Asymptotics of smoothed Wasserstein distances in the small noise regime
Authors: Yunzi Ding
Jonathan Niles-Weed
Venue: NeurIPS 2022 Accept (2022)
Keywords: Optimal transport
statistical estimation

Abstract: We study the behavior of the Wasserstein-$2$ distance between discrete measures $\mu$ and $\nu$ in $\mathbb{R}^d$ when both measures are smoothed by small amounts of Gaussian noise. This procedure, known as Gaussian-smoothed optimal transport, has recently attracted attention as a statistically attractive alternative to the unregularized Wasserstein distance. We give precise bounds on the approximation properties of this proposal in the small noise regime, and establish the existence of a phase transition: we show that, if the optimal transport plan from $\mu$ to $\nu$ is unique and a perfect matching, there exists a critical threshold such that the difference between $W_2(\mu, \nu)$ and the Gaussian-smoothed OT distance $W_2(\mu \ast \mathcal{N}_\sigma, \nu\ast \mathcal{N}_\sigma)$ scales like $\exp(-c /\sigma^2)$ for $\sigma$ below the threshold, and scales like $\sigma$ above it. These results establish that for $\sigma$ sufficiently small, the smoothed Wasserstein distance approximates the unregularized distance exponentially well.

Asymptotics of smoothed Wasserstein distances in the small noise regime
Yunzi Ding'
Jonathan 2
Courant Institute of Mathematical Sciences, NYU
"Courant Institute of Mathematical Sciences and the Center for Data Science, NYU yunzi dennziiingggmall com jnw0cims nyu. edu
Abstract
We study the behavior of the Wasserstein-2 distance between discrete measures μ and in Rd when both measures smoothed by small amounts of Gaussian noise. This procedure, known as statistically attractive alternative the unregularized Wasserstein distance. We give precise bounds on the approximation properties this proposal in the small noise regime, and establish the existence of phase transition: we show that, if the optimal transport plan from to is unique and a perfect matching, there exists critical threshold such that the difference between W2(n,, v) and the Gaussian-smoothed OT distance W2 No, .Na) scales like exp(-c/o" for below the threshold, and scales like above These results establish that for sufficiently small, the smoothed Wasserstein distance approximates the unregularized distance exponentially well.
optimal transport, has re-
cently attracted attention
1 Introduction: optimal transport
Optimal Transport (OT) has seen a recent surge of applications machine learning, areas such as generative modeling 22 16), image processing [13 27 31,, and domain adaptation 7 [8. A natural statistical question raised by these applications to estimate the OT distances with samples. These distances, known the Wasserstein distances, are defined by
(H, = inf Tell(H..) / ||x y (1ddd((,1))
where I1( v) denotes the set of joint measures with marginals μ and v, known as transport plans. It is well known that plug-in estimators for this quantity, obtained by replacing μ and with em- pirical measures consisting of i.idd samples, have performance in high dimensions, with rates of convergence typically of order n -2/d | 22
15 22 when 2p. Moreover, minimax lower bounds show that this curse of dimensionality is unavoidable in general 25 32).
The existence the curse of dimensionality for OT has led to a series of proposals to obtain better rates of convergence by imposing additional structural assumptions-such as latent low- dimensionality (25) smoothness (24] 33]-or by replacing W,, by better-behaved surrogate, such as an regulai-ee version with much better statistical and computational proper- ties 23 28
particularly intriguing option, developed by [18, consists in smoothing the Wasserstein distance by adding Gaussian noise. The following result shows the statistical benefits of this approach.
Proposition 1.1 ((I)). For d > 1 and a 0, denote by N, the centered Gaussian measure on Rd with covariance a² La. For any compactly supported probability measure μu in Rdd let 1 32, Tn
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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be i..dd. samples from μs, and define the empirical measure
jin Σ 8(m)).
Then there exists constamt c = e(p,, a, d) such that
EW2 * No. No) S en 1/2
18 call this framework coussian-smootleen optimal transport (GOT), and follow up work has shown that it possesses significant statistical benefits, with fast rates of convergence and clean limit
laws 19 20 38
To leverage the beneficial properties of the GOT framework, it is necessary to understand how well the smoothed distance W2 v*+,,) approximates the standard Wasserstein distance W2 (p,, v). An application of the triangle inequality [8, Lemma 1] shows that
(2( W NN))
Indeed, the triangle inequality implies [W2(u*No, v*N W2(u,, S W2(,, + vs N)) and the latter two terms are of order at most In general, this upper bound is unimprovable, as we show below. On the other hand, can also be very loose: μ is a translation of v, then W2(n No, , V * NN) does W2( * No, v/ * No) approximate W2 (1, ()) when a is small, and how does the answer to this question depend on the measures μ and //?
