A novel notion of barycenter for probability distributions based on optimal weak mass transport
Authors: Elsa Cazelles
Felipe Tobar
Joaquin Fontbona
Venue: NeurIPS 2021 Poster (2021)
Keywords: Optimal transport
Optimal weak transport
Barycenter of probability measures

Abstract: We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass by Gozlan et al. (2017) and Backhoff-Veraguas et al. (2020). We provide a theoretical analysis of this object and discuss its interpretation in the light of convex ordering between probability measures. In particular, we show that, rather than averaging the input distributions in a geometric way (as the Wasserstein barycenter based on classic optimal transport does) weak barycenters extract common geometric information shared by all the input distributions, encoded as a latent random variable that underlies all of them. We also provide an iterative algorithm to compute a weak barycenter for a finite family of input distributions, and a stochastic algorithm that computes them for arbitrary populations of laws.  The latter approach is particularly well suited for the streaming setting, i.e., when distributions are observed sequentially. The notion of weak barycenter and our approaches to compute it are illustrated on synthetic examples, validated on 2D real-world data and compared to standard Wasserstein barycenters.

A novel notion of barycenter for probability distributions based on optimal weak mass transport
Elsa Cazelles IRIT, Université de Toulouse, CNRS elsa. fr
Felipe Tobar
IDIA & CMM, Universidad de Chile ftobar@dim. uchile.
Joaquin Fontbona
CMM, Universidad de Chile fontbona@dim. uchile.cr cl
Abstract
We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass [26], [9]. We provide a theoretical analysis of this object and discuss its interpretation light of convex ordering between probability measures. In particular, rather than averaging the input distributions in a geometric way (as the Wasser- stein barycenter based on classic optimal transport does) weak barycenters extract common geometric information shared by all the input distributions, encoded as latent random variable that underlies all of them. We also provide an iterative algorithm to compute weak barycenter for a finite family of input distributions, and stochastic algorithm that computes them for arbitrary populations of laws. The latter approach is particularly well suited for the streaming setting, i.e., when distributions are observed sequentially. The notion of weak barycenter and our approaches to compute it are illustrated on synthetic examples, validated on 2D real-world data and compared standard Wasserstein barycenters.
the show that,
Introduction
Optimal transport (OT) [41] has had tremendous impact in the machine learning (ML) community recently, as it provides meaningful and implementable distances between probability distributions [34], thus advancing many aspects in the field, see e.g. [7, 43, 33). The space of probability measures on Rd with finite second moment can be metrised with the Wasserstein-2 distance, the computation of which amounts to finding transport plan that minimises the quadratic average cost of transporting mass from a source probability measure onto target one. In this context, a natural method for averaging finite family of probability measures is to compute their Fréchet mean, with respect to the Wasserstein-2 distance, which corresponds to the Wasserstein barycenter introduced in [1].
The goal of the present work is explore theoretical features and potential applications to ML of barycenters probability measures analogously defined terms of optimal weak transport (OWT, [26]) or more precisely quadratic barycentric transport costs. In nutshell, for source measure μ and target measure v. the OWT problem aims to transport mass so that the conditional spatial mean of target support points y, given their source support points ær is close to finding an intermediate measure 17, possibly more concentrated than // in the sense of convex ordering of probability measures, which is close μ with respect to the Wasserstein-2 distance.
in average. This amounts
The main motivation of our work to investigate the effect and meaning of combining a family of probability measures using OWT instead of OT. To that end, will define the weak barycenter of
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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this family through an optimisation problem, and discuss some of its properties, Importantly, we will see that, rather than averaging the imput distributions metric sense, solving weak barycenter problem corresponds to finding probability measures that encode geometric or shape information shared across all of them. In fact, the weak barycenter problem will be interpreted as finding a latent random variable common to all the input distributions. Implications of this latent variable interpretation, in terms of robustness to outliers, will also be drawn in our work.
A second motivation for our work is to develop and implement computational methods for weak barycenters, capitalising on the fact that the optimal weak coupling between any pair of distributions, with finite second moments, is always realised by a unique optimal map. This property is in sharp contrast to standard OT, where the absolute continuity with respect to the Lebesgue measure of the source or target measure is typically needed to grant the existence and uniqueness of a map- so-called Monge map- realising the optimal coupling between them. This map is often required in different ways to compute Wasserstein barycenters (see [5], [44] or [35]).
Similarly to the Wasserstein barycenter problem, we will develop a fixed-point formulation of the weak barycenter problem, based on OWT plans. This allows us to, following [5, 44], construct an iterative procedure to compute weak barycenter for finite family of distributions and analyse its convergence properties. We will also define and study the so-called weak population barycenters, that deal with a population of probability measures distributed according to given law Q supported on the Wasserstein-2 -2 space, as in [30] for the OT case. Extending ideas from [10], we will then propose an iterative stochastic algorithm for online computation of the weak population barycenter, from a stream of probability measures sampled from Q We will then provide numerical simulations using this proposed method, order to illustrate the geometric meaning of the weak barycenter, and we will compare it with related objects obtained with standard OT or its entropy-regularised counterpart.
Organisation of the paper. Sec. 2 documents the background on OWT and the assumptions underlying our work. Sec. 3 analyses the weak barycenter problem, interprets it in the light of convex ordering and latent variable model and addresses the case of an infinite population of distributions. Sec. 4 introduces two algorithms for computing the weak barycenter in the finite or population settings. See. 5 and 6 present the experimental setting and validation of our proposal respectively. Lastly, Sec. 7 discusses our findings and future research questions. The Appendix contains all the proofs, additional details or our simulations and the code of our experiments.
2 Background: : optimal weak transport and Wasserstein barycenter
The optimal transport (OT) problem [42] aims to find the lowest cost to transfer the mass from one probability measure onto another. Therefore, OT is a natural way to compare two probability distributions in terms of their geometric information. In particular, the Wasserstein-p distance W.,, associated with the Euclidean cost in IR metrises the space Pp (Bd) of probability measures on R with finite p-moment. Precisely, for μl, E (R
1/P
Wy(uvv) min [ (ETT(GP)) x xR
1 da-yyl'drae (I)))(()))))
(I)
where π is a transport plan between f and v, that is, an element of the set I(u v/) of probability
measures on the product space Ed x Rd with marginals μ and v. For p 2 and j absolutely continuous (a.c.), the unique optimal plan is concentrated on the graph of a measurable map called Monge map such that v = T T## μr, see eq. (I4) in Appendix A.I.
Optimal weak transport. We consider here the optimal weak transport (OWT) problem introduced in [26] and in particular the special case of barycentric transport costs. The OWT problem is then defined for μr, E Pa (IR.) by
/ 1| 11-- yd yd J
((ule)= = inf
(2)
sEl((r))
where n. the disintegration of the transport plan π with respect to the first marginal μ., i.e. T (drdy)- = nz (dz)) As our work strongly leans on OWT theory, we recall in Appendix A.2, Th. 6, that V continuous with respect to the Wasserstein metric [9]. Additionally, the two following results from [8] (stated for our specific setting) lay the ground for proposed weak barycenters.

