Distributional Convergence of the Sliced Wasserstein Process
Abstract
Motivated by the statistical and computational challenges of computing Wasserstein distances in high-dimensional contexts, machine learning researchers havedeﬁned modiﬁed Wasserstein distances based on computing distances betweenone-dimensional projections of the measures.
Different choices of how to aggregatethese projected distances (averaging, random sampling, maximizing) give rise todifferent distances, requiring different statistical analyses.
We deﬁne the Sliced
Wasserstein Process , a stochastic process deﬁned by the empirical Wasserstein distance between projections of empirical probability measures to all one-dimensionalsubspaces, and prove a uniform distributional limit theorem for this process.
As aresult, we obtain a uniﬁed framework in which to prove sample complexity anddistributional limit results for all Wasserstein distances based on one-dimensionalprojections.
We illustrate these results on a number of examples where no distributional limits were previously known.
Introduction
The Wasserstein distances have become useful tools in machine learning and data science, withapplications in transfer learning [6, 32], generative modeling [4, 18], statistics [8, 19] and variousscientiﬁc domains [35, 43].
Despite the popularity of these distances, they suffer from seriousdrawbacks in high dimensions.
From a statistical standpoint, estimating the Wasserstein distancesfrom data suffers from the curse of dimensionality , with convergence rates degrading sharply as thedimension increases [15, 28, 36, 42].
From a computational standpoint, despite recent algorithmicadvances [1, 7], the best algorithms for approximately computing general Wasserstein distancesbetween distributions supported on n points in R d for d ≥ 2 have running times scaling quadraticallyin n , which is prohibitive on very large data sets.
These deﬁciencies have motivated the developmentof modiﬁcations of the Wasserstein distances which reduce the high-dimensional case to a series ofone-dimensional problems.
Given two compactly supported probability distributions P and Q in R d , we write P u and Q u forthe projections of P and Q onto the one-dimensional subspace spanned by u , for any u ∈ S
d − 1 .
Explicitly, if X ∼ P , we let P u denote the law of u (cid:62) X .
The measures P u and Q u are probabilitydistributions on R obtained by collapsing P and Q to the one-dimensional “slice” in the directionof u .
Crucially, no matter how large d is, the Wasserstein distance W pp ( P u , Q u ) between the onedimensional measures is always easy to work with: it can be estimated from data at the rate that isindependent of the dimension, and if P u and Q u are supported on n points, then W pp ( P u , Q u ) can becomputed in nearly linear time by a simple sorting procedure.
This observation has given rise to a number of different proposals for deﬁning a distance between
P and Q by aggregating the one-dimensional distances, the most prominent of which is the sliced
Wasserstein distance [3, 31]:
[Equation]
where σ denotes the uniform measure on S d − 1 .
Other options include:
• Discrete Sliced Wasserstein distance: (cid:100)
SW p p ( P, Q ) :
= 1 L (cid:80) [Equation] =1 W pp ( [Equation]
u
[Equation]
Q u [Equation] [Equation] u
[Equation] ⊆ S [Equation] • Max-Sliced Wasserstein distance: MSW pp ( P, Q ) :=
max u ∈ S d − 1 W pp ( P u , Q u ) [14, 28, 30]. • Distributional Sliced Wasserstein distance: DSW pp ( P [Equation]
Q [Equation]
= [Equation] W pp [Equation]
P u [Equation] Q u ) d[-− 1 [26].
-]
[Equation] u ) , where P C is a subset of probability measures on S d − 1 [27].42 Though the details of these techniques differ, they can be put on a common footing: if weview the function W : u (cid:55)→ W pp ( P u , Q u ) as a bounded function on S d − 1 , then each of these sliceddistances takes the form F ( W ) for some function F : (cid:96) ∞ ( S d − 1 )
→ R .
Since these distances are all based on one-dimensional projections, it is natural to conjecture thatthey enjoy improved statistical performance.
This conjecture has been veriﬁed in certain specialcases [24, 26, 28]
However, the analysis of these distances has largely been conducted separately,with different arguments tailored to each distance.