(2(p,)) 2/) for all
0. These examples raise natural question: how well
The main goal if this paper to give sharp answer to this question for finitely supported measures. We focus the finite support case for two reasons. First, when μ and are finitely supported, j * N and v * N₂ are each finite mixtures of Gaussians, and the behavior of Wasserstein distances for such measures is topic of active research [6] [O]. Second, as our results indicate, the behavior of this quantity for finitely supported measures is unexpectedly rich, with sharp dichotomy in rates depending the structure of the optimal transport plan between μ and y: we show that when the unique optimal transport plan between j and v is a perfect matching, then there exist positive o, and such that
W2( W2(u * No, v No) -
Yo E (0,
In other words, for sufficiently small o, the GOT distance approximates the standard W2 distance exponentially well, substantially sharpening ( More strikingly, we establish the existence of phase transition: for o < o., the gap is exponentially small, whereas for a > a,, the gap scales linearly. By contrast, if the optimal transport plan between μ and is not unique or is not a perfect matching, then no phase transition appears: the upper bound of is tight even in a neighborhood of a 0.
Our work provides a precise understanding on how GOT resembles vanilla OT in the vanishing noise (a J. 0) regime. These results complement those recently obtained by (5] in the large noise regime, who show that if fu and v have n matching moments, n 2 1, then W2( Na * NN,) O Along with results in (5), our work completes the limiting picture of the Euclidean heat semigroup acting on atomic measures under the Wasserstein distance. All the relevant rates are presented in Table I
We note that our work leaves open the question of characterizing the rates for non-atomic measures. It is possible to show that, for general measures, there are measures exhibiting polynomial rates in- termediate between and -</a² however, these rates appear to depend delicately on the geometry of the measures and their support. Giving a full characterization of the rate for general probability measures is attractive open question.
2 Preliminaries and main results
We are concerned with the optimal transport problem between discrete measures
a,)();),
Br0(us)
j=1

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Regime
Condition
lim (im(W2(11 No, Vo, xN))
Rate
Reference
Unique perfect matching
W2(u,
c/o"
Theorem 4.1
a 0
No unique perfect matching
W2(H,
o
Theorem 4.4
a f 00
I and y agree up to 1ith moment
o
Table 1: Limiting behavior of W2(p No, * No) for atomic measures μ and V.
in the space Rd equipped with the squared Euclidean cost function cx v) | y|ا Here {ar)/=1 and {Bg}}=1 are positive numbers such that Σi=1 a; = Σ Br = 1, and the sets :=
{ax) and y == (リz} consist of distinct elements of IR (Note that we do not require that X'y = 0.) Explicitly, we may write
(2, inf xell(u..)
110-vyl y1 /ll- min mell(a)))
Σ dE[K],jE(E
I -
(2)
We call minimizer in an optimal coupling.
We shall show that the behavior of the quantity W2(u= = No. NG) as 0 depends strongly the structure of the optimal couplings between μ and v. We rely on the following definition.
Definition 2.1. The measures μ and possess unique perfect matching there exists unique solution गt to (2) and if the support of this unique solution is a perfect matching, i.e., the set {x E N.y
< y) 0} bijection between X and y.
Our main results (Theorems and 5.1 show that the GOT distance approximates the Wasser- stein distance exponentially well for small a if and only the measures possess a unique perfect matching. To obtain these bounds, we show that if a is small and μ and v possess a unique perfect matching. then the optimal plan for μ and is also approximately optimal, in an appropriate sense, for the convolved measures * No and // * No. By contrast, if and v do not possess a unique perfect matching, then we explicitly exhibit an alternate coupling between μ * N and * N, with significantly smaller cost, therefore showing that Wz(μ No, V NN) is smaller than (2(p, v) by an amount that scales linearly in Ø.
To identify the range of a for which the exponential error bound holds, we introduce a robust version of optimality for x² in the perfect matching case. This definition, strong cyclical monotonicity, tends the classical cyclical monotonicity criterion from optimal transport [see, e.g. (77. and captures how sensitive the optimal plan is perturbations of the source and target measure. We show that this notion is closely related to the strong convexity and smoothness of the dual optimal solutions to the optimal transport problem, known as "potentials." The strong convexity of these potentials has previously been explored in computational and statistical contexts [26 36), but to our knowledge its connection the stability of discrete optimal tranport plans is new.