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Theorem ([8), Theorem 1.2). The problem (2) admits a unique minimiser
This first result strongly differs from the classical OT setting, for which the uniqueness of an optimal transport plan is not guaranteed for arbitrary measures. The optimisation problem in Eq. (2) can also be reformulated thanks to the Brenier- Brenier-Strassen theorem [25]. [8], through the notion of convex ordering. We denote by 1 Se the convex order of measures, meaning that odn ødv for any convex function & that is nonnegative or integrable with respect to + V. By Strassen's 's theorem [40], two distributions are in convex order if and only if there exists a martingale coupling between them. The following theorem is generalisation of the result originally proved in [25], Th. 1.2.
Theorem ([8), Theorem 1.4). Let μ E (2 and V E (R()). There exists a unique 1f" Se V such that
W2(4 = inf W2 7) -(ulv). 4ScN
(3)
Moreover, there exists a convex function dh RRd - IR class C2 with Vij being 1-Lipschitz, such that Vio#p= = Finally, the optimal coupling N.. E <IT(,,)) veriftes J yd (v) Vx)(z) j.a.s.
The measurable map, or barycentric projection, (x) := Ja (y) associated to the plan nH*. achieving the minimum in Eq. (2) is consequently uniquely defined and will be called optimal barycentric projection. From this notation, we can write the OWT cost in terms of an OT cost according to (ulv) = W (n, u)). We emphasise that 5 is directly related to the optimisation problem (2), whereas applied works such as [38, 36] make use of a barycentric projection constructed from transport plan solving an OT problem between and (often regularised) as a substitute for the Monge map, which may not exist (more details on S are displayed in Appendix A.3).
Last, let us note that OWT is somehow also related to the martingale OT problem developed in the stochastic finance community [13, 2, 27], which puts the focus on the optimal transfer of mass between distributions assumed to be in convex order themselves.
Wasserstein barycenter. The classical Wasserstein barycenter problem for set of probability
measures PI,.... Pn. E (R2) with weights Ai...., An in the simplex (i.e. A; 2 0 and >x A is defined [1] by
arg min (EP)
Σ (u,
(4)
The Wasserstein barycenter has been extensively studied both theoretically and numerically [30, 44, 5, 15]. Regarding the numerical part, [39] focuses on the computation of Wasserstein barycenters for a fixed number of measures and a stream of observations per measure; additionally, [32] proposed an alternative via stochastic optimisation for computing the Wasserstein barycenter of a.c. distributions only from observations. Constrained by their assumption of a.c., [44] computes the Wasserstein barycenter by smoothing the observed empirical distributions. Furthermore, [21] compares the complexity of both the sample Wasserstein barycenter and a stochastic approximation to estimate population barycenter (discrete measures and entropic regularisation). Finally, the authors of [3] recently proposed an algorithm to compute the barycenters in polynomial time.
3 Optimal weak transport barycenters and latent variable interpretation
3.1 Definition and basic properties
In similar fashion, based on the weak transport cost in Eq. (2), we propose the following variant: Definition 1. The set of weak barycenters of a finite family of measures {v:h= E (2( with weights {>..e n in the simplex defined as
arg min (R
(5)
Thus, a weak barycenter averages, with respect the Wasserstein metric, optimally chosen set of probability measures .n...., .h) which are more concentrated than the corresponding vs in the sense that 77 Se 14 for each 1 5 < The existence of solution is established as follows:
3

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Proposition 1. The weak barycenter problem in Eq. (5) admits minimiser μ E P2(Rd))
See Sec. B of the Appendix for the proof of the above Proposition (which relies on Prokhorov's theorem) and all the proofs for this Section. Uniqueness is in general not granted: we next show that the set of solutions is indeed an interval, with respect to the partial order of convex ordering of probability measures.
In the following, 1 i i n, and 8ş the Dirac measure supported on a E RR.
denote by X and Y; random variables with respective laws μ and vs, for
Lemma 1. If is weak barycenter of (=:)s= and yl Se μ4, then p' also is weak barycenter. In particular. the Dirac measure supported on En (X) is always a weak barycenter. Moreover, a Dirac distribution 8g is weak barycenter if and only if a Ei=d,, (Yi).
consequence of the above lemma is that for any weak barycenter 14,
(x) Σ AclEa, (Yi),
(6)
and the value of the weak barycenter problem is given by
a
inf (vV())) Σ A1IE(())1 - Σ A,E())
(EP2(R4)
We also derive the following characterisation on the set of weak barycenters:
Proposition 2. A measure μt E ((RR)) is a weak barycenter of {vi]imi..... if and only if its mean satisfies (6) and ja So holds for all 1 < n, where denotes the centered version of law v.
For instance, in the case of one dimensional Gaussian distributions v; = (mm, of), the set of weak barycenters includes {u (m(,,o²) ) S o² minigign o
natural question is whether a "maximal" weak barycenter exists, in the sense of convex ordering (up to translation by the mean). For d = 1, the answer is affirmative. When the means E(Y;) are equal, this follows from the complete lattice property of the set of probability measures with respect to the convex ordering (see [29]): the general case can then be reduced to the latter using Proposition general not true and the answer depends on the family {v:)i=1...n.
For d 2 2, this property
In the particular case of a.c. input measures, we can bound the distance between the Wasserstein and weak barycenters by the variances of the distributions (v:) The barycenters are then closer the more concentrated each Vi is.
Lemma 2. Let VI,,..,
Vn. E (2RR))) be a.c., at least one of them with bounded density, Let p. and respectively denote the weak and the Wasserstein barycenters. Then
(2, <
X (BI |2)
3.2
Weak barycenters as latent variables
The weak barycenter encodes common geometric information present in all the input measures considered, therefore, it can intuitively and rigorously interpreted as being the distribution of latent variable underlying the realisations of random variables of laws v for all 1 S i S n.
Theorem 3. Let fu be a weak barycenter of {y:ji=i..... Then, for each 1 S S n, a random variable Y, ~ Vs can be realised
Y, = + EY; EX)+Y,
where X μ and Y, Y; X(Y is centered conditionally X. Moreover, has (X) X (EY, EX) for all 1 ......M. n. Finally, we have B(Y; EY, X EX) X EXX or, equivalently; y i Sa Dú with ja and in the laws X E. and Y, EY, respectively.
That is to say, each Y, i= ......P and distributed according to the weak barycenter μ,, translating that value by RY,, E.X and adding cluster-specifee component Y, or idiosyncratic noise, centered conditionally on X.
can be realised by sampling a random variable X common to all

