the RHS, (P Pn) G,SWp(m, r)1, it can be seen from Proposition 3.11 For that reason, derive an overall complexity in the case of p 1.
Corollary 3.13. The sample and projection complezities of G,SW v) reads as (raplexity(G,SSW) = + L-1/2). If we consider the number of projections as L = [nd] for some 3 E (0,1) then the overall complezity
= O(n Onn-1/2).
3.3 Noise-level dependencies
The parameter a of the Gaussian smoothing function N₂ may significantly influence the attained privacy level. Hence, we provide theoretical results analyzing the effect of the noise level on the induced -aussian-smoothed sliced divergence.
3.4 Order relation
We first show that the noise level tends to reduce the difference between two distributions as measured using G,SD" (H, provided the base divergence D satisfies some mild assumptions.
Proposition 3.14. Let u,v E (R)(R)) and consider the noise levels T1, 01.02 such that 0 S or S 02 < x. Assume that the base divergence D satisfies D(d + *No,, v/ *No.) S D(p' * *No, v'''''NN,, NNi)) for any μ¹ / E P(R). Then, Grg (H,,)) < Go, SDD
Note that the assumption for the base divergence incquality holds for the Gaussian-smoothed Wasscrstein distance Nietert et al. (2021). While we conjecture that it holds also for smoothed Sinkhorn and MMD, we leave the proofs for future works. Based on the property in Proposition 3.14 we show some specific properties of the metric with respect to the noise level a.
Proposition 3.15. G,SDD (u, v) is decreasing with respect and we have lim,- G,SD" (14, v) D),
The proof of Proposition 3.15 comes straightforwardly from Proposition 3.14 by taking 02 = a and letting 01 - This property interestingly states that the G,SD" recovers the sliced divergence when the noise level vanishes. We end up this section by providing a relation between distances under two noise levels.
sliced Wasserstein
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10
Approximation Error
SWD
GS SWD
MMD
20
SKD
10
GS SKD
Projections
Figure 2: Absolute difference between the approximated Monte Carlo approximation of all divergences compared to the true one (evaluated with 10, 000 number of projections). The two sets 500 samples 1850 are randomly drawn from (O, The Gaussian-mmoothed sliced divergences are parameterized with a =3
SW (1, v) - Go,, SW (1, v) and Proposition 3.16. 0 S 01 S 02 be two noise levels. Then, one has SW
GG, SW (H, (1, v) - G. sw, (1, ( 5 (2'- - -,, (14 +) + -of),
in particular for p 1, GoSW(p,
(20 - oî).
3.4.1 Continuity
Now we analyze the continuity properties of some G,SD" w.r.t. the noise level. Proposition 3.17. For any two distributions μ and v for which the sliced Wasserstein is well-defindd, the Gaussian-smoolled sliced Wasserslein dislance continuous w.r.l. a.
Proposition 3.18. Assume that the kernel defining the mazimum mean discrepancy (MMD) divergence is bounded. Then the maussian-smoother sliced G,MMD continuous w.r.t. a.
The above propositions show that most distribution divergences are continuous with respect to a under mild conditions.
4 Numerical Experiments
In this section, we report on a series of experiments that support the established theoretical results. We also highlight the usefulness of the findings context of privacy-preserving domain adaptation problem.
4.1 Supporting the theoretical results
Sample complexity. The first experiment (soc Figure base divergences. It shows that the sample complexity stays similar the one of their original and sliced counterparts up to constant (see Proposition 3.8). For this purpose, have considered samples in Rd randomly drawn from Normal distribution (N 1). For the Sinkhorn divergence, the entropy regularization has been set to 0.1 and for MMD, we used Gaussian kernel for which the bandwidth has been set to the mean of all pairwise distances between samples. The number of projections has been fixed to L = 50 and we perform 20 runs per experiment. For the first study, the convergence rate has been evaluated by increasing the samples number up to 25,000 with fixed dimension d = 50. For the second one, we vary both the dimension and the number of samples.
! analyzes the sample complexity of different
Figure I shows the sample complexity of some sliced divergences, respectively noted as SWD, SKD and MMD for Sliced Wasserstein distance, Sinkhorn divergence and Maximum Mean discrepancy and their Gaussian-smoothed sliced versions, named as GS SWD, GS SKD and GS MMD. On the top plot, we can see that all Gaussian-smoothed sliced divergences preserve the complexity rate with just a slight to moderate

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divergence
10.1
05 SMD 0
SMO.
-0
-5
10
540. +5
SCD. =1
107
SCD.
nb samples
Figure 3: Mcasuring the divergence between two sets of samples in R:50 drawn from (O,, We plot the sample complexity for different Caussian-smoother sliced divergence at different level of noises.
overhead. The worst difference is for Sinkhorn divergence, while MMD almost comes for free in term of complexity. From the bottom plot where sample complexities for different dimensions d are given, confirm the finding that Gaussian smoothing keeps the independence of the convergence rate to the dimension of sliced divergences.