3 Strong cyclical monotonicity
In this section, we consider transport plans in the form of a perfect matching between {x;} and {.r}. By relabeling the points, we may assume without loss of generality that the optimal transport plan between f and y is the matching given by
r {(11, V1), (I2, v2) ...., (#x, yk)}
In this section, develop robust notion of optimality for P. This notion is based on strength- ening of the classic optimality condition for optimal transport, based on cyclical monotonicity. We recall the following definition.
Definition 3.1 (See, e.g., 30). A set S C I2d x Rd is cyclically monotone if for any (1),b)..... (a,, bn) E S, have
Σ 1=1
|a b1ll2 <
|a; - b
where we set bn+1 == by-
3

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The significance of this notion is the following fundamental result.
Theorem 3.2 (See 37 Theorem 5.10). II(, ) has cyclically monotone support, then it is an optimal transport plan between and v.
Our main definition strengthens this characterization by requiring the inequalities in the definition of cyclical monotonicity to be strict.
Definition 3.3. We say f [k] x [e] + R>0 is positive residual function on [k]] if f(i, 0, f( > 0 for i + j. and f(1.j) = i() for all i,j E [M]
Definition 3.4 (Strong cyclical monotonicity). For positive residual function f on [k], we say that
I' is f-strongly cyclically monotone, if for any S n S. k and distinct T(1,, T(), (22) E [X] (with the convention (n + 1) r(1)), we have
n
[0+(1) vr(al S
Σ
[+(() 3-(1+1)ll
Σ 7(+(1)
or equivalently,
Σ (2r()) (r(() (r(+1))
Σ ((((),
Strong cyclical monotonicity indicates that the optimal plan with support is superior to any other plan by positive margin in its transport cost. The importance of Definition 3.4 that is equivalent robustness of the optimality of I' under small perturbations of the points {2t} and {y: To make this connection precise, we make the following definition.
Definition 3.5. For 2 O, we say I' is e-robust, for any distinct any (x))), (2( (r((n) E Rd such that
E [k]] and
max ||=())| Se,
there holds
n
[,+(1 - V~(0l12 <
[((+(() + aa(()) - (Br(+11) + aT(11))
i=1
We write
R(T) = sup fe 2 0 : r is e-robust).
The quantity R(T), which we call "robustness of optimality," captures the behavior the optimal plan when the supports of and are slightly perturbed. As the following proposition indicates, robustness in this sense is equivalent to strong cyclical monotonicity.
Proposition 3.6.
strongly cyclically monotone if and only R(T) > O.
Proposition 3.6 may be viewed robust analogue Theorem 3.2 if TT(,, has strong cyclically monotone support, then it is a robustly optimal transport plan between and v, in the sense that it remains an optimal plan even when μ and are corrupted by noise. As we establish in Section 4 this observation is central to the analysis of GOT.
3.1 Implementability and explicit bounds on R(T')
Despite the mathematical simplicity of Definition 3.4 it not clear how to verify for a particular set I', nor how to establish that this property holds for optimal plan between μ and и. To this end, we propose another condition, strong implementabiliry which is equivalent to strong cyclical monotonicity but is more amenable to analysis. We also show that both conditions are equivalent to and y/ possessing a unique perfect matching.
29 introduced the notion implementability and established it as equivalent condition of cyclical monotonicity. parallel to the results [29]. we also introduce the following stronger condition of implementability.

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Definition 3.7 (Strong implementability). For positive residual function on [k], we say that I is f-strongly implementable, if there exists a potential function y,, such that for any i, E [Ri), we have
(Ii, y/4 4/) (p(x)) - (2(,,) + f(i.j).
Analogous the equivalence result in 229,. show that strong cyclical monotonicity and strong implementability are both equivalent to the uniqueness and optimality of
Proposition 3.8. The following three statements are equivalent:
(i) D is f-strongly cyclically monotone for some f;
(ii) D is f-strongly implementable:
(iii) T is the unique optimal transport plan from {xx) to {yk).
Remark 3.9. We can imply from the direction (i) to (iii) of Proposition 3.8 that, if the directed bipartite graph with vertex set {æ1, .2...., Tke Y1, "2..... y.] and ares between 2, and y; with weight ||, - Yy does not possess an alternating cycle of zero total cost, then I is the unique optimal transport plan from {xi} to {yi One may use this sufficient condition to verify uniqueness of an optimal transport plan I in practice.
