Distributional Convergence of the Sliced Wasserstein Process
Authors: Jiaqi Xi
Jonathan Niles-Weed
Venue: NeurIPS 2022 Accept (2022)

Abstract: Motivated by the statistical and computational challenges of computing Wasserstein distances in high-dimensional contexts, machine learning researchers have defined modified Wasserstein distances based on computing distances between one-dimensional projections of the measures. Different choices of how to aggregate these projected distances (averaging, random sampling, maximizing) give rise to different distances, requiring different statistical analyses. We define the \emph{Sliced Wasserstein Process}, a stochastic process defined by the empirical Wasserstein distance between projections of empirical probability measures to all one-dimensional subspaces, and prove a uniform distributional limit theorem for this process. As a result, we obtain a unified framework in which to prove sample complexity and distributional limit results for all Wasserstein distances based on one-dimensional projections. We illustrate these results on a number of examples where no distributional limits were previously known.

a series of Distributional Convergence of the Sliced Wasserstein Process
Jiaqi Xi and Jonathan Niles-Weed
Courrnt Institute of Mathematical Sciences, New York University, NY 10012 Center for Data Science, New York University, NY 10011
Abstract
Motivated by the statistical and computational challenges of computing Wasser- stein distances in high-dimensional contexts, machine learning researchers have defined modified Wasserstein distances based on computing distances between one-dimensional projections of the measures. Different choices how to aggregate these projected distances (averaging, random sampling, maximizing) give rise different distances, requiring different statistical analyses. We define the Sliced Wasserstein Process, stochastic process defined by the empirical Wasserstein dis- tance between projections of empirical probability measures to all oie-dimensional subspaces, and prove uniform distributional limit theorem for this process. As result, we obtain unified framework which to prove sample complexity and distributional limit results for all Wasserstein distances based on one-dimensional projections. We illustrate these results on a number of examples where no distribu- tional limits were previously known.
Introduction
The Wasserstein distances have become useful tools machine learning and data science, with applications in transfer learning [6. 33], generative modeling [4, 19], statistics [8, 20), and various scientific domains [36, 44]. Despite the popularity of these distances, they suffer from serious drawbacks in high dimensions. From statistical standpoint, estimating the Wasserstein distances from data suffers from the curse of dimensionality, with convergence rates degrading sharply as the dimension increases [16, 29, 37, 43]. From computational standpoint, despite recent algorithmic advances [1, 7], the best algorithms for approximately computing general Wasserstein distances between distributions supported
points in Rd for d > 2 have running times scaling quadratically in n, which is prohibitive on very large data sets. These deficiencies have motivated the development of modifications of the Wasserstein distances which reduce the high-dimensional one-dimensional problems.
Given two compactly supported probability distributions P and Q in Rd, we write Pa and Qu for the projections of P and Q onto the one-dimensional subspace spanned by u, for any u E Sd-1 Explicitly, if X ~ P. we let P. denote the law of u X. The measures P., and Qu are probability distributions on IR obtained by collapsing P and Q to the one-dimensional "slice" in the direction of Crucially, matter how large d is, the Wasserstein distance WP(PP, Qu) between the one dimensional measures is always easy to work with: it can be estimated from data at the rate that is independent of the dimension, and if P,, and Qu supported points, then WT (P,, Q.) can be computed in nearly linear time by a simple sorting procedure.
This observation has given rise to a number of different proposals for defining distance between P and Q by aggregating the one-dimensional distances, the most prominent of which is the sliced
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Wasserstein distance [3. 32]:
SWP(P.Q) = Wp (a)(o.)
(I)
where a denotes the uniform measure on 3d-1, Other options include:
Discrete Sliced Wasserstein distance: SW; p Q) = + Wp. u. Q..). {u.} C - [3].
Max-Sliced Wasserstein distance: (P, :- g8 WP, Qu) [15, 29, 31].
Distributional Sliced Wasserstein distance: DSWP = sup-EPo J WW. Q.)
where Pe is subset of probability measures on s [28].
Though the details of these techniques differ, they can be put common footing: if view the function W : u +> (P Qu) as a bounded function on Sd-1 then each of these sliced distances takes the form F(W) for some function F: (22 R.
Since these distances are all based on one-dimensional projections, it is natural to conjecture that they enjoy improved statistical performance. This conjecture has been verified in certain special cases [25, 27. 29] However, the analysis of these distances has largely been conducted separately, with different arguments tailored to each distance. This raises the following fundamental question: is there unified approach to the analysis of these distances, which provides statistical guarantees for all of them simultaneously?