Two other experiments on the sample complexity and identity of indiscernibles are also reported in the supplementary material.
Projection complexity. We have also investigated the impact of the number of projections when estimating the distance between two sets of 500 samples drawn from the same distribution, (N, Figure 2 plots the approximation error between the true expectation of the sliced divergences (computed for a number of L 10, 000 projections) and its approximated versions. We remark that, for all methods, the error ranges within 10-fold when approximating with 50 projections and decreases with the number of projections.
Performance path the impact of the noise parameter. Since the Gaussian smoothing parameter is key in privacy preserving context, as it impacts on the level of privacy of the Gaussian mechanism, we have analyzed its impact on the smoothed sliced divergence. We have reproduced the experiment for the sample complexity but with different values of a. The number of projections has been set to 50. Figure 3 shows these sample complexities. The first very interesting point to note is that the smoothing parameter has almost no effect on the GS MMD sample complexity. For the GS SWD and GS SKD divergences, instcad, the sioothing Lends increase the divergence at fixed number of samples. Another interpretation is that to achieve given value of divergence, getting given divergence value = 5, one needs almost 10-fold more samples for 15). This overhead of samples needed when smoothing increases is properly described, for the Gaussian-smoother sliced SWD in our Proposition 3.8 as the sample complexity depends on the moments of the Gaussian.
needs more far samples when the smoothing is larger (í.e. for
As for conclusion from these analyses, we highlight that the Gaussian-smoother sliced MMD seems to present several strong benefits: its sample complexity does not depend on the dimension and seems to be the best one among the divergence we considered. More interestingly, it is not impacted by the amount of Gaussian smoothing and thus not impacted by a desired privacy level.
4.2 Domain adaptation with G,SW
As an application, we have considered the problem of unsupervised domain adaptation for a classification task. In this context, given source examples X, and their label y. and unlabeled target examples X,, our goal is to design classifier h(-)) learned from the source examples that generalizes well on the target ones. classical approach consists in learning representation mapping g(-) that leads to invariant latent representations, invariance being measured distance between empirical distributions of mapped source and target samples.
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JEPS > VWIST
Office 31 D
Office 31 -A--
SWD
VND
Figure 4: Domain adaptation performances using different divergences on distributions with respect to the Gaussian smoothing. (Left) USPS to MNIST. (Middle) Office-31 Webcam to DSLR. (Right) Office-31 Amazon to Webcam.
USPS -> MNIST
Office 31 W
Office 31 A- W
0.075
0050
accuracy
0.775
0.750
GS SWD
SWD
SWD
20
20
30
40
50
10
50
10
20
50
Figure 5: Domain adaptation performances using different divergences on distributions with respect to the Gaussian smoothing using one-epoch-fine-uuee tuned models. (Left) USPS to MNIST. (Middle) Office-31 Webcam to DSLR. (Right) Office-31 Amazon to Webcam.
Formally, this leads to the following problem
min (C.(t(((X.))) D(g((X). 9.h
where L. can be the cross-entropy loss or a quadratic loss and D divergence between empirical distributions, in our case, will be any Gaussian-smoothed sliced divergence. We solve this problem through stochastic gradient descent, similarly distance Lee et al.] (2019). Note that, in practice, using smoothed divergence preserves the privacy of the target samples as shown by (Rakotomamonjy & Ralaivola 2021).
many approaches that sliced Wasserstein distance as distribution
When performing such model adaptation, privacy/utilifg trade-off that has to be handled. In practice, would prefer the most private model while not hurting its performance. Hence, one would seek the largest noise level ø > ( to use while preserving accuracy larget domain. Hence, it is useful to evaluate how the model performs on a range of noise level (hence, privacy level). This can be computationally expensive at it requires to fully train several models on hundreds of epochs. Instead, we leverage on the continuity of our GDSD to employ a fine-tuning stralegy: we train a domain adaptation model for the largest desired value of a (over the full number epochs) and when decreased, we just fine-tune the lasted model by training on only one epoch.
Our experiments evaluate the studied Gaussian-smootthed sliced divergences in classical unsupervised domain adaptation. We have considered two datasets: handwritten digit recognition (USPS/MNIST) and Office 31 datasets.
In our first analysis, we have compared our GD performances with non-smoothed divergences. The first one is the sliced Wasserstein distance (SWD) Lee al. (2019) and the second one is the approximation based adversarial approach, known DANN Ganin & Lempitsky (2015). For all methods
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and for each dataset, used the same neural network architecture for representation mapping and for classification. Approaches differ only on how distance between distributions have been computed. Here for each noise value we have trained the model from scratch for 100 epochs. Results are depicted in Figure 4 For the two problems, we can see that performances obtained with the Ganssian-smoothed sliced Wasscrstein or MMD divergences are similar those obtained with DANN or SWD across all ranges of noise. The smoothed version of Sinkhorn less stable and induces a slight loss of performance. Owing to the metric property and the induced weak topology, the privacy preservation comes almost without loss of performance in this domain adaptation context.