This raises the following fundamental question: isthere a uniﬁed approach to the analysis of these distances, which provides statistical guarantees forall of them simultaneously?
In this work, we develop such a uniﬁed approach.
In addition to generalizing prior works, ourtechniques allow us to prove new distributional convergence results for the sliced Wasserstein distanceand its many variants.
These results make it possible to construct asymptotically valid conﬁdenceintervals 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) [24]or for the sliced and max-sliced Wasserstein distances between discrete distributions [29].
Obtaining distributional limits for empirical Wasserstein distances is an active area of research.
In theone-dimensional case, fundamental contributions were made by [10, 11 ? ], and further progress hasbeen made in the case where one or both of the measures are discrete [13, 37, 39].
Multi-dimensionallimits were recently obtained by [9, 12], 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 [25].
In this work, we draw ontechniques recently proposed in [21] to obtain central limit theorems by exploiting duality.
We consider compactly supported probability measures P and Q in R d with connected supports, andthe Wasserstein distances W pp for p > 1 .
To analyze the empirical behavior of the sliced Wassersteindistance and its variants, we deﬁne a stochastic process
[Equation]
where P n and Q n consist of i.i.d.
samples.
We may view G n as a random element of (cid:96) ∞ ( S d − 1 ) ,which records the deviation of the Wasserstein distance from its population counterpart along everydirection simultaneously.
We call G n the Sliced Wasserstein Process .
Our main result shows that
[Equation]
where G is a tight Gaussian process on S d − 1 .
That is, the collection of random variables 72 √ n ( W pp ( P nu , Q nu ) − W
pp
( P u , Q u )) indexed by elements of S d − 1 enjoys a uniform central limittheorem.
As is well known in the statistics literature [40], uniform central limit theorems of this typegive rise to distributional limits for any sufﬁciently regular functional on (cid:96) ∞ ( S d − 1 ) via the functionaldelta method—in particular, we directly obtain distributional limit theorems for the sliced Wassersteindistance and its many variants as a 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.
Main Result
Throughout, P and Q denote two probability distributions in R d with compact supports, contained ina closed ball B (0 , R ) around the origin.
We ﬁx a p > 1 , and consider the Wasserstein distance oforder p :
[Equation]
where the inﬁmum is taken over all couplings between P and Q .
It is well known (see [41]) that thisproblem possesses a dual formulation:
[Equation]
where f c denotes the c -transform: f c ( y )
=
inf x ∈ B (0 ,R ) (cid:107) x − y (cid:107) p − f ( x ) .
It can be shown (e.g.,[23,
Lemma 1 & 5]) that the supremum in this dual formulation is achieved, and that without loss ofgenerality we may assume that f satisﬁes f (0) = 0 and (cid:107) f (cid:107) Lip ≤ pR p − 1 .
We denote the class ofsuch functions by C , and call any maximizer a Kantorovich potential .
In order to obtain Gaussianlimits, we adopt the following assumption:
( CC ) For all u ∈ S
d − 1 , the support of P u or Q u is an interval.
For p > 1 , assumption ( CC ) guarantees that the supremum in (5) is achieved at a unique Kantorovichpotential in C [34, Proposition 7.18].
In the absence of this uniqueness, Gaussian limits fail to holdfor the optimal transport problem, even for discrete measures [37].
We now state our main result.
Theorem 2.1.
Suppose that P and Q are two probability distributions in R d whose supports arecontained in the closed d -ball B (0 , R ) for some R > 0 .
Assume that P and Q satisfy ( CC ).
Let P nand Q m denote empirical measures consisting of n and m i.i.d.
samples from P and Q , respectively.
If n/ ( n + m ) → λ ∈ (0 , 1) as n, m → ∞ , then(cid:114) nm
[Equation]
) where G is a tight zero-mean Gaussian process on S d − 1 with covariance function
[Equation]
where f u , f v ∈ C are the unique Kantorovich potentials for ( P u , Q u ) and ( P v , Q v ) , respectively.
Theorem 2.1 formally includes the one-sample case as well, by taking λ = 0 , 1 . Remark 2.2.