The equivalence in Proposition i.rssitionn.8hhooo for any positive residual function f: however, in the context of optimal transport with the squared Euclidean cost, it is most natural to focus the quadratic case. The positive residual function constructed in the equivalence between (iii) and (i) in Proposition (388 of the form fi,, = 2 llu - y 1|? for some À > 0, in which case the implementability condition reads
(o,, y - サ,} 2 ((v)) p(u,) + 2 3: - 11
Quadratic residual functions are closely connected to convex analysis and to the theory of optimal transport. This condition is equivalent the existence of a X-strongly convex potential şp satisfying Vp(y)) = H; for all i E [k] [35), or, equivalently, the existence of a Lipschitz Brenier map from f to 4). The regularity of Brenier maps is a deep question in analysis [see, e.g. [4]. Proposition 3.8 establishes that, in the finite- finite-support case, this question is equivalent to the uniqueness of the optimal transport plan for μ and v.
More generally, we have the following theorem characterizing the properties of strongly imple- mentable plans with residual functions of quadratic type.
Theorem 3.10. The following conditions are equivalent:
(i) For some 0 S < 3, there exists potential function yp Rd + Rd which is c-strongly convex and 3-smooth, such that x; = V.p(yi) for all i [k].
(ii) T is strongly implementable for
f(i.)) 2(3 -a) ai) (112, 2111² + a:8lyy - 2 20 -alyy Vj, H; -2,)); (3)
or equivalently, there exists (5(y:) C Rd, such that for all i,j E [k] (i + jj.
(o,, Vi -b)) 2 ((v)) - (1,))
(1|:; xz a:311 43 22x Mj 34 xzi) (4)
2(B a)
Proof. This is a direct application of Theorem 4 in (35]. The condition (i) in Theorem 3.10 is equiv- alent to the set 1( ((y))) being -nteepooeoelel in Definition 2 of (35] and the condition (ii) is equivalent to equation (4) in Theorem of [35]
Theorem 3.10 also formally encompasses the choice 3 to0 when &p strongly convex but not smooth, in which case (3) reads
s(i.j) = - yı 11,12
(5)
which recovers the positive residual function used in the proof of Proposition 3.6
5

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Remark 3.11. We should emphasize that the defined in Theorem 3.10 indeed a positive residual function given O: <. B, since Cauchy-Schwartz gives
(xx Uj 24 xz) - ||: =2 a² a21ly: 125 1 xyyl" 0:31|ye ygll
As direct consequence of direction (ii) to (i) in Theorem 3.10 if is strongly implementable for a positive residual function f which is quadratic in yi yj and Zz. there exists smooth and strongly convex potential function verifying implementability.
Corollary 3.12. Suppose I is strongly implementable for
f(i.j) = (Mrxllz; 2,ll² Agullys vjll 27 (36 Vj - 23))
where Awar Aay and Auy are nonnegative numbers which satisfy Nay + Aay Aeaday Then there exists potential function yp Rdd - Rd which is T; V for all i [k]-
Aay Ann
-strongyy convex and Ava -moothh such that Tay
By the Smith-Knott optimality criterion for optimal transport 34 the conclusion that r; Vip(y;)
for all E [k] is equivalent to the potential function solving the following dual version of 22:
inf O3S (x;) + Σ 350(U;),
(6)
where o* denotes the Legendre conjugate.
Finally, we show that the above characterizations give rise to lower bounds R(T), which are easy to compute Okk)) time. The following bound gives a quantitative link between robustness of optimality and the residual function f.
Proposition 3.13. Suppose I is strongly implementable for positive residual function f. Then r e-robust for
inf [(i.j) x3 + 11
2
|
This implies that
1 inf R(T)=
f(i,j)
||.
|| + 11:: yı
By combining this bound with Theorem 3.10 we obtain a simple lower bound when f is of quadratic type. Proposition 3.14. When the equivalence in Theorem 3.10 holds, I is e-robust for
| - x251? 11l/,aallu) ally/ w)
max
inf 2 (2)
(8)
| xz51 + 11 yill
This implies that
max 1 2z 11y 1/
R(T)
2 193
|| - 2z || + 11 -
Remark 3.15. When condition (i) Theorem 3.10 holds, a-strong convexity and 3-smoothhess implies
1 ||. xzyll
|0 xzll l1u Usll
Thus the condition 8 may be replaced by the bound
{ i 1 1 x,,1 3(1 + (1)
inf max (T)
11
yj
which is more interpretable. We should emphasize that, to check the e-robustness of I with either ( or (9) requires prior knowledge on the parameters a and which are inherent to the optimal transport plan l.

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4 Exponential rates for unique perfect matchings
Our main results show that the robustness of optimality R(T) controls the gap between W2(44, v) and We(p Na, N.). Theorem 4.1. If o. R(T)) > O, then for a E (0,0.).