In the second analysis, we have studied the privacy/utilityy trade-off when fine-tuning models, using only one epoch, for decreasing values of Results shown in Figure 5 They highlight that depending the data and the used smoothed divergence, performance varies between one percent for Office 31 to four percent for USPS to MNIST. Note that except for the largest, value of we are training model using only one epoch instead of a hundred. A very large gain in complexity is thus achieved for swiping the full range of noise level. Hence depending on the importance this slight drop in performance will have, is worth using a large value of and preserving strong privacy or go through a validation procedure of several (cheaply obtained) models.
5 Conclusion
This work provided the properties of Gaussian-smoothen sliced divergences for comparing distributions. We derived several theoretical results related to their topological and statistical properties and showed, under mild conditions their base divergences, the smoothing and slicing operations preserves the metrie property. From statistical point of view, we introduced the double empirical distribution and focused on the sample complexity of the smootthed sliced Wasserstein distance and we proved that it converges with rate O(n-1/2p). We furhter analyzed the behavior of these divergences on domain adaptation problems and confirm the fact that using those divergences yiclds only to slight loss of performances while preserving privacy. Note that in the obtained bound we use upper bound higher moments of the smoothing distribution. An important direction for future research considering non Gaussian smoothing distribution enjoying this properly.
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A
Proofs
In the following sections, we give the proofs of the theoretical guarantees given the main of the paper.
A.1 Proof of Theorem 3.1 G, proper metric on P,(Rd) (R
Before starting the proof, we add this notation: the characteristic function of a probability distribution μ P(Rd) is (x(t) Eu 'j. Given this definition, similarly to the Fourier transform, the characteristic function of the convolution of two probability distributions readsas (uru(() (u(() (t).t)
Non-negativity (or symmetry). The non-negativity (or symmetry) follows directly from the non-negativity for symmetry) of DD, see Definition 2.3
Identity property- If the base divergence D2 satisfies the identity property in one dimensional measures, then for any J. E (P and u E g one has that DR(Ru * No, Ruft * *Na) 0, hence, by Definition 2.3. GoSD, (p, ju) = 0. Let us now prove the fact that for any μ, E (P()))) (H, = 0 entails μ = a.s. On one hand, G,SD,
= gives the fact that (p(Ruj *Na Rn NN)) for ug-almost every gd-1, Following the techniques in proof
u gd-1 hence Ry No R2 No for ug-almost every of Proposition 5.1.2 in Bonnotte (2013), for any measure 17 P(RT) (with 2 1), F(n](-) stands for the Fourier transform of 77 and given as Flil(v) = fr.. e-ss dn)(s) vdn))) for any v R". Then
FlRu * Nol(v)
= E
int d(Ruk * NN)(
JR
dR
(by the definition of the convolution operator)
JR
( du(s) (by the definition of Radon Transform)
Jdd JR
[ -(((w.))))
= FIN(())F(((((u)
Since for ua-almost every u S Rull No Ruy No, and hence FiRul No] F(Ruv Nol + F[N,]Fuul F[N (by the Fourier transform of the convolution) + Flui = Flu]. Since the Fourier transform is injective, we conclude that v.
Triangle inequality. Assume that D is metric and let p,u, (RR)). We then have
ga-1 GoSDp(H,D)= (s D"(Ruu+ + No. RuV No)ua(u)du)
1/P
(1/- gu (D(Rou* Run * No) + D(R,n) No, RuV * Na)) ua(u)du) 1/P
1/8 + (Ser (D"(Rujl * No. Ruy * No (u)dd)))
((fo D' * No. Ruv R "a(u)du) 1/P
-GSDD,(u,)) + GoSD,
where inequality in (x) follows from the application of Minkowski inequality.