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The observations of each class (i.e. input measure) can be interpreted as outliers with respect the (translated) law of the weak barycenter, which are statistically different and are thus left aside of its support. This way, the weak barycenter is robust to outliers, it tends to discard them, by construction. Furthermore, this "robustness" property results the stability of weak barycenter upon perturbation of class with larger noise (or more scaftered, outlying values). More precisely, if a class is corrupted in such a way that their observations result in stochastically larger distribution than the original one, a weak barycenter computed in terms of the original (stochastically smaller) class will still be a weak barycenter in the new corrupted setting. An intuitive and simple way to illustrate this point follows by considering a weak barycenter i of one dimensional and centered family of input distributions {V/ 1VsYee.....n. By Proposition 2. must verify p. Se Vi Vi all = 1,..., ......... In particular, from Theorem 3.A.1. in [37], we have that 1- P(X > u)du S S P(Y; > u)du for all TR where X μs. and Y; V4- Therefore, I is likely to avoid outliers. Another supportive intuition in terms of robustness is that a maximal weak barycenter would be one that includes the most possible points of all classes (or distributions) in its support (all this, after re-centering) and leaves out only "outliers". A non- non-maximal weak barycenter is then more conservative, meaning that it counts on fewer points and leaves out more possible oulliers.
Remark 1.
3.3 Extension for the population barycenter
The population Wasserstein barycenter introduced in [30] and [4] extends the definition of Wasserstein barycenter for an infinite number measures. This formulation particularly relevant for the construction of an iterative algorithm to compute the barycenter for the streaming case, that is, when the measures are received online. The proofs are reported in Section C of the Appendix.
Let us consider probability measure Q E (R))) meaning that Q is supported on set of measures with finite moments of order 2, such that for some (and thus all) μ E (2R()) we have that SPA(Re) W (m, v))((v)) < oc.
Definition 2. We define the set of weak population barycenters of distribution Q E P2((((R))) as
arg min MEP2(R P2(Rd)
((ulv)d((v).
(8)
The following lemma guarantees that the map (I, + S (x) appearing in Eq. (8) through V (nlp)
| (1)))))) is well defined.
Lemma 3. The function (J,,V) E (P2(TR))) + THA E P2 (RR2d) mapping (H, v) to the unique optimal plan The" realising V (ulv) in Eq. (2) is continuous. As consequence, for each p. E P2(Rd) the function (x, v) E Id x (R + 5%(z) (x) is measurable.
Using similar arguments those of Proposition and the fact that any probability measure can be approximated by a sequence of probability measures with finite support, the following proposition confirms that the weak population barycenter problem is also well defined.
Proposition 3. The minimisation problem in Eq. (8) admits a solution.
4
Algorithms via fxed-point representations
4.1 Weak barycenter
For the Wasserstein barycenter problem in Eq. (4), the authors in [1] proved that if at least one of the measures Vi...., Vy, is a.c., the Wasserstein barycenter unique. Furthermore, all the v4 's are a.c., and at least one of them has bounded density, then the unique Wasserstein barycenter is also a.c. and verifies a fixed-point equation. This last property has been thoroughly studied by [5] and [44] and leveraged to compute an approximation of the barycenter via an iterative algorithm based on Monge maps, whose existence and uniqueness are guaranteed by the a.c. of the measures involved.
Akin to the fixed-point methodology in the classical Wasserstein scenario, we define an iterative procedure based on the barycentrie projection computed in the optimal weak transport problem in Eq. (2), that is valid for arbitrary distributions. Therefore, we consider the following iterative rule for
5

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probability measures V1, V1....: Un E P2
n2 Σ S1S
Mk+1 = G(us), with G(n)=
#1
where for each i L, ...., n, the optimal barycentric projection is given by Spi : x + J yd "i (v). for T/hM [1 14) achieving the minimum in the OWT problem in Eq. (2). The proposed iterative procedure is presented in Algorithm 1.
A fundamental difference between the fixed-point computation of the Wasserstein barycenter [5] and a weak barycenter is that the optimal Monge map T. in the OT problem verifies # = V, whereas the pushforward measure # in the OWT setting still depends on μ.. We will then prove that the iterative algorithm in Eq. (9), based on the maps S admits converging subsequences. A convenient result is the continuity of the functional G in Eq. (9), which be proven using Arzela-Ascoli theorem on set of barycentric projections as well as the Skorohod's representation theorem.
Theorem 4. The function μ -> (p)) defined Eq. (9) Wg-continuous from P2 (Rd) to P2((R)).
Using an approach similar to [5] for the Wasserstein barycenter, we can state the following results for the proposed ixed-point procedure.
Proposition 4. If μ is weak-burycenter, that is a solution of problem (5), then G(u)) f i.e. Ei X (1)), p(x))aa...
The inverse implication of Proposition 4 is not necessarily true, that is, some fixed points may not be weak barycenters. However, a Dirac delta , w E Rdd that meets the fixed-point condition 5_. = G(&)), is weak barycenter (see Lemma 1).
Proposition 5. Let (μt)) be the sequence defined by the iterative procedure pk+1 = G(())) and starting from Ho E (22(TR). Then (μK))x tight and every converging subsequence must converge to fixed point of G.
We observe that these results also hold for the classical Wasserstein barycenter of a.c. mea- sures {1:)i=1...m such that at least one of them has bounded density. Moreover, the inverse implication, namely if j is fixed-point then it is a barycenter, in the Wasserstein barycenter case, for which one considers the fixed-point equation given by = (2:-) Xr HH, with T the Monge map verifying v, = T # Indeed, [1] prove that if Ja checks r = Ei= ( (x) for every I E Rd, not only μ-almost everywhere, then is a Wasserstein barycenter. Also, [44, Theorem provide additional conditions for this to be true by essentially invoking more smoothness on the distributions {ii.i....... Additionally, they only conjecture that under the same assumptions, the fixed-point is unique. Our method, however, includes arbitrary probability measures. Therefore, we do not expect to obtain similar results as in the Wasserstein barycenter case, for which smoothness is required.
not straightforward even
4.2 Weak population barycenter
Based on [10], we construct a stochastic iterative algorithm for computing the weak population barycenter in Eq. (8). We clarify that [10] is constrained to probability measures Q supported on
distributions that are a.c., whereas in our setting these distributions only need to belong to P2 (IR.). Let us notice that our algorithms can be interpreted as geodesic gradient descent in [10] and [17], however, OWT is not metric and its potential geodesic structure is so far unknown. Therefore, the proposed algorithm only aims mimic Riemmanian gradient descent. Our fixed-point result for the weak population barycenter problem is stated in the following Lemma:
Lemma 4. If μ a weak population barycenter of Q, then x f ())) p(x)-a.s.
As in the finite case, the inverse implication is difficult to obtain. In particular, this has not been proven for the classical population Wasserstein barycenter in [10], where it boils down to prove the uniqueness of an absolutely continuous fixed point of - (j TV (Q(v)) where T is the Monge map between j and v. As explained in [10], the uniqueness of such fixed points has also been studied under some strong assumptions in [15] by considering parametric classes of random probability measures with compact support. This result is expected to true by again invoking more smoothness on the distributions at hand. As our method focuses (in particular) on discrete probability measures,