W2(n, D) W2(n u=
The proof depends on the following simple lemma, which shows that robustness to optimality im- plies that the Wasserstein distance is unchanged when μ and v are conrupted by small noise. Lemma 4.2. If o. R(T) > O, then for any measure Q in IRd supported on B(0, o.).
W2(p. W2(u Q, Q).
Proofs of these results appear in the supplementary material.
In the regime where a does not exceed R(T), the above theorem tells that the GOT distance is an excellent approximation of the OT distance. Our second main result is converse that statement, showing that if a goes beyond R(T), the loss W2(μ, v) - W2(n Na, V .N₂) is bounded below by linear function of a. We start with the following proposition, which quantifies "violation of cyclical monotonicity" under possibly large perturbations in the sources and targets.
Proposition 4.3. IfT is an optimal transport plan, for any M 0, denote
G(M) sup Σ |0+(0 v-(0
[I ar(1)) (Wr(142) + a.(++1))l ||a+coll M Σ
Then G(M) is concave functiono of M for M E [0, +oc).
Note that G(M) vanishes for M o.. The next theorem shows that as long G(M) is not negligible for 2 o., the approximation loss for a 2 ax is linear in ø. Theorem 4.4. If R(T) 0, then
W (1, v) D
sup e /² G(M). M>a.
Here G(M) is defined as in Proposition 4.3
In particular, if G(3o.) 2 coo for an absolute C(co) > 0 such that for (0.20.).
constant co > then there exists a constant
W3 (1, v) W3 (1x No, v No) 2 Coe -0//02 Cae-oo/o2
The proof of Theorem
4.1 Lemma
Proposition
4.3 and Theorem can be found in the
supplementary material.
Beyond perfect matchings
In the case that R(T) O, or equivalently by Proposition 3.8 and Proposition 3.6 that the optimal
transport map between fr and v is not perfect matching. Theorems 4.1 and 4.4 are not applicable. In this situation, we are able to show that the approximation emoor is linear, even in a neighborhood of zero. fact, this holds whenever there exists an optimal transport plan between j and v which not a perfect matching.
To analyze this case, we return to the setting of general discrete measures:
a,8(I,), V Σ 8,8ly,)
(10)
1=1
Theorem 5.1. Let μ and be as in 10P. If u and v do not possess a unique perfect matching, then there exists Co such that for E (0,co)
(2, Wh No, No) a
7

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Theorem 5.1 tells that, unless the optimal transport plan between j and is unique and perfect matching, the loss from approximating the OT distance with the GOT distance at least linear in σ. We derive Theorem 5.1 from the following lemma, which shows that Theorem 5.1 holds in the special case where 4l is single point mass, and is uniform on two points.
Lemma 5.2. Let x:, y1 and y2 be different points in RRd. For μi == 6(x) and vo = 20(11) + +
there exists co O, such that for E (0, co), we have
W vb) W No, Vo No)
(11)
We obtain the full strength of Theorem 5. by reducing the special case of Lemma 5.2 particular subset of the support of and v. Full details appear in the supplementary material.
Numerical example
In this section, we present numerical example demonstrate different regimes of the rate W2(H, w/) ( No, V * No), in respect of Theorem 4.1 and Theorem 4.4 For the sake of clarity, we consider atomic measures y and both defined 1 One of the simplest cases where a coupling I' has R(T') = 0 is
1 [8((-1, ((1,
1 2 18((-1,1)) + 5((1 (1)]
is easy to see that the optimal transport plan from jJ to v is not unique, which is also a consequence of Proposition 3.8. Proposition 3.6 and the fact that R(T) 0 for the map
r {[[ [[[[[11))))) - (-1, ((1, (1,--1))) (1,-1)))
that achieves the optimal cost. We also consider the family of perturbed measures
p(p) ()( -))], E [0,1]
The source and target distributions corresponding to p k/10 10 for k: 1,2,3, 4 are depicted in Figure 1. For each k, the unique optimal transport plan from μıx m(k/10) to is given by
Is TA= -1=(((1,-1++
k 10 (-1 1)), ((1,
k To (1, 101--11 }
For each of these GOT tasks, we draw 200 samples from the source distribution μ* No and target distribution * No. and use the empirical W2 distance as an estimate of the true W2(m No, / No). We repeat the process 100 times and report the mean, shown in the following figures.
By Theorem 4.1 and Theorem 4.4 we expect W3 v) - W2 * NN. No) to be of seale e-//a² for a E (0, R)) and W v) W No. NG) 2 for 2 R(Tx). This transition from exponential to linear is visible in Figure 2.