A.2 Proof of Theorem 3.2 SDSD, metrizes the weak topology
The proof is done by double implications and the technical material relies on the continuous mapping theorem (Athreya Lahiri (0066) and bounded convergence theorem for the first direct implication " The second one, "=", is based the fact that weak convergence is equivalent to the convergence corresponding to Lévy-Prokhorov distance (Huber 2011
"=" Assume that μ4 > p. Fix gd-1 the mapping u + R2 is continuous from Rd R, then an application of continuous mapping theorem (Athreya & Lahiri 2006) entails that Rul4k Ruft. By Lévy's continuity
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theorem (Athreya & Lahiri 2006 Rulkk No Ruff No. Therefore, limg-+0o D(Rullk: Rull No) 0. Since we suppose that the divergence D is bounded, then there exists K 2 0 such that for any k D' (Rulin Ruft No) I K. An application of bounded convergence theorem yields
lim Go SDDD = -
1/p = 0. lim DRRRup No, Ruf * No No)ua(u)du) 1-500
"<"" (By contrapositive). Suppose that doesn't converge weakly to and assume that lim BoSD"((u..4 = 0. On one hand, since R is a complete separable space then the weak con- vergence equivalent to the convergence corresponding to Lévy-Prokhorov distance defined as: The Lévy-Prokhorov distance (nn, {) between 7,55 E ()E, (space of probability mcasures on measurable metric space) is given by:
A(g. () = inf (n(A) <((A)+ ((A) < ((A)) + +e, for all A E T), where A far B: p(z A) < <e). =20
Hence there exists & > 0 and a subsequence {Ma(K)KKEN such that ((L(K).))) > One the other hand, we have lima+x0 Go = 0, that is equivalent to (D(RuHs(1) * No, Ruy * No)h& con- oo). Since the LP-convergence tails the woint-wise convergence (Khoshnevisan 2007p, there exists subsequence (Ha(ee)) la such that lim D(Ru)s(((()) NN. Rul * No) 0 almost everywhere for all s Recall that the divergence D
verges to 0 in LP(( (f gd-1 + R fgg- f(u)ua(u)du
metrizes the weak convergence in P(R) then RuHa(((R) * *N, + R almost everywhere for all u Sd-1 Therefore, Rull(((())) = Rull almost everywhere for all u E Sd-1, Using Cramér-Wold device (Huber 201 we get Me(c(()) = μ.. Since the Lévy-Prokhorov distance metrizes the weak convergence, it entails that lim A(Ma((())) (R)) = 0, that contradicts the fact that M(Ua())) M) s. We then conclude by contrapositive k000
that H4 > p.
A.3 Proof of Proposition 3.3 SD p is lower semi-continuous
Recall that the base divergence D is lower semi-continuous w.r.t. the weak topology in P(RR,, namely for every sequence of measures {ukKEN and {1/ in P(R) such that 1e = μí and 1/2 = v', one has D(H', 1/) < lim inf D(JL), 1/).
k00
Now, let {au and {urheeN are two sequences of measure in PR(R)) such that Pa = r and + V. By continuous mapping theorem (Bowers & Kalton 2014) and Levy's continuity theorem, we obtain Rull N, = Ruju* NN and Ruy *N,, = Rud * No a for all u E gd-1, Since the base divergence D is a lower semi-continuous with respect to weak topology in P(R), then
DP No. Ruv * No) & (lim inf D(Ru)4 * No, Ruys * NNa) & liminf D * No. Ruy Na). R-000
R-000
It gives
GoSD,(H,V) (L
lim inf D2 (Rufti No. Ruvk * No No No)ud(u)du) 1/2 R-000
Furthermore, by application of Fatou's lemma Bowers & Kalton 2014). we get
(Lf D" No. Rudk * No No)ua(u)du)
1/P
G,SD,(u, lim inf k-+00
lim inf k0000
which is the desired result.
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A.4 Proofs of statistical properties
A.4.1 Proof of Lemma 3.5 Rufn No an average of Gaussian mixture
Straighforwardly, for every Borelian I E B(R), we have
Rufhn * (oo(1) = 1.1 (r + s)df Σ T x, (r)
22
1,(u
X1 Lx(uxx
(f
1r(s') - x.)),ss'
1r(s')[N(u (x (since SNC (8' - -u Xx) = [N((" (x)r))
N(u X (T).
Thanks to Theorem of Cramér and Wold Cramér & Wold 1936). conclude the equality between the measures Ruβn NN - M("_ x²)
A.4.2 Proof of Proposition 3.8
Let us give first the overall structure of the proof. We we use frequently the triangle inequality for Wasserstein distances between the quantities Ån, No (u Xi, and Rull * No. We then obtain two quantities, I and (see below for explicit), bounding 2 Lv C.,SWppA,, p)]. To control I bound, we use a well known converging bound in Fournier & Guillin (2055) of Wasserstein distance between empirical and true measure. For II bound, we consider maximal TV-coupling in Villani (009g] and use result the 2p-moment of absolute
Gaussian random variable founded in Winkelbauer (2014).