In this work, we develop such a unified approach. In addition to generalizing prior works, our techniques allow us to prove new distributional convergence results for the sliced Wasserstein distance and its many variants. These results make it possible to construct asymptotically valid confidence intervals for variants of the sliced Wasserstein distances and to guarantee the validity of the bootstrap. Prior to our work, such results were only known for the standard sliced Wasserstein distance (1) [25] or for the sliced and max-sliced Wasserstein distances between discrete distributions [30].'
Obtaining distributional limits for empirical Wasserstein distances is an active area of research. In the one-dimensional case, fundamental contributions were made by [10, 11, 12], and further progress has been made in the case where one or both of the measures are discrete [14, 38, 40]. Multi-dimensional limits were recently obtained by [9, 13], but these are not centered at the population-level quantities, making them of limited utility for inference. However, when the distributions are very smooth, there exist estimators with distributional limits with good centering [26]. In this work, we draw on techniques recently proposed in [22] obtain central limit theorems by exploiting duality.
We consider compactly supported probability measures P and Q in Rd with connected supports, and the Wasserstein distances WT for p 1. To analyze the empirical behavior of the sliced Wasserstein distance and its variants, we define a stochastic process
G,) (WT((Pnn Quu) WP(Pu, Qw))
U E Sd-1
where P., and Qn consist of i.i.d. samples. We may view Gn as random element of go0 (Sd-1), which records the deviation of the Wasserstein distance from its population counterpart along every direction simultaneously. We call Gn the Sliced Wasserstein Process.
Our main result shows that
G ~ GE E (
(3)
where G is a tight Gaussian process on gd-1, That is, the collection of random variables Vn(WT (Prus Qua) WP Qw)) indexed by elements of gd-1 enjoys uniform central limit theorem. As is well known in the statistics literature [41], uniform central limit theorems of this type give rise to distributional limits for any sufficiently regular functional on (s via the functional delta method-in particular, we directly obtain distributional limit theorems for the sliced Wasserstein distance and its many variants as special case. Our results likewise give techniques for proving the consistency of the bootstrap for any of the mentioned functionals as a consequence of general results for uniform central limit theorems.
'Conurrently and independently of our work. [21] proved distributional limits for the sliced and max-sliced Wasserstein, but not for other variants, a byproduct of general results for distributional limits for Wasserstein distances.

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2 Main Result
Throughout, P and Q denote two probability distributions in IR with compact supports, contained in a closed ball B(,, R) around the origin. We fix a p > 1, and consider the Wasserstein distance of order p:
/ || - vll dr y)
QP)
inf =EI(P,.)
(4)
where the infimum is taken over all couplings between P and Q. It is well known (see [42]) that this problem possesses dual formulation:
sdp+ / g dQ, pdQ
WF
sup
(5)
(:B(O,
where fe denotes the c-transform: (1) inl =E ||x
can be shown (e.g..124, Lemma & 5)) that the supremum in this dual formulation achieved, and that without loss of generality we may assume that f satisfies f(0) 0 and Il I|tip S p1tp-1 We denote the class of such functions by C, and call any maximizer Kantorovich potential. In order to obtain Gaussian limits, adopt the following assumption:
(CC) For all E sd-1, the support of Pa, or Qu is interval.
For > 1, assumption (CC) guarantees that the supremum in (5) is achieved at a unique Kantorovich potential in C [35, Proposition 7.18]. In the absence of this uniqueness, Gaussian limits fail to hold for the optimal transport problem, even for discrete measures [38].
We now state our main result.
Theorem 2.1. Suppose that P and Q are two probability distributions in Rd whose supports are contained in the closed d-ball B(0, R) for some R > O. Assume that P and Q satisfy (CC). Let P,. and Qm denote empirical measures consisting of n and i..dd samples from and Q, respectively. n/ m) + X E (0, 1) as n. m - oc, then
(WP(Pa, Qm.) WF(P.Q.)) G in (GA-1)))
(6)
where G is a light zero-mean Gaussian process on Sd-1 with covariance function
(G(())(V)) =(1 - 2) f fuluTa)fo x) ()) +2 / (i(TTu))soo' ()
(1 21 / (a(uTa))P(a)) / fo(0
- > sa(T aQ(a) / ( (3
where fur fo E are the unique Kantorovich potentials for (Pur Qu) and (Pu, Qu), respectively.