Using Proposition 3.14 we obtain a lower bound on R(T's), which we plot with vertical dashed line in Figure 2. Exponential decay is visible the left of the dashed lines, as anticipated. Figure 3 shows this behavior more clearly on logarithmic scale, where we observe that log( - log (W(,,)) Wz(u No, NN))) is linear in log(o) ) for small Ø.
7 Conclusion
This paper develops approximation results for Gaussian-smoothed optimal transport, showing that GOT approximates the Wasserstein distance exponentially well for small a when μ and и possess a unique perfect matching. By contrast, if and do not possess unique perfect matching, then the gap between W2(n, v) and W2(u * No, v N,) is linear a as a - 0. The difference between these two behaviors can be traced to the fact that if and possess unique perfect matching, then

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Source (original) and Target
Source (perturbed) and Target
MT
1.00
mu: p 0.0
TIJ
mu: - 0.1
0.75
mu: p 0.2
mu: p 0.3
0.50
0.5
0.00
-0.50
-0.75
-1.00
-1.00
-0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
1.00
Figure Source and Target distributions
betweer GDT and dittrrees
Cap tetween GOT and OTW dittnnes
116
116
623
075
60
bglegrai
Figure 3: Rate of log(- log(-log(Cs (og(G))) versus log(o) in the vanishing noise regime. Here Go W2(,, v) W2(u No. No.v*N))
Figure 2: Rate of W2(,vv) W2( vx in the vanishing regime.
the optimal transport plan between them is stable under small perturbations of and v. In particular, for noise distributions with sufficiently small support, W2(u,v) W2 Q v Q). On the other hand, if and y do not possess unique perfect matching, then their optimal plan is not stable, and can change even under infinitesimal perturbation.
Our techniques are based on a new notion of stability for discrete optimal transport plans, which we call robustness of optimality and characterize by developing strong variants of the classic notions of cyclical monotonicity and implementability for optimal transport plans. Just as cyclical monotonic- ity is closely connected the convexity of the dual potentials for the optimal transport problem, show that strong cyclical montonicity is equivalent to strong convexity. This characterization shows that a discrete optimal transport plan is robust perturbations of the source and target measures and only if its support lies in the subdifferential of a strongly convex function. We anticipate that this characterization will have statistical and computational implications for the estimation of optimal transport plans between discrete measures.
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12


Reviews:
[{"review_id": "pauFCTpQRZn", "reviewer": "Reviewer_Wajc", "summary": "The paper provides an approximation rate for Gaussian-smoothed Wasserstein distances for discrete measures. It shows that the approximation rate can be exponential in the perfect-matching plan case (with a phase transition to linear rate if $\\sigma$ is large) and is linear otherwise.", "questions": "1. The results in the paper are interesting in that the approximation rates for the \"perfect-matching\" case are much faster than other cases (exponential compared to linear). To complete the story, can you present some sufficient conditions to have perfect matching in OT?\n\n2. The technique used in the paper seems not to be restrictive to discrete measures only. I suggest the author extend these results to arbitrary $\\mu$ and $\\nu$, as the \"perfect-matching\" is easier to attain for absolutely continuous probability measure. If it is not possible to do this extension, then it would be helpful to have a paragraph to discuss why. \n\n3. In the statement of Theorem 3.3., what do you mean by $G(\\sigma^*)$ is asymptotically greater than $\\sigma^*$? Because $\\sigma^*$ here is fixed, I can not see the asymptotic greater sign here with respect to what variables. ", "limitations": "Because of the restriction of the setting in the paper, it is not obvious to see implications in using Gaussian-smoothed Optimal Transport for general measures. I recommend considering the problem in a more general setting and trying to see if the current approach still works. ", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strength**: The results in this paper are interesting and all the theoretical results are technically correct. \n\n**Weakness**: The setting considered in the paper appears to be restrictive. Both measures of interest are assumed to have finite number of support points. It makes the developed theory difficult to apply to general setting of Gaussian-smoothed Optimal Transport. \n\nSome points in presentation that can be improved:\n\n1. In Proposition 1.1., the constant $c$ is dependent on $d$ (dimension) as well. The author should make this point clear so it is coherent with the argument about the curse of dimensionality of the Wasserstein distances. \n\n2. In the presentation of Section 3 (Case I): We need reasons why you choose to present Proof of Theorem 3.3. but not of other results (like Theorem 3.1, which appears to be one of the main results). If the proof of Theorem 3.1. is not presented in the main text, then why do you present Lemma 3.4.? In general, the writing style in Sections 3 and 4 needs to be improved so that the reader can know what proof is going to be presented and what is deferred to the appendix.  \n\n3. The order of results in the appendix is not consistent. Dividing those results into several small sections may help to keep track of this issue.", "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": {}}, {"review_id": "oaEiuisVmd", "reviewer": "Reviewer_KjJi", "summary": "This work concerns the asymptotic behaviours of Gaussian-smoothed 2-Wasserstein distance. In particular, the authors provide bounds on the difference between Wasserstein distances of a pair of discrete measures and their smoothened versions when the variances of Gaussian kernels are small. The authors consider two scenarios, when a perfect matching between two discrete measures exists and when it does not. For the former case, they show that the asymptotic gap decays exponentially in some near-zero regions, and linearly otherwise. For the latter case, they show that the gap is linear even in a region around zero.", "questions": "Minor Comments:\n- In the proof of Theorem 3.1 in the Appendix:\n  - Should the truncated kernel be $\\mathcal{N}(0, \\sigma^2 I) \\cdot \\mathbf{1}\\\\{\\\\|X\\\\| \\le \\epsilon_*\\\\} + (1 - p) \\delta_0$?\n  - At the end of the proof, for the second inequality, it would be more clear if the authors state which couplings are used to bound the Wasserstein distances.\n\nQuestions:\n- In the toy experiment, if we slightly modify $\\nu$ to $\\nu'$ using $k \\approx 0$, then there is an unique optimal transport plan between $(\\mu, \\nu')$, but should the asymptotic behaviors of both pairs $(\\mu, \\nu')$ and $(\\mu, \\nu)$ be approximately similar?\n- If we look at the Gaussian-smoothened version of a discrete measure as a Gaussian mixture, then does the separability between the means (i.e., the discrete support) play some role in the asymptotic behaviors?\n- Do you think that the appearance of the uniqueness of the optimal plan in the study of asymptotic behaviors is related to the use of 2-Wasserstein? For example, if we consider the 1-Wasserstein distance between two sets of points on the real line, and suppose there is some overlapping between the sets, then the optimal plan is not unique, but we can simply remove the overlapped points in both sets without affecting the distance (and consequently the asympotatic behavior) and return to the uniqueness case?\n\nMinor Error:\n- In the figures, $p$ should be $k$.", "limitations": "The authors have adequately addressed the limitations and potential negative societal impact of their work.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The paper is well-written and well-presented.\n- The addressed questions are of theoretical importance, and the given answers to these questions are novel and complete to the best of my knowledge. The authors covers all possible scenarios, and the tools used for proofs (stronger notions of cyclical monotonicity and implementability, robustness of optimality and their relations) are interesting to me.\n\nWeaknesses:\n- In the experiment, the toy example is indeed beneficial for understanding, but I think the authors should also verify their theory in large-scale settings (i.e., large $m, n$ - maybe in the one-dimensional space if the design of points statisyfing the uniqueness of optimal plan is an issue).", "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": "VSTjqCez0qZ", "reviewer": "Reviewer_jBuP", "summary": "This paper presents an analysis of the Gaussian-smoothed Wasserstein distance (GOT) in the framework where the Gaussian kernel parameter $\\sigma$ is \"small\". It is already known that (GOT) approximates the true Wasserstein distance and that the difference between the two is of order $\\sigma$. The objective of this paper is to refine this bound and to show, under certain assumptions of uniqueness of the transport plan, that this bound can be improved and that GOT approximates Wasserstein exponentially well in a certain regime. More precisely, the authors show that there is a phase transition on $\\sigma$ such that, below, the bound is exponential, and, above, it is linear in $\\sigma$. This paper completes the understanding of GOT with respect to $\\sigma$, which has already been studied in the $\\sigma \\to +\\infty$ regime.", "questions": "- In Remark 2.14 it is said that condition (5) in the context of residual functions of quadratic type is \"easier to verify in pratice\". It is not really clear to me how (6) can be verified easily because it depends on $\\alpha,\\beta$. To check that this condition does not hold we should do it for all $\\alpha <\\beta$ am I correct ? Can the authors detail this point ?\n- The fact that $\\Gamma$ is f-strongly implementable seems to be really related to the article [PdC20] of the paper, i.e. to the fact of imposing that the Monge map is the gradient of a smooth and strongly convex potential. Can the authors discuss this connection in more detail? \n- The proof of Theorem 2.7 is rather succinct: \"This is a direct application of Theorem 4 in [THG17]\". Could the authors give more details about the connection between \"$\\Gamma$ is strongly implementable\" and the results of [THG17] ?", "limitations": "The authors did not discuss the potential negative societal impacts of their work, however this is not really relevant in this context as the article is quite theoretical and specific. Concerning the limitations: I find that the authors could discuss how, in practice, can we check if we are in the regime of fast approximation (i.e. when the transport plan is unique). \n\n\n------- AFTER