The assumption of compact support guarantees that the set of Kantorovich potentialscorresponding to P u and Q u for any u ∈ S d − 1 is uniformly Lipschitz, and is therefore a subset of a
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 1-dimensional projections of P and Q as well as thecost function, (see e.g. [21, Theorem 5.2]) that do not hold for p-Wasserstein distances ( p > 1 ) ingeneral. Remark 2.3. In the proof, assumption (CC) is only used to guarantee that for each u ∈ S d − 1 ,there exists a unique Kantorovich potential achieving the supremum in the dual formulation of
W pp ( P u , Q u ) .
It is therefore possible to replace (CC) by any weaker assumption know to guaranteeuniqueness [38, 44], but we adopt (CC) because it is the simplest such assumption we are aware of.
In particular, Theorem 2.1 holds for p = 1 under the additional assumption that, for each u ∈ S d − 1 ,the Kantorovich potential for W 1 ( P u , Q u ) is unique.
As alluded to above, Theorem 2.1 gives rise to a wealth of statistical theorems as easy corollaries.
To describe these implications, we return to the abstract setting described above: denote by W : 116 S d − 1 → R the function W ( u )
=
W pp ( P u , Q u ) , and consider any functional F : (cid:96) ∞ ( S d − 1 )
→ R .
Then the distances we consider take the form F ( W ) .
By different choices of F , we obtain thesliced Wasserstein distance, the max-sliced Wasserstein distance, and the other variants describedabove.
Theorem 2.1 will allow us to compare F ( W ) to its empirical counterpart F ( W nm ) , where
W nm ( u ) =
W pp ( P nu , Q mu ) .
We recall the deﬁnition of directional Hadamard differentiability [33]: we say that F is directionally
Hadamard differentiable at Φ if for all sequences h n (cid:38) 0 and Ψ n → Ψ ∈ (cid:96) ∞ ( S d − 1 ) , the limit
[Equation]
exists.
We verify in Section 3 the directional Hadamard differentiability of several examples.
Underthis assumption, we have the following.
Corollary 2.4.
Assume F is directionally Hadamard differentiable.
Under the assumptions of
Theorem 2.1,
[Equation]
Proof.
See [33].
We also obtain a consistency result for the bootstrap, which we state for simplicity in the n = m case.
Corollary 2.5.
Assume that F is directionally Hadamard differentiable, and adopt the assumptionsof Theorem 2.1. {+Let P n = 1 n (cid:80) ni =
+}
δ X i and Q n = 1 n (cid:80) ni =1 δ Y i , and for k (cid:28) n denote by P ∗ and Q ∗ bootstrap empirical measures consisting of k i.i.d. draws from P n and Q m , respectively, and set
W ∗ = W pp ( P ∗ u , Q ∗ u ) .
If k → ∞ and k/n → 0 , thensup h ∈ BL
[Equation]
[Equation] where BL (1) is the set of functions with bounded Lipschitz norm 1.
Proof.
See [16].
Finally, when the functional F : (cid:96) ∞ ( S d − 1 )
→ R has a linear Hadamard derivative, the resultingstatistic will again be asymptotically Gaussian.
The following uniform convergence result shows thatwe can consistently estimate the covariance function of G from data, which can be used to obtainasymptotic conﬁdence intervals in this setting.
Theorem 2.6.
Under the same assumptions as Theorem 2.1, there exists an estimator { ˆΣ u,v } u,v ∈ S d − 1for the covariance functions { Σ u,v } u,v ∈ S d − 1 of the limiting process G in the sense that
[Equation]
Remark 2.7.
Theorem 2.6 can be used to obtain asymptotic conﬁdence intervals via Slutsky’stheorem. For instance, if the functional F has a linear derivative F (cid:48) W which is of the form F (cid:48) W (Φ) = 143 (cid:82) Φ( u ) d τ ( u ) for a Borel measure τ , then Theorem 2.6 implies that ˆ σ 2 := (cid:82) (cid:82) ˆΣ u,v d τ ( u ) d τ ( v )converges in probability to var( F (cid:48) W ( G )) , and therefore that F ( W nm ) ±
ˆ σz δ/ 2 (cid:113) n + m nm is an asymptotic(1 − δ ) conﬁdence interval for F ( W ) .