Theorem 2.1 formally includes the one-sample case as well, by taking X = 0,
Remark 2.2. The assumption of compact support guarantees that the set of Kantorovich potentials corresponding to P. and for any E 5d-1 is uniformly Lipschitz, and is therefore a subset of Donsker class. If the supports of P and Q were unbounded, in order to deduce the Donsker property, we would need additional assumptions on the I-dimensional projections of P and Q as well as the cost function, (see e.g. [22, Theorem 5) that do not hold for p- p-Wasserstein distances (p > 1) in general.
Remark 2.3. In the proof, assumption (CC) only used to guarantee that for each E $d-1 there exists unique Kantorovich potential achieving the supremum in the dual formulation of (Pu: ). It is therefore possible to replace (CC) by any weaker assumption know to guarantee uniqueness 139, 45], but we adopt (CC) because it is the simplest such assumption we are aware of In particular, Theorem 2.1 holds for p = under the additional assumption that, for each u E 8d-1 the Kantorovich potential for W (1(P.s Qu) is unique.
3

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As alluded to above, Theorem 2.1 gives rise to a wealth of statistical theorems as easy corollaries. To describe these implications, we return the abstract setting described above: denote by W gd-1 - IR the function Wuu) = (P. Qu), and consider any functional F 8 (Su-1) IR. Then the distances we consider take the form F(W). By different choices of F, we obtain the sliced Wasserstein distance, the max-sliced Wasserstein distance, and the other variants described above. Theorem 2.1 will allow us to compare F(W) its empirical counterpart P( nm), where Wam(u) (u) = WP((nns Qma).
We recall the definition of directional Hadamard differentiability [34]] we say that is directionally Hadamard differentiable at & if for all sequences fn > and vn v e00 (Sd-1), the limit
F hn vn) F(Φ) (
lim
n R-000
hn
exists. We verify in Section the directional Hadamard differentiability of several examples. Under this assumption, we have the following.
Corollary 2.4. Assume F is directionally Hadamard differentiable Under the assumptions of Theorem 2.1,
TUTI
(F(Wam) -F(W)) (G).
n n m
Proof. See [34].
We also obtain consistency result for the bootstrap, which we state for simplicity in the case. Corollary 2.5. Assume that F' is directionally Hadamard differentiable, and adopt the assumptions of Theorem 2.1. Let Pn. E ôx, and Qn -i=1 dy, and for k < n denote by P and Q' bootstrap empirical measures consisting of k i..dd draws from Pn and Qms respectively, and set W (PP(P. Qt). If + co and k/nn O, then
sup B[A BIN(K((F(W")) ((W,))) REBL(1)
Xuy Y YI,..,, Ynl
E(M(T((((WW) F(W))) 20,
where (L(1) is the set of functions with bounded Lipschitz norm 1.
Proof. See [17].
Finally, when the functional F (S + R has a linear Hadamard derivative, the resulting statistic will again be asymptotically Gaussian. The following uniform convergence result shows that we can consistently estimate the covariance function of G from data, which can be used to obtain asymptotic confidence intervals in this setting.
Theorem 2.6. Under the same assumptions Theorem 2.1, there exists estimator ( J u for the covariance functions (Eug)) u of the limiting process G the sense that
EpQ sup [|E.,e -u, + as 71 + w
Remark 2.7. Theorem 2.6 can be used to obtain asymptotic confidence intervals via Slutsky's theorem. For instance, if the functional F has a linear derivative Fw which the form Fly (T) ʃ (u) dr(u) for a Borel measure т, then Theorem 2.6 implies that à² = =ff En,o dr (u) dr(v)
converges in probability to var( Fly. (G)), and therefore that F(W (mm)) Tō22]22 ntm is asymptotic
(1 - 6) confidence interval for F(W).
Applications
In this section, we focus on three of the variants we discussed-the sliced, max-sliced, and distribu- tional sliced Wasserstein distances-and show how our main results obtained in the previous section be used to obtain accurate asymptotic inference for these quantities. For notational simplicity, focus on the case where m, and rescale the resulting Gaussian process by factor of /2.