REBUTTAL -------\n\nAs written below I am satisfied with the authors' answer so that I change my score to 7", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Overall I find this article quite well written, the thread of definitions and proofs is clear, the ideas are well linked. From a purely theoretical point of view the results are, I think, really interesting. They complete the statistical understanding of GOT with respect to $\\sigma$, which is a nice contribution. Moreover, the ideas/tricks introduced and the theorems go far beyond the study of the GOT distance and can certainly be used to establish other theoretical results in optimal transport. In particular I think that the notions of strong implementability/strongly cyclically monotonic and robustness of the transport plans are useful and rich.They allow us to establish the uniqueness of the transport plan, which is, in the discrete case, a key property that is not much addressed by the community as far as I know. The fact of having cleared all these properties around the strong implementability is for me a contribution that is useful in itself. The phase transition of GOT opens also the door to other studies, notably on sample complexity or on approximations of the Wasserstein distance. \n\nThe main criticism I would have is that this article focuses on a really specific and technical problem, related to a sub-problem of optimal transport. For a reader who is interested in GOT the contributions are certainly really interesting, but the article does not discuss the potential applications for optimal transport in general neither for machine learning. To be more precise, the article strings together theoretical results without giving much insight, nor discussing the propositions and theorems. It also lacks a conclusion that could perhaps bring some perspectives around this work. In this context, I think it would be interesting to save space by moving the proof of Theorem 3.3 to an appendix to make a conclusion/perspective part and for discussing the different results and their implications. For example it could be interesting to discuss, even informally, the possible generalization to the case of continuous measures or to explain their implications for the Wasserstein distance approximation. \n\nMoreover, the numerical results are very succinct and, I find, difficult to read. I find for example that the phase transition, which is the center of the contributions, is not really visible on Figure 2. I think that this part should be more complete, by better illustrating this phase transition or for example the notion of robustness of a transport plan.\n\nFor these reasons I rather recommend a weak-accept, but I am ready to change my mind depending on the authors' answer.\n\nSmall remark: \n- In terms of notation the $\\sigma$ is both used to define a permutation and for the $\\sigma$ of the Gaussian kernel.\n- Moreover the fact that $\\sigma$ is a permutation in Definition 2.4 is not clearly stated in the article.\n- Typos: $T$ instead of $\\Gamma$ in Proposition 2.12 and 2.13. ", "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": "4FBPA290PFI", "reviewer": "Reviewer_AtL7", "summary": "This paper studies the approximation of the 2-Wasserstein distance between two discrete probability measures $\\mu$ and $\\nu$ by the 2-Wasserstein distance between the same distributions smoothed by a Gaussian measure, that is called Gaussian-smoothed optimal transport. In particular, they prove the existence of a phase transition in the small noise regime of the variance parameter, which depends on the existence of a perfect matching between the distributions $\\mu$ and $\\nu$.\n", "questions": "- Could you transpose your results to the case of entropy regularized optimal transport? In this case, the uniqueness of the transport plan is guaranteed but the perfect matching does not make sense anymore for large regularization parameter since the mass of the regularized transport plan is spread out.\n\n- In Proposition 1.1, why is the condition that probability measures are compactly supported essential?\n", "limitations": "The authors decided to limit their study to finitely supported measures, which is well justified.\n", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Studying the behaviour of Gaussian-smoothed OT distance is particularly interest as it approximates the true transport and does not present a curse of dimensionality in the sampling complexity. In particular proving the curious result of the existence of a phase transition in the context of finitely supported measures $\\mu$ and $\\nu$ is very nice. Moreover, this paper is well written, the problem and presentation are clear and the choices made for the study (the finitely supported framework) are precisely justified. The proofs, based on strong cyclical monotonicity and perfect matching, are quite elegant. The simulation study is also concise and convincing.\n\nOverall, the results constitute a modest (because limited to Gaussian-smoothed OT distance) but very interesting contribution to the study of approximations of the classical Wasserstein distance.\n\nMinor comments:\n- The notations for the parameter $\\sigma$ of the Gaussian and the permutation $\\sigma$ can be confusing.\n- The order and presentation of proofs in the appendix could be improved.", "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": {}}]

Author rebuttal:

Decision: Accept