On one hand, using triangle incquality of Wasserstein distance, we have
SW (Rn, (An, [)] = E Peen [1/gen wp(dn, Ruj * No (Nouu*NNo)aate No)ua(()du)) 1/2 "r]
E Non w. Won
Wr(dn, Ruju * NN (ua(uuu(u) 1/v
psn 1 8a-1
1/P 1o (WP(mn, Ruj N Na)laa())du)
ga-1
< (I+ (I+II) 1/P
where
I A 2P-1
E w
E
WP i N(u xx, ua(()du
ggd-1
and
II A 20-1
E
E
WP -
Σ
N(u o2, Ruf * Na)) va(u)du
3d
1=1
The proof is based on two steps to control the quantitics and II. Step 1: Control of Let us state the following lemma:
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Lemma A.1 (See proof of Theorem in Fournier & Guillin (015)). Let n E P(R) and let p 2 1. Assume that (n) < oc for some > There evisls constant Cp.9 depending only p. q such that, for all n 2 1,
El m)l I Cp.q (n
where
n 1/21 1-11/19220 9>2p: n 1/2 log 1 1=20
(n(p.))
n nn (n-l1-p)/11 1 pp
We note that in is an empirical version of the Gausian mixture E Nolu" x2, (() Then, by application of Lemma A.1 we get
E Wo [W (pus
1 EEN X,, o")]] s Cp.g Mpplo {
Σ N(u [(TTx,,,,)))
n
An An(p.))
Let us first upper bound the q-th moment of Mj 1 Niu (( for all q 22 1. For all u E Sd-1, we have
N(u
( n
N(u [((xxx.00))) = |1|aa( n
Σ _ N(u X3, x2,o2))(()
" Σ
Mg
n
where Zi ~ NNuu X,, o)). By Equation (17) in Winkelbauer (014
we have
Ma ( Σ N(u X(, x2))
29/200 09 V
r("
9-
1
Σ iF
1
-uuTx;)
2
2'2
20²
1=1
Since XJ,.... X,, are i.dd samples from ,, it yiclds
E En (Mpll ( N(u x,, ((,x),,0)))]
20/2 09
7 11
1
xx2
D(
EEp
[i
)] (X~
VT
2
2'
202
29/2 0"
r
+
4)
(-1)+
Ep [[( T x)22
VT
2
(1)
(2o2)
km0
29/2 0
q +
00
(-1)*
r
M2r (1))
π
2
(5),
(202)
k=0
Setting 22 we have (n q)
Jogn then
22
2p + 1
p)
(-1)*
log(n)
I 2 1
M2a(u)
2
(202)
k=0
Step 2: Control of II.
We follow the lines of proofs of Proposition 1 in Goldfeld et al. (020)) and Theorem in Nietert et al. 2022
Using coupling i and Ruj * N₂) via the maximal TV-coupling (see Theorem 6.15 in Villani 2009 |], the
control of the total variation the Wasserstein distance, we get for any fixed u E gd-1
WP 1 Σ N(u No,Tx,,07,, Rult No)) < 20-1 / [(Pl/m,,u()) Iu (t)ld)
1=1
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Published in Transactions on Machine Learning Research (11/2024)
where hn,u and Iu are the densities associated with My and Ru/ No, respectively. Let fa,0 the probability density function of Na,0, ie, (a, Cauchy-Schwarz incquality gives
V2z(0002 Mooay for i > 0 to be specified later. An application of
E
W P
N(u T 02) Rull * Nol1]
Rup
|t|" for(t)) Mmn(()) ga(()] en LN.0
$ 20--1 E
dt
Vlaa(1)
10p ((10P( (
(km,ult) - ( pu(t)) dt)
S 20 E
4 NO
Sa,0(1)
S 2" (B 11 (19% ( (Sa poompoo (Knult)) 1 gu(t))² Sa,(())
1/2
d1l)
Note that J& 1111pP0 (t)dt is the 2p-th moment of (No equals to (see Equation (18) in Winkelbauer (2014)
11 20 [o,o(1) 20
2p +p 1 r(
VT
2
Moreover,
1 Σ dN(u x xz)
1 Σ So u X,),
un (t)
n1
It is clear to see that (t) is a sum of i.i.d. terms with expectation (u(4), which implies
E
Buan [(hn,u(t) - gu(t)) = V pOn Σ
- u ft Xx)
1=1
Vulsa u Tx
n
(So, 11 x)
(2TO²)- -1
Epp (e (t-u x)9..
n
Now
-it-u Xx "2
(t-u" Tx)² x)² dd(x)) || 4
e (1-u 2) du(z).
Ey (e
||
Remark that when || < 1g, then (--uxy² > 1112 lu To 1 -| (since ||/||2 1). We get u--uTxaa > Iel2 and hence
JInssy te-u x)² du(x)= < 0 dd((z) This gives,
= (t-u a)²
du(z) < [[I |d|
and
| l
E JR
(hn,u(t) guu())) (t)
dt (2TT²)- (
(/
Ezoore i dt + /
2 -2
exicoi P
||
| 2
dt
n
20

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Published in Transactions on Machine Learning Research (11/2024)
Note that the integral Ja e n dt SR= e- -3-- 2 dt is finite if and only if # 0 namely d> v2 and its value is given by
-t2
-t2
2
4
((00) e:o
1)2
2 5z
For the second integral
2 Tooo
|8|
00
22
1
'ldt
222
P
|||
>
dt = 2
Ez
P
|||
2
0 P
||| > E)de
2
Then,
S n 1112p/p-11 100-1/0--((22005 (22o0)-(((2n))))
(2
2p +pp+1 1
4TT
22
[[IX| <lde)²
[((
(1
f +
VT
this gives the desired result using the fact that (a + b) a b1/p, for a, b.b20. 0.