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3.1 Sliced Wasserstein Distance
Asymptotic and finite- sample inference for the sliced Wasserstein distance (SW) has already been thoroughly studied by [25]. We show that we can recover some of their results from our techniques. Their main focus was on a robustification of the SW distance, the "trimmed"" SW distance, defined as
finite-sample
, s.,.s(P.Q) = ga Je (1)
( G (1)P
where a denotes the uniform probability measure on Sd-1 and F G are the (pseudo-)inverses of the CDFs of Pa, Qu respectively. When O. SW reduces the original sliced Wasserstein distance SW,. p- The trimmed SW distance SW,, p.5 discards the mass of Po and Qo above and below the 1 8 and å quantiles (hence the term "trimmed"), and therefore more robust in the face of outliers. This robustification is necessary when P and Q are no longer assumed to have compact supports. [25] derive Gaussian limits and bootstrap consistency for this functional.
To see how their results for the standard sliced Wasserstein distance (i.c., = 0) can be derived under stricter assumptions from Theorem 2.1, we denote by F (Sd-1) - R the integration functional:
F = / s do(u).
The dominated convergence theorem immediately implies that F is Hadamard differentiable, with derivative
(V)) - V(u) do(u).
We obtain the following.
Theorem 3.1. Suppose that two compactly supported probability distributions P and Q in Ted satisfy (CC). Then
Va(SW -.SS::R...)--SS((RRO s [ Jga-1 (())
The random variable S is Gaussian, and by integrating (7) it can be shown that its limiting variance agrees with the expression in [25].
3.2 Max-sliced Wasserstein distance
Unlike integration, the supremum functional is not smooth and does not possess linear Hadamard derivative. Write MSW3(P,, Q) suPwesa-1 WF Qu), and note that MSWF((P, Q) w(W), where 600 (sd-1) IR the supremum functional. It is shown in Theorem 2.1 of [5] that is Hadamard directionally differentiable with the derivative
w (g) = lim sup g(æ), <l0 AEA.(f)
where A.()) := (x: f(x) 2 sup f eJ. Moreover, if f and are continuous on Sd-1 with respect the standard Euclidean distance, then lim sup g(x) sup g(x) where An(f) (x =EAa(f)
<10
f(x) = sup f}. (See Corollary 2.3 of [5])
Applying the functional delta method to and the uniform weak convergence (6), note that the limiting process has continuous samples paths a.s., so the limiting distribution of MSW can be written
(w)(w)(G)) (G)=
sup G(v).
(10)
me (EW)(W)(W))
In the spiked transport model (STM), this expression can further simplified. The STM was introduced by [29] to formalize the situation where two distributions differ only in a low dimensional subspace of IRd. We describe the special case of one-dimensional spike here. Fix some e E Sd-1 and let X, Y L span(v) be two random variables with different laws. Let Z be another random variable independent of Y) and supported on the orthogonal complement L of L.
5

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Then we define two distributions in Rd by P == law X + Q := law(Y (Y + 2). It is shown there that (SWW(P..)
W, (law law (Y)) = QP) In addition, in this model the set goes down to 0. fact, the Hadamard derivative
Ac AW (P.,(P.CC (P). shrinks to the singleton set {w) can be reduced to the random variable G(u), i... the marginal of the limiting Gaussian process G along v. We summarize the result in the following theorem.
Theorem 3.2. Suppose that two compactly supported probability distributions P and Q in R fi the spiked transport model with spike u E ga-1, Assume furthermore P and Q also satisfy (CC). Then,
Vn (MSWP, p. (P., Qn.) MSW. (.,,1))P G(o).
(11)
Remark 3.3. Note that it is not necessary for P and Q to satisfy the spiked transport model in order to deduce the CLT of max-sliced Wasserstein. Namely, even if the set nu E sd-1 WPp(Pu Qu) = MSW PP, Q)} is not singleton, the same proof still works but the limiting distribution is the supremum of the Gaussian process G over the set {u gt-1; (P Qu) M SW (P, Q)3 which is not necessarily Gaussian. This example shows that certain finctionals give rise to non- Gaussian limits, even though the limit in (6) is Gaussian. We given an example of this behavior the supplementary material.
3.3 Distributional Sliced Wasserstein Distance
Proposed in [28]. the distributional sliced Wasserstein distance a generalization of sliced Wasser- stein distance. Formally, given two probability measures /' and Q on R' with finite p-th moments where p > 1 and subset of probability distributions Pc on g such that Eg =0,0~~~100 0' S C for all T E Pe for some constant C O, the distributional sliced p-sliced Wasserstein distance between P and Q defined by
(fe WP(PP Q0) (0) (B)
1/2
DSW, (P,Q()) == sup rEPe
We may therefore write DSWP(P,Q;CC = wc(Fe(W)), where we L (Pc) + is the supremum functional and Fc 800 (5 Pc) is defined by Fc( = Sga. F(u) d (u).