In this section, we focus on three of the variants we discussed—the sliced, max-sliced, and distributional sliced Wasserstein distances—and show how our main results obtained in the previous sectioncan be used to obtain accurate asymptotic inference for these quantities.
For notational simplicity, wefocus on the case where n = m , and rescale the resulting Gaussian process by a factor of √ 2 .
3.1 Sliced Wasserstein Distance
Asymptotic and ﬁnite-sample inference for the sliced Wasserstein distance (SW) has already beenthoroughly studied by [24].
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
[Equation]
where σ denotes the uniform probability measure on S d − 1 and F − 1 u , G − 1 u are the (pseudo-)inversesof the CDFs of P u , Q u respectively.
When δ = 0 , SW p,δ reduces to the original sliced Wassersteindistance SW p .
The trimmed SW distance SW p,δ discards the mass of P θ and Q θ above and belowthe 1 − δ and δ quantiles (hence the term “trimmed”), and is therefore more robust in the face ofoutliers.
This robustiﬁcation is necessary when P and Q are no longer assumed to have compactsupports.
[24] derive Gaussian limits and bootstrap consistency for this functional.
To see how their results for the standard sliced Wasserstein distance (i.e., δ = 0 ) can be derivedunder our stricter assumptions from Theorem 2.1, we denote by F : (cid:96) ∞ ( S d − 1 )
→ R the integrationfunctional:
[Equation]
The dominated convergence theorem immediately implies that F is Hadamard differentiable, withderivative
[Equation]
We obtain the following.
Theorem 3.1.
Suppose that two compactly supported probability distributions P and Q in R d satisfy( CC ).
Then
[Equation]
The random variable S is Gaussian, and by integrating (7) it can be shown that its limiting varianceagrees with the expression in [24].
3.2 Max-sliced Wasserstein distance
Unlike integration, the supremum functional is not smooth and does not possess a linear Hadamard derivative.
Write MSW pp ( P, Q ) =
sup u ∈ S d − 1 W pp ( P u , Q u ) , and note that MSW
pp ( P, Q ) =
ω ( W ) , where ω : (cid:96) ∞ ( S d − 1 ) → R is the supremum functional.
It is shown in Theorem 2.1 of [5] that ω is
Hadamard directionally differentiable with the derivative
[Equation]
where A (cid:15) ( f ) := { x : f ( x ) ≥ sup f − (cid:15) } .
Moreover, if f and g are continuous on S d − 1 with respectto the standard Euclidean distance, then lim (cid:15) ↓
sup A (cid:15) ( f ) g ( x ) = sup x ∈ A 0 ( f ) g ( x ) where A 0 ( f ) = { x :f ( x ) = sup f } .
(See Corollary 2.3 of [5])
Applying the functional delta method to ω and the uniform weak convergence (6), we note that thelimiting process G has continuous samples paths a.s., so the limiting distribution of MSW can bewritten as
[Equation]
In the spiked transport model (STM), this expression can be further simpliﬁed.
The STM wasintroduced by [28] to formalize the situation where two distributions differ only in a low dimensionalsubspace of R d .
We describe the special case of one-dimensional spike here.
Fix some v ∈ S d − 1and let X, Y ∈ L := span ( v ) be two random variables with different laws.
Let Z be anotherrandom variable independent of ( X, Y ) and supported on the orthogonal complement L ⊥ of L .
Then we deﬁne two distributions in R d by P := law ( X + Z ) , Q := law ( Y + Z ) .
It is shownthere that MSW pp ( P, Q ) =
W p ( law ( X ) , law ( Y ))
=
W 2 ( P, Q ) .
In addition, in this model the set
A (cid:15) ( W pp ( P · , Q · )) shrinks to the singleton set { v } as (cid:15) goes down to 0 .
In fact, the Hadamard derivativecan be reduced to the random variable G ( u ) , i.e. the marginal of the limiting Gaussian process Galong v .
We summarize the result in the following theorem.
Theorem 3.2.