A.4.3 Proof of Proposition 3.11
Using triangle inequality, we have
Wp(An, 0n) S Wplnn, Ruj. + No) + Wp(R No, Ruv No) + Wp(Rnus D.)
and then
Wi(h 5 s 3-- Wp](n,
( Rujt x No) W? No, Ruv= + WP No. 2n)
This implics that
E penn No. E Eoo [G,SW, (pn: 2n)
3-- HE +
(G,SW, IG.. u]] + +-- + pu (G,SW, (B,, v)).
By application of Proposition 3.8 it yields This gives that
p3.00 1/2p + ( + Y p..o,)) (log (logn)>>> nl/v
E, JN E. GG, SWS (pn, 0n)) S 3' 3'-3 3
This ends the proof of the first statement in Proposition 3.11 For the second one, we also use triangle inequality
Wg No. Ry NG) < 3p-1 WK(R No. N A.)) + wg(dn, 2n) + WB(On), RBy No).
Then we control each term as we did before.
A.5 Proof of Proposition 3.12: projection complexity
Using Holder's incquality, we have
P (H,, - GoSD, (H,D,PP,,,u) [n, S E sc GGSD,,
E e1
GISD, (1,, v) - G,SDD(u, 13) 1/2
(v W GG,D, P (1,
1/2
A(p,o)
L1/2
21

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Published in Transactions on Machine Learning Research (11/2024)
A.6 Proof of Corollary 3.13
overall complexity (p 1)
By application of triangle incquality, one has
G,SW (An: P.)) G,SW(H, p) [G,SW(An, GoSW(An 2n)) -
(GoS((A,, - G,SW(1,))
Using Proposition 3.12 we have
oz Ga&SS Pr.)) GRS Pw)) S As =
(V [W (n)))///
VL
VL
Using Proposition 3.11 for p p=1 get,
+ (Tiow + 11,,,u) log
nn W E . N
Pr)) - G,SW(H, $3 Vn
n
Therefore, by applying the expectations with respect to the projection and sampling obtain
E 22 E non IN:" E w..
[ Bn)) GHSW(H,
1
1
+ (Mi,a + Ti,a,) log
E
onn wo E pon Ive [Ao] + 331, Vn
V
By Jensen inequality, we have
E R o E Epon [Ao] { 1 No
E.e.eeE.E.nnnn Even (Vu-ay ( 1n]1 1/2
A.7 Proof of Proposition 3.14
For all u E gd-1 we have Ruf, Ruy E P(R). By application of the inequality of noise level satisfied by D in one dimension we get
D'(Rujt+ * No., Ruv No.) I D2 (Raf * No., Ruy No.).
Then, computing the expectation over the projections u since the divergence is non-negative concludes the proof.
A.8 Proof of Proposition 3.16: relation between G,SW?( under two noise levels
First, using the contractive property of convolution (sce Lemma 3 in Nictert et al. (2021) ), stating that for any probability measure P(R), W Wp(++a, v*aa) S Wy(u,v) v). Hence Wo[u*Nap v*Noo) W
Now using Proposition 3.14 of the oreder relation satisfied by G,SW" yields
Go SW,SW,,u, S Go, SW,
In the other direction, have that. No, Nos *N oo (similarly for No,). Setting the following random variables: Xu RuftYY Yu Ruy, Zx ~ No. Zy ~ No. Z'x ~ N Vog Zy N
Vo3-o? The sliced
Wasserstein distance WP(Rn, * Noz, Ruy * Nag) given as a minimization over couplings (X,, Zx, Zxx) and (Y,, Zy, 2y-), namely
Wg No., Ruy * No2)
inf 2x. 2xx
E[[((,, + Ex) (Yn (Y++Zy)) + (2x -1x)111
Yu Yy.
Using the inequality E| -2 2 for any random variables V,V, W L, integrable, we obtain,
2 2--1E 7x) (Ya Zr) (Z'x 7p > [[ 7x) (Y. -- -x)}1]]
22

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Published in Transactions on Machine Learning Research (11/2024)
Hence,
2 WP (Rufl No Ruv 2 inf [[ 2x) - (Yu + 2y) 1 21))))
* Ruv No,) 22 sup E 11'x 21)1
2 + Ruv No.) 2 sup E [Z'))1)
Hence,
So,SW SW,,,,, v) 22' Go,SW ((1, + 4 (suu {((2)))))
Finally, for any p 2 1 the p-th moment of [ool satisfies El|,o |i 2T(((+1)/22 02h S 220 then
r(1/2)
Go SW (,, < 2 Go (w, +2 -f))
and concludes the proof.