The function Fe is trivially Hadamard differentiable following the same argument for the standard sliced Wasserstein distance given above. The supremum functional on (Pc) is also Hadamard directionally differentiable by Theorem 2.1 of [5]. Since the composition of Hadamard differentiable functions are still Hadamard directionally differentiable (see e.g. [2, Proposition 2.47J). under the assumption (CC), may conclude that we F& is Hadamard directionally differentiable. i.c
Vā (DSW Q DSW, [DSSVCC..P.C0)--SV3....... $ lim sup (EA.
dr)).
94-1
(12)
4 Simulation Studies
We illustrate distributional limit results in Monte Carlo simulations. Specifically, we investigate the speed of convergence of the sliced Wasserstein distance and the max-sliced Wasserstein distance. We also investigate the convergence speed of the amplitude, which provides an example of functional not covered in prior work.
We also illustrate the accuracy of the approximation using the re-scaled bootstrap. All simulations were performed using Python. The Wasserstein distances as well as the sliced Wasserstein distances were calculated using the Python package POT [18] and the max-sliced Wasserstein distances were approximated by the Riemannian optimization method proposed [23].
4.1 Sliced Wasserstein Distance
We present an example that concerns two different distributions with connected projections along all directions. Consider simple model of transport where source and target distributions P, Q are uniform on unit sphere 8 and the unit sphere s (1,1.1) centered at (I, respectively.

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We first give explicit representation of the theoretical limit of the example given in section 3.1. Fix any point 0 S², the projections of P and Q along are uniform over (-1, and (-1- ag. 1 1+ ag) respectively where no := 01 + 0g + 03. Then the unique Kantorovich potential that achieves Wasserstein distance between Po and Qo is (x) -20 Hence, we have
vn ( Qn W2(P.. ~ G
where G is the mean-zero Gaussian process indexed by S with covariance functions
8 (u, u).
EG(u)G(v) 3
follows from Theorem 3.1 that the limiting distribution of the empirical 2- Wasserstein distance is the centered Gaussian S with variance
Var(S) () = 8g { / anar(u, v) do (v) 0.832.
We sample i.i.d. observations . Xn P and YI...., Y, Q with size = 50, 100, 500. This process repeated 500 times. We then compare the finite distributions distance with the theoretical limit given in section 3.1. We demonstrate the results using kernel density estimators Figure along with the corresponding Q-Q plots. We see that the finite-sample empirical distribution gets closer to the limiting Gaussian distribution in (9) as the sample size increases. addition, we simulate the re-scaled plug-in bootstrap approximations by sampling 1000 observations of P and Q. Fix some empirical SW Vnsw3 (Prs Qn), we generate B 500 replications of VI(S (P Qi) SW2 (P,, Qn)). The distributions of the replications with various replacement numbers L, compared with the finite-sample empirical distribution and the theoretical limit, are shown in Figure We observe that the naive bootstrap
L-Wasserstein
n) better approximates the finite sample distribution compared to fewer replacements (1 n²//2 n 3/4). This is consistent with the observation of inference on finite spaces. [30]
paraplo side 30
sonsple slze -- 200
barpil size 200
0c-
54
03
0.-
sumple sicc
Sumple
sumple slcc
Figure I: Top:
Comparison of the finite sample density (pale turquoise) and the limit distribution of the empirical sliced distance (pink). Bottom: The corresponding Q-Q plots where the red solid line indicates perfect fit.
4.2 Max-sliced Wasserstein Distance
We present an example that simulates the behavior of the max-sliced Wasserstein distance when y 2. We take P to be the uniform distribution on the unit sphere $2 and Q to be uniform on the surface of ellipsoid x²2 22/a² + 222 + z² where 8.5. We sample i.i.d. observations with size 50, 100, 500 and this process is repeated 2000 times. The estimation plotted in the top part of
7

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ieglacmment m1%)
replecemeer 12
62
03
Figure 2: Bootstrap for the empirical sliced distance. Illustration of the re-scaled plug-in bootstrap approximation (n - 1000) with replacement E {n,, n//4 n2/2j. Finite bootstrap densities (pale green) are compared the corresponding finite sample density (pale turquoise) and the limit distribution (pink).
0992
04.0
D007
Figure 3: Comparison of the finite sample density (pale turquoise) and the limit distribution of empirical max-sliced Wasserstein distance (pink).
Figure 3 indicates that the finite sample distributions approximate the limiting Gaussian distribution derived in Theorem 3.2 very well even when the sample size is small.