Suppose that two compactly supported probability distributions P and Q in R d ﬁt thespiked transport model with spike v ∈ S
d − 1 .
Assume furthermore P and Q also satisfy ( CC ).
Then,
[Equation]
Remark 3.3.
Note that it is not necessary for P and Q to satisfy the spiked transport model in orderto deduce the CLT of max-sliced Wasserstein.
Namely, even if the set { u ∈ S d − 1 : W pp ( P u , Q u ) =
MSW ( P, Q ) } is not a singleton, the same proof still works but the limiting distribution is thesupremum of the Gaussian process G over the set { u ∈ S
d − 1 : W pp ( P u , Q u ) =
MSW ( P, Q ) }which is not necessarily Gaussian.
This example shows that certain functionals give rise to nonGaussian limits, even though the limit in (6) is Gaussian. We given an example of this behavior in thesupplementary material.
3.3 Distributional Sliced Wasserstein Distance
Proposed in [27], the distributional sliced Wasserstein distance is a generalization of sliced Wasserstein distance.
Formally, given two probability measures P and Q on R d with ﬁnite p -th momentswhere p > 1 and a subset of probability distributions P C on S d − 1 such that E θ,θ (cid:48) ∼ τ | θ (cid:62) θ (cid:48) | ≤ C forall τ ∈ P C for some constant C > 0 , the distributional sliced p -sliced Wasserstein distance between
P and Q is deﬁned by
[Equation] We may therefore write DSW pp ( P, Q ; C ) = ω C ( F C ( W )) , where ω C : (cid:96) ∞
( P C ) → R is thesupremum functional and F C : (cid:96) ∞ ( S d − 1 → P C ) is deﬁned by F C (Φ)( · ) : = (cid:82) S d − 1 Φ( u ) d · ( u ) . The function F C is trivially Hadamard differentiable following the same argument for the standardsliced Wasserstein distance given above. The supremum functional on (cid:96) ∞ ( P C ) is also Hadamarddirectionally differentiable by Theorem 2.1 of [5].
Since the composition of Hadamard differentiablefunctions are still Hadamard directionally differentiable (see e.g. [2, Proposition 2.47]), under theassumption ( CC ), may conclude that ω C ◦ F C is Hadamard directionally differentiable,
i.e. [Equation]  Simulation Studies We illustrate our distributional limit results in Monte Carlo simulations.
Speciﬁcally, we investigatethe 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 a functionalnot covered in prior work.
We also illustrate the accuracy of the approximation using the re-scaled bootstrap.
All simulationswere performed using Python.
The Wasserstein distances as well as the sliced Wasserstein distanceswere calculated using the Python package POT [17] and the max-sliced Wasserstein distances wereapproximated by the Riemannian optimization method proposed in [22].
4.1 Sliced Wasserstein Distance
We present an example that concerns two different distributions with connected projections alongall directions.
Consider a simple model of transport where source and target distributions P, Q areuniform on unit sphere S 2 and the unit sphere S 2(1 , 1 , 1) centered at (1 , 1 , 1) respectively.
We ﬁrst give an explicit representation of the theoretical limit of the example given in section 3.1.
Fixany point θ ∈ S 2 , the projections of P and Q along θ are uniform over ( − 1 , 1) and ( − 1 + a θ , 1 + a θ )respectively
where a θ :=
θ 1 + θ 2 + θ 3 .
Then the unique Kantorovich potential that achieves2 -Wasserstein distance between P θ and Q θ is φ θ 0 ( x ) = − 2 a θ x .
Hence, we have
[Equation]
where G is the mean-zero Gaussian process indexed by S 2 with covariance functions
[Equation]
It follows from Theorem 3.1 that the limiting distribution of the empirical 2 -Wasserstein distance isthe centered Gaussian S with variance
[Equation]
We sample i.i.d.
observations X 1 , . . .
, X n ∼ P and Y 1 , . . .
, Y n ∼ Q with size n = 50 , 100 , 500 .
This process is repeated 500 times.
We then compare the ﬁnite distributions of 1 -Wassersteindistance with the theoretical limit given in section 3.1.