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the conditions under which the inverse implication holds are beyond the scope of our work. However, from the experimental results in Section 6, we believe our method presents practical advantages.
We next present an iterative scheme converging towards a distribution verifying the fixed-point equation in Lemma 4. This scheme is illustrated helow in Algorithm 2. To prove its convergence, will need a technical assumption on Q
There exists
> and R > 0 such that Q gives full measure to the set |x)2+<dp(x>) R).
(A)
KG = {u = /
Definition 3. Let μo E Ka Ltd. Q Qand 7he > We define the following iterative procedure:
p8+1 (1- 7/ -i)ddd + vusyyt] #ue 2 0,
(10)
where pp is the optimal barycentric projection between μa and wa and id the identity operator.
The following standard conditions on the steps 7/k will also be assumed:
the < 0
and
7h 00,
(1I)
km
Theorem 5. Assume Conditions in Eq. (11), (A) and moreover that every measure verifying the fixed-point equation = (K)))DQ( p(x))aass a weak barycenter. Then the sequence (J&))R in Eq. (10) is a.s. relatively compact w.r.. W2 and every limit point is weak barycenter.
The proof. provided in the supplementary material, inspired by the standard Wasserstein barycenter case studied in [10], [35]. Assumption (A) grants that the sequence in Eq. (10) remains in some compact set, and can be replaced by more general conditions (see Remark in Appendix).
Algorithm 1: Weak barycenter
Weak population barycenter Input: number of steps K; initialise distribution 140 ~ Q for 0, ,..,.... K do
Algorithm 2:
Input: distributions VI,..., Una # steps K;
initialisation: 140 = v/;
for k 0, 1,...... K do
for i = 1,2, ... do
Sample n² Q
Solve the OWT problem between Fa and v2 to obtain TuaM S. yd 24 (y)
Update Tk:
Solve the OWT problem obtain m
S ydm (y):
end
pk+1 = [11 rxdid + + Sk #pus
Mk+1 (E S.HHKK
end
end
5 Computational aspects
Setting and computation of OWTs. Both Algorithms and 2 require the computation of the optimal barycentric projection associated to the OWT problem in Eq. (2). For two discrete measures μ Ei a,on, and / = Σe= bydy,. this boils down to solving the following quadratic programming problem
min TEBTX
Σ as
TT
Tij 2 0, m1 =1=
= a, -T x
b
(12)
which can be solved using a solver such as cvxpy. We also propose to solve the OWT problem in Eq. (12) with proximal algorithm. The optimal barycentric projection is then constructed as sy The details and examples are presented in Appendix E.I.
Comparison setting. In the next section, we compare our proposed computation for weak barycenters in Definition (Algorithm 2) to the classic Wasserstein barycenter in particular for a stream of a.c. measures. Namely, will run Algorithm by, following [19, 36), replacing optimal barycentric
7

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60
OJ but Weak bar
0.
-15
-15
-000
-10
33
Figure 1: (left) Empirical Gaussian distributions and their OWT (black) and OT (red) barycenters for Gaussian observations (crosses) and corrupted observations (dots). (right) Empirical distributions supported on two ellipses and their OWT (black) and OT (red) barycenters.
projections by the barycentric projections associated either to i) an optimal plan in the Kantorovich problem (I), or ii) the optimal Sinkhorn plan in the entropy regularised OT problem [18] given by
/ ||x dzl +KK[(Tlu &v),
arg min
(13)
=ell(u.))
where KL. denotes the Kullback- Leibler divergence. The associated barycenters will be referred to as OT barycenter and OT Sinkhorn barycenter respectively. The optimal plans for OT and regularised OT problem were computed using POT toolbox [23]. Notice that what we call OT barycenter (resp. OT Sinkhorn barycenter) is not solving a Wasserstein barycenter problem (resp. a regularised Wasserstein barycenter problem). Therefore, in the literature (see Section 2) in that i) can process a stream an unknown number of measures, úi) does not require the measures to be a.c., and iii) does not appeal to additional regularisation of the measures or the Wasserstein metric.
method for barycentric computation differs from previous ones
Experimental results
This section is devoted to the empirical validation of our proposal on both synthetic and real-world data. We first focused on Algorithm since multiple algorithms to compute a Wasserstein barycenter for a fixed number of distributions are already available [19. 39j. We present two robustness to outliers experiments, then we validate our OWT barycenter on synthetic dataset and real-world ones. The overall conclusion of experiments is that the weak barycenter is more likely maintain the common (or shared) geometric features of the measures involved, as expected from Theorem 3. Additional experiments are presented in Appendix E.2, including the comparison of the energy for the computed weak barycenter in Algorithm against the approximated optimal energy (using (q. and the plug-in estimator).
6.1 Robustness to outliers
OT's sensitivity to outliers a well-known problem that can be addressed e.g. with unbalanced OT [11]. We observed that OWT also allows to deal with outliers, which is coherent with the latent variable interpretation (see Remark 1). We illustrate this with two experiments. In Fig. (left), we consider 50 sets of 20 - 30 observations from different 2D Gaussian measures, where each observation may be corrupted by random translations (Bernoulli = 0.05) thus producing outliers. We show the resulting barycenters (dots), and barycenters without outliers (crosses) for Wasserstein barycenter (red) and weak barycenter (black), which shows robustness to outliers. In Fig. 1 (right), consider distributions supported pair-of-ellipses, and 120 observations per distribution. Again, each observation may be corrupted by random translations (Bernoulli p = 0.05). The weak barycenter (black) shows better preservation of the shapes than the Wasserstein barycenter (red), in particular, the red dots are more often located outside the ellipses.
6.2 Synthetic distributions
We implemented the proposed sequential computation of weak barycenters (Algorithm 2) on two examples of synthetic distributions: Gaussians and spirals. In each case, we sampled observations from a random distribution at each step, and considered K steps (and thus K measures for each case).