In terms of the re-scaled bootstrap, the accuracy of the bootstrap approximation seems to be good for all replacement numbers in this case, which again consistent with the observation of the case when the underlying distributions are supported in finite sets. [30] See Figure 4 for the simulation.
4.3 Amplitude
In this section, we give example of a new functional, the amplitude, to which our theory applies. For f E (Sd-1), we write amp(f) == sup f-inf f. When Q chosen to radially symmetric reference distribution, e.g., uniform the sphere, the quantity
QM)(W)(PP.)))
sup W2(P., Qu) uES4-1
inf W2 Qu)
wES
is the natural measure of the radial homogeneity of P-if amp (.). is small. then P differs from Q by similar amounts in each direction. The amplitude functional defined on g ( is
replacement no2244
replacement 12
Figure 4: Bootstrap for the empirical max-sliced Wasserstein distance. Illustration of the re-scaled plug-in bootstrap approximation (n. = 1000) with replacement / E [n, 3/3 n n//2). Finite bootstrap densities (pale green) are compared the corresponding finite sample density (pale turquoise) and the limit distribution (pink).

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Smplitao
620
005
22
-15
-10
Figure 5: Comparison of the finite sample density (pale turquoise) and the limit distribution of the empirical PRW (pink). Left: P U 4y 22
Right: U( + 4 = 2 U(S?).
Hadamard directionally differentiable. [5]
Consider P being uniform over the surface of the ellipsoid {x²/4 + 4y² + =1} and Q uniform on S Applying Corollary 2.4 to P. Q and amp, we obtain
vn (amp (P 2x.)) 5/4) 4, ((1,0,0))
We simulate the density of the amplitude of empirical Wasserstein distances of 1d projections. The finite sample density generated by n 600 samples and theoretical limit are given in Figure 5. Let P be uniform on (x.² / + 2² + 2 16 }} and keep Q unchanged. Then
Vn (amp(W2(()) (P Qn.)) 8/3) 4 G((,,,,
0, 1)) - G(
We generate n = 600 samples according to P and Q and the result is also shown in Figure 5.
Both finite sample densities indeed converge the theoretical Gaussian limits as the sample size increases.
Conclusion
This paper defines the Sliced Wasserstein Process, stochastic process indexed by elements of the unit sphere Sd-1 in IRd, and shows that under regularity assumptions on / and Q, this process converges tight Gaussian process 800 (Sd-1). This convergence result, which can be viewed as uniform central limit theorem for the empirical Wasserstein distance along all directions simultancously, immediately implies distributional convergence and bootstrap consistency results for the sliced Wasserstein distance and its variants, thereby unifying and streamlining existing proofs in the literature and providing distributional limits for variants of the sliced Wasserstein distance for which no such results were previously known.
An important question left open by our work whether similar result holds under weaker assump- tions on P and Q. We conjecture that the compact support assumption can be lifted, though doing so would likely require making relatively stringent tail conditions. Avoiding assumption (CC) is more subtle, as some assumption of uniqueness of potentials is required to obtain Gaussian limits.
also be applied to entropically regularized variants of the Wasserstein distance, where empirical process theory arguments have also been central proving both sample complexity and distributional limit results. We leave this extension to future work.
Finally, we anticipate that our techniques

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12

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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]
(b) Did you describe the limitations of your work? beginning of section 2.
[Yes] See the discussion at the
(c) Did you discuss any potential negative societal impacts of your work? [N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] See the statements of theorems along with discussions and remarks before or after the results,
(b) Did you include complete proofs of all theoretical results? [Yes] See appendix.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main exper- imental results (either in the supplemental material or as a URL)? [Yes] We have included the code and instructions in the supplemental material.
(b) Did you specify all the training details (c.g.. data splits, hyperparameters, how they were chosen)? [Yes] See section 4.
(c) Did you report error bars (e.g., with respect the random seed after running experi- ments multiple times)? [Yes] See section 4.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, cloud provider)? [Yes] See section 4 and the related reference.
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) your work uses existing assets, did you cite the creators? [Yes]
(b) Did you mention the license of the assets? [Yes]
(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]
See the reference.