We demonstrate the results using kerneldensity estimators in Figure 1 along with the corresponding Q-Q plots.
We see that the ﬁnite-sampleempirical distribution gets closer to the limiting Gaussian distribution in (9) as the sample size nincreases.
In addition, we simulate the re-scaled plug-in bootstrap approximations by samplingn = 1000 observations of P and Q .
Fix some empirical SW √ nSW 22 ( P n , Q n ) , we generate B = 500 replications of √ l ( SW 22 ( ˆ P ∗ n , ˆ Q ∗ n ) − SW 22 ( P n , Q n )) .
The distributions of the replicationswith various replacement numbers l , compared with the ﬁnite-sample empirical distribution andthe theoretical limit, are shown in Figure 2.
We observe that the naive bootstrap ( l = n ) betterapproximates the ﬁnite sample distribution compared to fewer replacements ( l = n 1 / 2 , n 3 / 4 ).
This isconsistent with the observation of inference on ﬁnite spaces.
4.2 Max-sliced Wasserstein Distance
We present an example that simulates the behavior of the max-sliced Wasserstein distance whenp = 2 .
We take P to be the uniform distribution on the unit sphere S 2 and Q to be uniform on thesurface of ellipsoid x 2 /a
+ y 2 + z 2 =
where a = 8 . 5 .
We sample i.i.d.
observations with sizen = 50 , 100 , 500 and this process is repeated 2000 times.
The estimation plotted in the top part of
Figure 3 indicates that the ﬁnite sample distributions approximate the limiting Gaussian distributionderived 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 forall replacement numbers in this case, which is again consistent with the observation of the case whenthe underlying distributions are supported in ﬁnite sets.
[29] See Figure 4 for the simulation. 4.3 Amplitude
In this section, we give an example of a new functional, the amplitude, to which our theory applies.
For f ∈ (cid:96) ∞ ( S d − 1 ) , we write amp( f ) :
= sup f − inf f .
When Q is chosen to be a radially symmetricreference distribution, e.g., uniform on the sphere, the quantity [Equation] is the natural measure of the radial homogeneity of P —if amp( W 2 2 ( P · , Q · )) is small, then P differsfrom Q by similar amounts in each direction. The amplitude functional deﬁned on (cid:96) ∞ ( S 2 ) is
Hadamard directionally differentiable.
Consider P being uniform over the surface of the ellipsoid { x 2 / 4 + 4 y 2 + z 2 = 1 } and Q uniformon S 2 .
Applying Corollary 2.4 to P, Q and amp, we obtain
[Equation]
We simulate the density of the amplitude of empirical Wasserstein distances of 1d projections.
Theﬁnite sample density generated by n = 600 samples and theoretical limit are given in Figure 5.
Let P be uniform on { x 2 / 4 + y 2 / 4 + z 2 / 16 = 1 } and keep Q unchanged.
Then
[Equation]
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 to the theoretical Gaussian limits as the sample sizeincreases.
Conclusion
This paper deﬁnes the Sliced Wasserstein Process , a stochastic process indexed by elements of the unitsphere S d − 1 in R d , and shows that under regularity assumptions on P and Q , this process convergesto a tight Gaussian process on (cid:96) ∞ ( S d − 1 ) .
This convergence result, which can be viewed as a uniformcentral limit theorem for the empirical Wasserstein distance along all directions simultaneously,immediately implies distributional convergence and bootstrap consistency results for the sliced
Wasserstein distance and its variants, thereby unifying and streamlining existing proofs in theliterature and providing distributional limits for variants of the sliced Wasserstein distance for whichno such results were previously known.
An important question left open by our work is whether a similar result holds under weaker assumptions on P and Q .
We conjecture that the compact support assumption can be lifted, though doing sowould likely require making relatively stringent tail conditions.
Avoiding assumption (CC) is moresubtle, as some assumption of uniqueness of potentials is required to obtain Gaussian limits.
Finally, we anticipate that our techniques can also be applied to entropically regularized variants ofthe Wasserstein distance, where empirical process theory arguments have also been central in provingboth sample complexity and distributional limit results.
We leave this extension to future work.
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