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2D Gaussians (r = 100 & K = 15). We considered distributions (m,, I), with m uniformly distributed on (-3.3) x (-5, and the identity matrix. Fig. (left) shows the empirical distributions together with the OWT and OT barycenters, the weak barycenter being the less spread out as expected. The three remaining plots illustrate the behaviour of the barycenters constructed as stated in Sec. 5. For small regularisation parameter & in Eq. (13), the OT and OT Sinkhorn barycenters are similar, however, as e increases the OT Sinkhorn (OTS) barycenter becomes closer to the weak barycenter and thus even more concentrated, meaning that its samples tend to be closer to each other. Critically, for a very large &, as the entropy tends to spread the mass in the regularised optimal plan, the associated barycentric projection will roughly move the mass to the spatial mean of the target distribution's 's support.
g OT Siinnnn 2H Weak serycunter
OT Fnitorr, CHO1 Weac barycenter
OT Sweten, g-d Weak barycuntar
Weat banycetter
Figure 2: (left) Empirical Gaussian distributions and their OWT (black) and OT (red) barycenters computed with Algorithm 2. Illustration of the weak (black), OT (red) and OT Sinkhorn (blue) barycenters for different values of 0.1, 1.5.
Spiral distributions. (r (200, 225)< & K 10). In this experiment, we consid- ered distributions supported on a spiral- see Fig. 3 (left), with random ratio in (0,3), The OT and OWT barycenters are presented in Fig. 3 (right). Again, the weak barycenter seems to better pre- serve the shape of the spiral than the OT barycenter.
-10
-30
-15
-10
-20
-0
35
20
33
-10
Figure 3: (left) Distributions supported on spiral. (right) OWT (black) and OT (red) barycenters computed with Algorithm 2.
6.3 Real-world dataset
MNIST dataset. We considered the well-known MNIST dataset [31] of grayscale images of hand- written digits. The images, of size 28 x 28 pixels. can be normalised and thus be interpreted as discrete probability measures supported on a two-dimensional grid of size 28 x 28. We computed the barycenters with steps of Algorithm 1 between two digits "8", that are noisy versions of the same digit with the aim to produce more stable barycenter. To produce noisy data, we randomly (Bernoulli p = 0.1) move pixels of the prototype digit displayed in Fig. (left). Fig. 4 (right) shows the barycenters using the OWT, OT, and entropic-OT (for 1). This example illustrates how OWT reduces dispersion, so that weak barycenter provides the best uniformly spread results among the barycenters considered, with the two loops of the "8" well shaped.
Cytometry dataset. In biotechnology, flow cytometry is measured through intracellular markers of single cells in biological sample with the objective of recognising common features across patients. However, these measurements are often disrupted by acquisition, rather than biological artefacts [28], thus hindering the identification of common features. To address this challenge, we compute the weak barycenter for the the flowStatt package of Bioconductor [24]). We considered K 15 patients and variable number of cells per patient between 88 and 2185. Fig. shows the 15 distributions (left) and the computed barycenters (right), thus confirming the ability of the weak barycenter to resolve the alignment of the dataset, while maintaining the expected diamond-shape. Moreover, the advantage of our proposed streaming procedure is fully exploited in this setting, since data from or several patient can arrive
light (FSC) and
light (SSC) cell's markers (using

---

Regularised
Prototype "8"
1st noisy "g"
2nd noisy "8"
OWT
OT
OT
8
6
Figure 4: Digit "8" (MNIST). From left to right: Prototype "8", first and second noisy versions of the prototype by randomly (Bernoulli 0.1) moving pixels, three barycenters constructed with Algorithm associted to the OWT plan, OT plan and the entropy regularised OT plan for 1.
sequentially. Though this setting has been addressed with the Wasserstein barycenter in [14), also in Fig. 5, such method required a fixed grid to compute the barycenter unlike our method, thus revealing the computational simplicity the weak barycenter.
OT Weak Luryenier
10
Figure 5: (left) Cytometry dataset for n 15 patients and FSC vs. SSC cell's marker. (right) The weak-barycenter (black) computed with Algorithm 2 and the OT barycenter (red). The data are represented with the same axis as the figure of barycenters.
7 Discussion
We have introduced the weak barycenter, which extracts common geometric information of probability measures on IRd based on optimal weak transport, and showed that it can be interpreted as latent variable model. From the fixed-point formulation defined in terms of optimal weak transport maps, irrespective of the regularity assumptions on the measures involved, we developed practical computation via an iterative algorithm with guaranteed convergence. algorithms do not require common grid on the sample space, when processing either observed data or samples from distributions. We have also proposed weak barycenters of a possibly infinite population of measures and developed stochastic procedure for computing it in the streaming data regime where distributions are processes into the weak barycenter as they arrive. This has critical implications for continual-learinng methods in the ML community.
particular, the proposed
Additional studies will focus on deepen the latent variable interpretation of weak barycenters, and its relationship to the aggregate information represented by the Wasserstein barycenter. Also, we identify two relevant theoretical aspects for further research: i) to exhibit general conditions on family of input measures (or on the law of the population) for the existence of weak barycenters that are not Dirac masses; and ii) to provide conditions on those input measures for a "maximal" weak barycenter (in terms of convex ordering) exist when d 2, among all the solutions of the weak barycenter problem (and, possible, a way of constructing it by regularisation most probably). The statistical behaviour of the weak barycenter can also be investigated, in particular when constructed from large empirical random samples of given distributions. Lastly, the weak population barycenter could also be used to construct predictive posterior in the context of Bayesian learning, was done for Wasserstein barycenters in [35].
Acknowledgments. We thank Julio Backhoff- Backhoff-Veraguas for his valuable insight during the writing of this paper. This work was funded by ANID grants: AFB170001 & ACE210010 (CMM), FB0008 (AC3E),
#3 #3190926 and Fondecyt-Regular #1210606.
10