(d) Did you discuss whether and how consent was obtained from people whose data you't 're using/curating? [N/A]
(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given participants and screenshots, if applicable? [N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent participant compensation? [N/A]
13


Reviews:
[{"review_id": "RXqFTdE_Yl", "reviewer": "Reviewer_2q9t", "summary": "The authors present a unified method for providing distributional limits for processes constructed from the empirical Wasserstein distance between one dimensional projections of empirical distributions. To do so, they rely on the theory of empirical processes and on the Hadamard directional differentiability of various functions.", "questions": "- Can you specify where the assumption of compactness of the support of the probability measure in $\\mathbb{R}^d$ is necessary?\n\n- As proved in [Nadjahi], the sliced-Sinkhorn divergences do not depend on the entropy regularization parameter, unlike the entropy regularized optimal transport (or Sinkhorn divergence). Therefore, considering sliced-Sinkhorn divergences could also be of interest. As similar empicial process methods have been used for entropy regularized OT, would it be possible to extend you result to this particular framework?\n\n- It could be made clear from the beginning that the proposed method is in any case limited to absolutely continuous probability measures, as their support must be connected.\n\n- Since your results apply for various functions F, it might have been interesting to propose a new divergence to assert the interest of your generalisation. I however acknowledge that it is not an easy task to propose a new satisfactory distance.\n\n- More could be said about the trimmed sliced Wasserstein distance, and the precise meaning of \"robustification\".\n\n- In the appendix B.1, you consider the case $p=1$, which is not included in the Wasserstein distance setting that you develop (Section 2). In particular Theorem 2.1 is stated for the case $p>1$. Are the results of appendix B.1 can be obtain directly? If so, this could be clarified.\n\n[Nadjahi] - \"Statistical and Topological Properties of Sliced Probability Divergences\", Nadjahi et al.", "limitations": "The authors have specified that their method is quite restrictive in the sense that it only applies to a rather small set of probability distributions.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This article follows a natural set of results that have been proven for both classical and entropic optimal transport. The presentation is clear and the article is easy to read. I develop the strengths and weaknesses further in the following.\n\nStrengths:\n- The paper proposes a unified method to prove distributional limits for distances based on sliced distributions and 1D Wasserstein distance. The method is based on a previous work [20], that addressed the case of empirical optimal transport.\n\nWeaknesses:\n- The novelty of this article seems to be limited as most of the major results have been proven in [19,23].\n- The assumption (CC) that the support of the projected measures is connected for all the vector in the unit sphere is very restrictive. I agree that the general case involves a complicated geometry of the distributions, however, the unified framework on which the paper is based concerns very few applied cases.\n- The method for proving the results of the paper uses classical tools of weak convergence of empirical processes and does not particularly differ from the case of classical optimal transport.\n\nOverall, the results presented here are restricted to probability measures under strict assumptions and are therefore rather limited: every one dimensional projection of probability distributions onto the unit sphere must have an interval as a support. Moreover, this paper seems to have been published in a hurry (see \"Minor comments\" below), there is notably no conclusion and no perspective of using such results for example.\n\nMinor comments:\n- The labels are visible in the margin.\n- The reference [28,29] appears twice.\n- l.94 : \"uniquness\"\n- In the proof of Theorem 2.1 and in Corollary 2.3, the random variables $X_i$ and $Y_i$ are not defined.\n- In section 3.2, it is question of MSW, $\\tilde{W}$ and WPP (which is not defined). Standardisation of terminology is necessary.\n- In Theorem 3.1, the dependence on $\\delta$ of the trimmed SW disappeared.\n- l. 190 : \"given in section 4.1\" should be removed/rewritten.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "C9TBUJ-yeW4", "reviewer": "Reviewer_84Jv", "summary": "This paper defines the Sliced Wasserstein process, a stochastic process defined by the empirical Wasserstein distance between slices of empirical probability measures. A uniform distributional limit theorem is derived. Experimental results demonstrate the convergence towards the limiting Gaussian process.", "questions": "- A question I have regarding the SW process is the following: G_n defined in Equation 2 is said to enjoy a uniform central limit theorem. My naive question is: given the fact that $\\mathbb{E}[W_p^p(P_{nu}, Q_{nu})]\\neq W_p^p(P_u,Q_u)$ (due to the permutation/transport plan of the Wasserstein distance), then G_n is defined according to the \"wrong expectation\"? (see Corollary 2 of https://proceedings.neurips.cc/paper/2020/file/eefc9e10ebdc4a2333b42b2dbb8f27b6-Paper.pdf for instance where you need to compute a sample complexity). Maybe what I am saying is not true because the authors are considering a single slice. I would like to have their opinion on this.