A.9 Proof of Proposition 3.17
continuity of the smoothed Gaussian sliced Wasserstein w.r.t. a
From Lemma 1 in Nietert et al. 2021). we know that the Gaussian-smoothed Wasserstein is continuous with respect to a, for any distribution Rn and Ruft. In addition, for any u, we have (p * N₁, Ru * NNa) S
WR(Rnu Ruji). Then by applying Lebesgue's dominated convergence theorem (Bowers & Kalton 2014) to the above inequality with W Wp(Ruz, Ruj) as a dominating function, that is ug-almost everywhere integrable because both measures are in Pa (R()), then conclude that the Gaussian-smoothed SWD continuous w.r.t. O.
A.10 Proof of Proposition 3.18: continuity of the smoothed sliced Souared-MMD w.r.t.
Let first recall the definition of the MMD divergence. Let k R x R R be a measurable bounded kernel on I and consider the reproducing kernel Hilbert space (RKHS) H4 associated with k and equipped with inner product < ".' >H. and norm || ||m.- Let PH, be the set of probability measures such that Sux V (1, ()dn(() < oc. The kernel mean embedding is defined as (n) = fg k(, ()dn(().. The squared- maximum mean discrepancy between 7) P(R) denoted as MMD: PH. (R) x PH. R T expressed
as the distance between two such kernel mean embeddings. It is defined as Gretton al. 2012
MMD2(1, (n,C) = (n) - +x(C)) = (r,Tr,,(((TTTT)
T')] 2e (KT R)] (R,
where T and T' are independent random variables drawn according to n, R and R' are independent random variables drawn according to S, and T is independent of R. We define the Gaussian Smoothed Sliced squared-MMD as follows:
G.MMD'(( f Jsan |10g (Na) Tx No) wa(u)du
=f (ET.T [K(T, T') TT Rw [K(T, R))
ERR, [k R'))) ua(udu..
From the definition of the smoothed sliced sMuared- we have
// =( No No((')
Er,, [N(T,T)) (MTT,
f1 ([ ((KU+)))))RM)N)NN..()) NNo
/1 ( u x,,TTt No)H
-11 PARR [T hit uu+u' -u,, " (oN
23

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Published in Transactions on Machine Learning Research (11/2024)
Similarly,
E (R, Rt'] = -JanaeJJeee Jerese k(r+u + u y, + +u Huljaoooou/////aaAuA..
and
(k I,, /1 RXB
// Jasi k(t+u' xux,r+ +u ())aalasdo(()aNNN000NNN).
Together the assumption of boundness of the kernel function & and the continuity of integrals, the three latter terms are continuous functions w.r.t. a E (0, oo). Again by the boundness of the kernel function k, there exists a positive finite constant C. such that
Ey [k(T,T") Be (k(T R)) + ER, (K,R, R')] s 4C₁.
We conclude the continuity a G.MMD (M, v) by an application the continuity of integrals.
B Additional experiments
B.1 Sample complexity on CIFAR dataset
We have also evaluated the sample complexity for the CIFAR dataset by sampling sets of increasing size. Results reported in Figure 0 confirms the findings obtained from the toy dataset.
SWD
10"
GS SWD
MMD
divergence
GS MMD
10
nb of samples
Figure 6: Measuring the divergence between two sets of samples drawn iid from the CIFARIO dataset, We compare three sliced divergences and thoir Gaussian smoothed versions with a a 3.
B.2 Identity indiscernibles
The second experiment aims at checking whether our divergences converge towards small value when the distributions to be compared are the same. For this, we consider samples from distributions and v chosen as normal distributions with respectively mean x 1a and s1a with varying (noted as the displacement). Results are depicted Figure 7 We can see that all methods are able to attain their minimum when s Interestingly. the gap between the Gaussian smoothed and non-smoothed divergences for Wasserstein and Sinkhorn almost. indiscernible the distance between distribution increases.
24

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Published in Transactions on Machine Learning Research (11/2024)
107
10F
divergence
SWD
GS SWD
MMD
10
GS SKD
10
displacement
Figure 7: Mcasuring the divergence between two sets of samples in 5000 one with mean 21g and the other with mean sla with increasing 8. We compare three sliced divergences and their Gaussian smoothed version with a = 3.