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References
[1] M. Aguch and G. Carlier. Barycenters in the Wasserstein space. SIAM Journal Mathematical Analysis, 43(2):9049924, 2011.
[2] A. Alfonsi, J. Corbetta, and B. Jourdain. Sampling of probability measures in the convex order and approximation of martingale optimal transport problems. Available SSRN 3072356, 2017.
[3] Jason M Altschuler and Enric Boix-Adsera. Wasserstein baryeenters can be computed in polynomial time in fixed dimension. J. Mach. Learn. Res., 22:44-1, 2021.
[4] P.C. Alvarez-Fstchan, E. Del Barrio, Cuesta-Albertos, and C. Matrán. Wide consensus for parallelized inference. arXiv: 151 2015.
[5] P.C. Álvarez-Esteban, E. Del Barrio, J.A. Cuesta-Albertos, and C. Matrán. A fixed-point approach to barycenters in Wasserstein space. Journal of Mathematical Analysis and Applications, 441(2):744- 762, 2016.
[6] L. Ambrosio, N. Gigli, and G. Savaré. Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media, 2008
[7] M. Arjovsky, S. Chintala, and L. Botton. Wasserstein generative adversarial networks. In Proceedings of the 34th International Conference on Machine Learning, volume 70, pages 214-223. PMLR, 2017.
[8] J. Backhoff- Backhoff-Veraguas, M. Beiglböck, and G. Pammer. Existence, duality, and cyclical monotonicity for weak transport costs. Calculus of Variations and Partial Differential Equations, 58(6):203, 2019.
[9] J. Backhoff- Backhoff-Veraguas, M. Beiglbück, and G. Pammer. Weak monotone rearrangement on the line. Electronic Communications in Probability, 25, 2020.
[10] J. Backhoff- Backhoff-Veraguas, J. Fontbona. G. Rios, and F. Tobar. Stochastic gradient descent in Wasserstein space. arXiv preprint arXiv: 2201 04232, 2022.
[11] Y. Balaji, R. Chellappa, and S. Feizi. Robust optimal transport with applications in generative modeling and domain adaptation. arXi::2010.0566,, 2020.
[12] A. Beck and M. Tcboulle, A fast iterative SIAM journal on imaging sciences, (1)):183-202 2009.
algorithm for linear inverse problems.
[13] M. Beiglbück, P. Henry-Labordère, and F. Penkner. Model-independent bounds for option prices-a mass transport approach. Finance and Stochastics, 17(3)447-501, 2013.
[14] J. Bigot, E. Cazelles, and N. Papadakis. Data-driven regularization of Wasserstein barycenters with application to multivariate density registration. Information and Inference: A Journal of the IMA, 8(4):7197755, 2019.
[15] J. Bigot and T. Klein. Characterization of barycenters in the Wasserstein space by averaging optimal transport maps. ESAIM: Probability and Statistics, 22:35-57, 2018.
[16] M.W. Botsko. An elementary proof of Lebesgue's differentiation theorem. The American Mathematical Monthly, 11009):834 838, 2003.
[17] S. Chewi, T. Maunu, P. Rigollet, and A.J. Stromme. Gradient descent algorithms for Bures-Wassersee Wasscrstein barycenters. arXiv: 2001.01700, 2020.
[18] M. Cuturi. Sinkhorn distances: Lightspeed computation of optimal transport. In Advances in Neural Information Processing Systems 26, pages 2292-22222 2300. 2013.
[19] M. Cuturi and A. Doucet. Fast computation of Wasserstein barycenters. Proceedings of the 31st Interna- tional Conference on Machine Learning, 32(2))685-693, 2014.
and [20] A. Dessein, N. Papadakis, and J.-L. Rouas. Regularized optimal transport and the ROT mover's distance.
The Journal of Machine Learning Research, 191):590-644, 2018.
[21] D. Dvinskikh. Stochastic approximation versus sample average approximation for population Wasserstein barycenters. arXiv e-prints, pages arXiv-2001, 2020.
[22] E.A. Feinberg, P.O. Kasyanov, and Y. Liang. Fatou's lemma in its classical form and Lebesgue's con- vergence theorems for varying measures with applications to Markov decision processes. Theory of Probability & Its Applications, 65(2):270-291 2020.

---

[23] R. Flamary, N. Courty, A. Gramfort, M.Z. Alaya, A. Boisbunon, S. Chambon, L. Chapel, A. Corenflos. K. Fatras, N. Fournier, L. Gautheron, N. T.H. Gayraud, H. Janati, A. Rakotomamonjy, I. Redko, Rolet. A. Schutz, V. Seguy, D.J. Sutherland, R. Tavenard, A. Tong, and Vayer. Pot: Python optimal transport. Journal of Machine Learning Research, 22(78): 2021.
[24] R.C. Gentleman, Carey, D.M. Bates, B. Bolstad, M. Dettling, S. Dudoit, B. Ellis, L. Gautier, Ge. J. Gentry, et al. Bioconductor: open software development for computational biology and bioinformatics. Genome biology, Soooogy.5(00).......ooo
2004.
[25] N. Gozlan and N. Juillet. On a mixture of Brenier and Strassen theorems. Proceedings of the London Mathematical Sociery, 120(3):334 463, 2020.
[26] N. Gozlan, C. Roberto, P.MM Samson, and Tetali. Kantorovich duality for general transport costs and applications. Journal of Functional Analysis, (73((1):3327-233) 2017.
[27] J. Guyon and P. Henry-Labordere. Nonlinear option pricing. CRC Press, 2013.
[28] F. Hahne, A.H. Khodabakhshi, A. Bashashati, C.-J. Wong, R.D. Gascoyne, A.P. Weng. V. Seyfert-Margolis,
K. Bourcier, A. Asare, T. Lumley, et al. Per-channel basis normalization methods for flow cytometry data. Cytometry Part A: The Journal of the International Society for Advancement of Cytometry, 77(2):21 2010.
[29] R.P. Kertz and U. Rösler. Complete lattices of probability measures with applications to martingale theory. Lecture Notes-Monographh Series, pages 153-177, 2000.
[30] T. Le Gouic and J.-M. Loubes. Existence and consistency of Wasserstein barycenters. Probability Theory and Related Fields, 168(3-4):901999)) 2017.
[31] Y. LeCun. The MNIST database of handwritten digits. htlp://yann. lecun. com/exal/nnistf/ 1998.
[32] L.. Li, A. Genevay, M. Yurochkin, and J. Solomon. Continuous regularized wasserstein barycenters. arXiv preprint arXiv:2008. 12534, 2020.
[33] X. Nguyen. Convergence of latent mixing measures in finite and infinite mixture models. The Annals of Statistics, 41(1):370-400, 2013.
[34] G. Peyré and M. Cuturi. Computational optimal transport. Foundations and Trends in Machine Learning, (1(56))355-6077 2019.
[35] G. Rios, J. Backhofft Veraguas, J. Fontbona, and F. Tobar. Bayesian learning with Wasserstein barycenters. arXiv preprint arXiv: /805 10833v4, 2018.
[36] V. Seguy, B.B. Damodaran, R. Flamary, N. Courty, A. Rolet, and M. Blondel. Large-scale optimal transport and mapping estimation. In International Conference on Learning Representations (ICLR). 2018
[37] Moshe Shaked and J George Shanthikumar. Stochastic orders. Springer Science & Business Media, 2007.
[38] Solomon, F. De Gocs, G. Peyré, M. Cuturi, A. Butscher, A. Nguyen, T. Du, and L. Guibas. Convolutional Wasserstein distances: Efficient optimal transportation on geometric domains. ACM Transactions on Graphics (TOG), 34(4):1-11, 2015. 2015.
[39] M. Staib, S. Claici, J. Solomon, and S. Jegelka. Parallel streaming wasserstein barycenters. In Advances in Neural Information Processing Systems, pages 2647-2658, 2017.
[40] V. Strassen. The existence of prohability measures with given marginals. Annals of Mathematical Statistics, 36(2):423-439 1965.
[41] C. Villani. Topics in Optimal Transportation. Number 58, American Mathematical Soc., 2003.
[42] C. Villani. Optimal Transport: Old and New. Springer Berlin Heidelberg, 2008.
[43] Ye, P. Wu, J. Z. Wang. and Li. Fast discrete distribution clustering using Wasserstein barycenter with sparse support. IEEE Transactions on Signal Processing, 65(9):2317 2332, 2017.
[44] Zemel and Panaretos. Fréchet means and Procrustes analysis Wasserstein space. Bernoulli, 25(2):932-976, 2019.
12