\n\n- How hard would it be to go beyond the compact case? At least Sub-Gaussian.\n\n- I suggest the authors should use the remaining space they have (1.5 pages!) to develop sketches of proof in the main paper, or do more experiments, maybe to show that their theory is valid for the unbounded case numerically?\n\nI am willing to raise my score accordingly if the answers to my questions are satisfying and the weaknesses addressed.", "limitations": "No potential negative impact", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n- This work build upon the mathematical techniques developed in [20] and adapt them well for the SW process. \n- The results seem correct (I just skimmed through the proofs).\n\nNeutral:\n- Numerical results are convincing but I would have expected more examples, maybe ones that show the theory also hold (or not?) in the unbounded case.\n\nWeaknesses:\n- The paper is not very clear: more space could have been given for defining mathematical objects; a sketch of proof of the main theorem could have been given knowing that there is a lot of space not used at the end of the paper.\n- Contrary to what is said in the Checklist, the code is not given (it is specified it is given in a footnote shown page 6), and I would have liked to look at it.\n- The authors limit themselves to compactly supported probability measures P and Q (I am not certain if it’s possible to go beyond compact, maybe sub-Gaussian?). At least the authors could have developed a bit on why they considered this assumption and where are the difficulty for the unbounded case.\n- The authors did not verified the documents they sent: all the labels of equations and sections are visible…\n- There are a lot of typos (for instance the title of the SM does not correspond to the paper's title).\n- I don't see any real applications for these results: I know NeurIPS accepts very theoretical results, but it would be useful to know how this could improve or help us understand SW distances better given the already known results on the theory of SW.\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "9xtqoZl2AD1", "reviewer": "Reviewer_mUtM", "summary": "Optimal transport distances (a.k.a. Wasserstein distance)  have recently drawn ample attention in statistics and machine learning communities as powerful discrepancy measures for probability distributions. One of the bottlenecks of Wasserstein distance consists of its expensive computational cost. An alternative to making this problem computationally tractable leads in the fact that it can be cast on averaging calculations of $1$-D Wasserstein distances. This latter approach is referred to as Sliced Wasserstein distance (SWD). In a nutshell, SWD relies on projecting the data sample on some $u$-direction from the unit sphere, ($u \\in \\mathbb{S}^{d-1}).$\n\nThis paper addresses a theoretical unified problem of proving a Central Limit Theorem for the empirical SWD distance under a compact supports condition satisfied by the distributions in question.  The authors show that  the asymptotic distribution is a tight-centered Gaussian process on the unit sphere $\\mathbb{S}^{d-1}$.\n\n", "questions": "\n- For the discrete SWD:\n    - can one get directly the central limit from the one verified by the Wasserstein distance?\n    - does this limit process uniform with respect to $L$, the number of $u$-random direction projections. Namely, if there is $L \\leq L'$ random projections, the limit is (approximately) the same?\n- In Theorem 2.4., can one derive an order of the convergence rate to zero?", "limitations": "This is not applicable to the paper.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is easy to follow, and it presents a unified framework for proving central limit theorem for a family of SWD distance, including vanilla SWD, Max-Sliced, Trimmed SWD, etc. This result in itself is solid, especially for statistical applications involving SWD. \n\n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "3GlabsjbJwT", "reviewer": "Reviewer_enEA", "summary": "The paper provides a Donsker-typed theorem for Projected Wasserstein distance in $l^{\\infty}$ norm over the projected directions ball $S^{d-1}$. From that, it derives Limit Theorems for some well-known Sliced Wasserstein distances. Some simulation studies are carried out to support the theoretical findings.", "questions": "1. Throughout the paper, the distributions of interest $P$ and $Q$ are assumed to be compactly supported. A natural question is \"can we extend the result to a more general setting where $P$ and $Q$ have arbitrary supports?\". It seems like the compact condition of the supports is only needed to show the existence of the potential and to bound the entropy number. If we can extract those refined conditions and put them together into a more general result (maybe in Appendix), it would be more impactful to OT research. If it is not possible, it is helpful to have a paragraph to explain why.\n\n2. It would be interesting to see some simulation studies about the limiting distribution of the empirical process of Max-Sliced Wasserstein distances, even in low-dimensional cases. It would be helpful to build some intuition about this non-Gaussian limit.", "limitations": "The paper is well written and has no major limitations. ", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength**: \n\n1. The results in this paper are important for Optimal Transport theory and application. The presentation of the paper is clean and easy to read. \n\n2. All the proofs are solid and written carefully. \n", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}]

Author rebuttal:

Decision: Accept