25


Reviews:
[{"review_id": "mD4GGss4zE", "reviewer": "Reviewer_cJ1H", "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\n- The results are well-motivated. I can see this being of interest to many people in the field. \n- The results appear to be rigorous \n- The writing is fairly clear and well-organized\n- The experiments support the theory\n\n# Weaknesses\n\n- In my view, the most interesting approximation result is the total complexity bound from Corollary 3.13. The paper would be stronger if this were stated with explicit constants and proved explicitly. \n- The proofs appear to be rigorous, but are quite terse and dense. A bit more discussion might help.", "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 presents theoretical analysis and some computational experiments on Gaussian-smoothed sliced probability divergences, which include Gaussian-smoothed Wasserstein distances as a special case. The theoretical results include general properties of the divergences and some approximation complexity results.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Get more explicit about the total complexity bounds.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "EuRL4fKohD", "reviewer": "Reviewer_btsZ", "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\n1. For someone not an expert in this area, I really enjoyed reading this paper. It was very clear from the start and the authors did a great job at making it clear where the contribution of this paper lies with respect to other work.\n\n2. Many of the theorems and propositions are clearly explained.\n\n3. For a mainly theoretical contribution, I appreciate that the authors also did some experimental analysis.\n\n**Weaknesss**\n\n1. The notation can get a bit overwhelming. I assume that it is almost necessary but if the authors in any way could make it a bit more easy to understand it would improve the clarity of the work.\n\n2. The introduction says \"In this work, we fill this gap by providing a theoretical analysis of the Gaussian smoothed sliced Wasserstein distance.\" However in actuality, the authors provide a theoretical analysis of Gaussian smoothed sliced divergences, which are more general. For that reason, I would think of removing this sentence or changing it because the authors provide results that are even more general and not specifically related to just Gaussian smoothed sliced Wasserstein distances! It almost felt like the authors were selling themselves short because the theory was a lot more general and did not just apply to the Wasserstein distance.\n\n3. In section 3.2, the authors quantify \n\n1. the convergence of the double empirical to GσSD(μ, ν) \n2. (ii) the convergence of \\hat{GσSD(μ, ν)}  to GσSD(μ, ν),when approximating the expectation over the random projection with sample mean.\n\nCould you explain what the difference is between the two? It remains a little unclear which of them (if not both) is the main useful result in practice (if there is one).", "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 works studies the theoretical properties of the Gaussian smoothed sliced Wasserstein distance as well as generalized versions, which are known as Gaussian-smoothed sliced divergences. The work provides a theoretical analysis of general Gaussian-smoothed sliced divergences. The analysis consists of\n\n1) Establishing the topological properties of general Gaussian-smoothed sliced divergences\n2) Proving sample complexity results of these divergences by introducing a double empirical distribution\n3) Proving that these divergences satisfy an order relation with respect to a noise level and are also continuous with respect to this parameter.\n\nThe novelty of this work is that it studies the theoretical properties of Gaussian-smoothed sliced divergences. Previous work has focused on the Wasserstein distance, sliced Wasserstein distance, and Gaussian smoothed Wasserstein distance, but NOT Gaussian-smoothed sliced Wasserstein distance. This work aims to fill in this gap.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1) If the authors could try to clean up notation, that would be appreciated\n2) Reword the introduction to make it clear that the results are general and not for just Gaussian smoothed sliced Wasserstein distances. It just happens to be that since the work applies to many Gaussian smoothed sliced divergences they are able to prove results for the popular Wasserstein distance.\n3)", "broader_impact_concerns": "No concerns", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Vp59LMFuIS", "reviewer": "Reviewer_vY8A", "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\n1. This paper is well-written and easy to follow.\n2. They introduce the double empirical distribution and establish the $\\mathcal{O} (n^{-1/2})$ sample complexity results of the smoothed sliced Wasserstein distance. \n\nQuestions:\n1. In [1], they develop the limit distribution theory for smooth $p$ Wasserstein distance. Intuitively, would similar results hold for smoothed sliced $p$ Wasserstein distance?  \n2. In [2], they show the convergence of transport plans. Can you derive similar results?\n\n[1] Goldfeld, Ziv, et al. \"Limit distribution theory for smooth p-Wasserstein distances.\" The Annals of Applied Probability 34.2 (2024): 2447-2487.\n\n[2] Nietert, Sloan, Ziv Goldfeld, and Kengo Kato. \"Smooth $ p $-Wasserstein distance: structure, empirical approximation, and statistical applications.\" International Conference on Machine Learning. PMLR, 2021.", "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": "In this paper, the author provides the theoretical properties of the Gaussian smoothed sliced Wasserstein distance, including its topology and sample complexity. They also extend some of these theoretical results to other divergences. A series of experiments have been done to support these results.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I suggest rewriting Proposition 3.16 as a stability result of $G_{\\sigma}SW^p$ as that in [2].", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: Thank you for your submission!  The reviewers all felt that this would make a solid addition to TMLR.
Decision: Accept (poster)