Reviews:
[{"review_id": "eyhTdbTG-lR", "reviewer": "Reviewer_KZXz", "summary": "This paper studies barycenters of probability distributions using the weak transport distance, and establishes a number of results which mirror the theory of standard optimal transport barycenters that has been established over the past 10 years. In particular, they investigate existence and uniqueness, the first order optimality criterion, as well as iterative methods to compute weak barycenters. They propose a proximal sub-routine for this iterative method, and evaluate the performance of weak barycenters on a number of examples, comparing with standard barycenters and Sinkhorn barycenters.", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "The paper is the first to study weak optimal transport barycenters, and establishes a number of desirable results for this new notion. In particular, the authors establish existence (in both finitely supported and non-finitely supported Q cases) and some optimality properties of weak barycenters, such as closure under the convex order. They also give a latent variable characterization of the weak barycenter and study first order methods for computing the weak barycenter. Finally, they provide convincing and clear empirical results that there is significant value in this weak barycenter notion.\n\nOverall, I believe this is a quality, well-written submission. Although the technical results are largely based off porting over proofs from the standard case, I think the convincing empirics demonstrate that this is certainly a valuable notion, and the breadth of technical results is of substantial value for the literature.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The authors adequately discuss limitations (including references for the standard barycenter version of their proofs plus valid directions for extensions) and societal impact.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "7z7ukseSYiU", "reviewer": "Reviewer_kNgK", "summary": "This paper introduces and study Fréchet means (aka barycenter) problem for optimal weak transport (OWT), a variant of the standard optimal transport problem which presents in particular the benefits of guaranteeing the existence of optimal transport *map* without needing regularity assumptions on the measures. They prove theoretical properties of these objects (existence, characterization) and, using a fixed-point equation that any (OWT) barycenter must satisfy, they design an interative algorithm that is guaranteed to converge toward such a fixed-point (which, however, may not be a OWT barycenter). ", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "Quality:\nThe paper introduces an interesting optimization problem and studies it properly. The theoretical content is solid, it comes with algorithms + guarantees, and numerical experiments - though not ground-breaking as restricted to dimension 2 - support the approach in an informative way.\n\nClarity: \nThe paper is well-written and pleasant to read. \n\nOriginality / Significance: \nOWT is a fairly new topic in the Optimal Transport field and this paper addresses a natural question when new OT models are proposed. As such, it may open the way for interesting theoretical and applied use of O(W)T in machine learning and other fields. Important questions remain open, in particular the existence (and perhaps the estimation) of a ''maximal'' OWT barycenter when $d \\geq 2$. In particular, the discrepancy between the existence of a \"trivial\" OWT barycenter (Lemma 1) and the (interesting) output of the proposed Algorithms suggest that some gap should be filled in the understanding of fixed-points of the map $\\mu \\mapsto G(\\mu)$. \n\n\nMinor comments and suggestions:\n- As (for once!) we exactly know the objective value of a (true) OWT barycenter (Eq 7), it may be enlightening to show how does the output of the proposed Algorithms perform empirically in terms of energy. Perhaps does it avoid spurious local minima (if non-adversarialy initialized) and actually converges toward a true OWT barycenter \"in practice\". \n- Figure 9 (Pair of ellipses) may be worth including in the main paper as it feels quite convincing. \n- It may be nice to present an experiment (perhaps the Pair of ellipses?) in the formalism of Theorem 3, showing that (hopefully) one retrieves (some valid) $X$ given the $Y_i$. \n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The paper mentions some of its limitations/questions left. I did not identify negative social impact that may be specific to this work.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "5"}, "extra_text": {}}, {"review_id": "2gcvXO5I1gD", "reviewer": "Reviewer_Ee1U", "summary": "Computing the barycenter of a finite set of probability measures is an important task in various statistical and machine learning problems, and is usually done by comparing the measures with a specific divergence. This divergence plays a significant role as it affects the properties and practical performance of the resulting barycenter, and should then be carefully chosen. Motivated by the attractive theoretical properties of the Wasserstein distance, which emerges from optimal transport (OT) theory, the Wasserstein barycenter has been introduced and extensively studied from a theoretical and practical point a view.\n\nThis paper introduces a novel barycenter of measures that is closely related to the Wasserstein barycenter, since the comparisons are made by solving OT problems: the authors introduce the \"weak barycenter\" (Definition 1), which is based on the recently introduced \"optimal weak transport\" problem, and study its theoretical properties. Specifically, they prove the existence of a solution to the optimization problem that defines the weak barycenter (Proposition 1), and they provide a characterization of the solutions (Lemma 1, Proposition 2). They also bound the deviation between the weak and Wasserstein barycenters (Lemma 2), and show that the weak barycenter is linked to a specific latent variable (Theorem 3). \n\nThen, the authors introduce the \"weak population barycenter\" to average an infinite set of probability measures (Definition 2), which can be used for online settings. This barycenter correponds to an optimization problem, which is obtained by extending the definition of the weak barycenter and is well defined (Lemma 3, Proposition 3). \n\nThe authors prove that the weak and weak population barycenters are necessarily fixed points of specific operators (Proposition 4, Lemma 4), and they develop two iterative procedures (Algorithms 1 and 2, supplementary document) that are guaranteed to converge to such fixed points (Proposition 5, Theorem 5). The computational requirements of the two algorithms are investigated in Section 6. Finally, several experiments are conducted to compare the empirical performance of the weak barycenter against the Wasserstein barycenter (Section 7).\n", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "This paper introduces a new methodology to average probability distributions, even if these are provided as a stream. Besides, it is based on optimal transport theory, which has received increased attention in recent years from the statistics, optimization and machine learning community. Therefore, this work is relevant for the NeurIPS community.\n\nThe contributions are obtained by adapting existing results from the literature on the Wasserstein barycenter (for example, the two algorithms and their properties are largely inspired by \"A fixed-point approach to barycenters in Wasserstein space\", Alvarez-Esteban et al.), and in that sense, they are not very original. This does not affect the quality of the work in my opinion: the theoretical findings are technically sound, and I appreciate that the authors provided some explanations on what they imply intuitively (e.g., Lemma 2 and Theorem 3). \n\nHowever, my main concern, which prevents me from giving a positive score for now, is that it is not clear enough why weak barycenters should be used in practice. Therefore, I strongly encourage the authors to emphasize more the significance of their contribution. \nOne clear advantage is that the weak population barycenter (and Algorithm 2) can process streaming data. However, results in Section 7.1 and 7.2 only illustrate that the weak population barycenter is more concentrated than the \"OT barycenter\". On the other hand, the experiment on the cytometry dataset (Section E.3) seems more promising in illustrating the benefits of Algorithm 2, thus could be more discussed in the main document. \nBesides, I am not fully convinced by the superior performance of the weak barycenter (and Algorithm 1), since Section 7.3 only shows qualitative results for a single example from the MNIST dataset; the analysis should present more examples supporting this conclusion (e.g., similarly to Section 4 in \"Continuous Regularized Wasserstein Barycenters\", Li et al.).\n \nWhile experiments in Section 7 also explore interesting properties that this paper do not study theoretically (the robustness to outliers and relation with OT Sinkhorn barycenters), I think it would be more relevant to illustrate more the consequences of the theoretical results established in Sections 3, for example Lemma 2 and Theorem 3 (the link between robustness to outliers and Theorem 3 is not trivial), as well as the computational aspects of the algorithms (as in Section 5 in \"A fixed-point approach to barycenters in Wasserstein space\", Alvarez-Esteban et al.). \n\nRegarding clarity, the paper is well written (some minor typos are listed at the end of my review), but quite dense. Besides, I suggest incorporating the following changes in the organization, for better readability:\n- A subsection on the Wasserstein barycenter could be added in the background section (Section 2), to recall the definition (l.105-106) and contributions from the literature that are relevant for this work (including Section 6.1).\n- Section 3 and 4 could be merged. Same suggestion for Section 5 and 6 (or 6 and 7).\n- The explanation on the algorithms for computing the weak barycenters would be much clearer if Algorithms 1 and 2 (or only Algorithm 1) are given in the main document, rather than in the supplementary doc.\n\nTypos:\n\nl.121: missing parentheses for the expectations of $Y_i$\n\nl.230: \"an iterative procedure that converge\" -> \"an iterative procedure that converges\"\n\nl.278: \"We first focused in Algorithm 2\" -> \"We first focused on Algorithm 2\"\n\nl.303: \"$p \\in (200, 225)$\" -> \"$r \\in (200, 225)$\"\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "The limitations of this work have been adequately addressed, but not the societal impact.", "review_text": "", "extra_scores": {"ethical_concerns": "I don't think this submission raises any ethical issues.", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "8"}, "extra_text": {}}, {"review_id": "0NWM580me_2", "reviewer": "Reviewer_qhuc", "summary": "The present manuscript introduces the natural extension of the concept of the Wasserstein Barycenter to the barycenter based on optimal weak mass transport.  Some analytical, statistical and computational properties of this object (the barycenter based on optimal weak mass transport) are discussed and shown. Some numerical experiments are shown too.", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "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": "Apart from some minor typos here and there ( e.g., eq. instead of Eq.) , the paper is well written, and I would say it is tailored for a rather advanced audience that is familiar with the problem of OT and its nuances. With that being said, the introduced concept of weak barycenter is a rather natural extension once once has the weak Wasserstein distance. \n\nI think this paper is very nice as it presents a number of properties for the studied object with some computational an statistical aspects of it. This is the strong part of the paper, and it provides a nice presentation of the problem. However, some of the weaknesses of the present form of  the manuscript need to be improved. Find them below in no particular order.\n\n-  While the weak barycenter is a nice object, I could not find a good or convincing motivation (either computational, statistical, or in applications) for which one would like to compute the weak barycenter. \n\n- In a couple of instances where claims are made with little or no justification, for example in the experiments it is read \" weak barycenter being the most 297 concentrated as expected.\" I have no reason to expect that the weak barycenter will be more concentrated.\n\n- While the relation to the outliers is nice, this comes out of the blue and with no theoretical justification why. Outlier detection is a rather developed field with its own tools to quantify its properties, and I feel this is not developed in this manuscript and read as some experiment to run, with no justification on why robustness is induced. For example, we know that the median is \"more robust\" than the mean because we need more than 50% of outliers to drive the mean to an arbitrary position. How can we quantify the robustness of the weak barycenter?\n\n- The numerical section is the reviewers opinion the weakest part of the paper. The plots are very nice and interesting, but little to no discussion is made to connect the numerical results to the theoretical properties of the weak barycenter. For example, what is the conclusion from the 8 plots in figure 4 and 5? It is very hard for me to really obtain any new information from them.  Some numerical experiments are needed that really highlight the statistical, computational, and analytical properties of the weak barycenter. I do no see that in the present numerical results. \n\n- Some minimal discussion about the computational complexity of the weak barycenter is needed. Traditional Wasserstein barycenter is know to be very computationally intensive to compute. is the weak barycenter easier or harder to compute? \n\nMinor\n\n-The term \"concentrated\" is used a couple of times as a property of a probability distribution. Even if this is an intuitive concept, some formal definition would be greatly appreciated.\n\n\n- ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "N/A", "review_text": "", "extra_scores": {"ethical_concerns": "N/A", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4"}, "extra_text": {}}]

Author rebuttal:

Meta-review: All the reviewers insisted on the quality of the exposition of the paper, which is able to import advanced notions from OT to the ML community. This being said, the paper only partially motivates (from both a theoretical and numerical perspective) the relevance of weak OT for ML. For this reason, and after discussing with the reviewers, I believe that the paper is borderline for acceptance. I nevertheless proposed acceptance. I strongly urge the authors to take into account some of the suggestions of the comments of the reviewers in the final version. 
Decision: Accept (Poster)