Tangential Wasserstein Projections
Authors: Florian Gunsilius
Meng Hsuan Hsieh
Myung Jin Lee
Venue: Submitted to ICLR 2023 (2023)
Keywords: Optimal Transport
Wasserstein
Generalized geodesics
Projection
Tangent Cone
Causal Inference

Abstract: We develop a notion of projections between sets of probability measures using the geometric properties of the $2$-Wasserstein space. It is designed for general multivariate probability measures, is computationally efficient to implement, and provides a unique solution in regular settings. The idea is to work on regular tangent cones of the Wasserstein space using generalized geodesics. Its structure and computational properties make the method applicable in a variety of settings, from causal inference to the analysis of object data. An application to estimating causal effects yields a generalization of the notion of synthetic controls for systems with general heterogeneity described via multivariate probability measures, as well as a way to estimate optimal weights jointly over all time periods.

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TANGENTIAL WASSERSTEIN PROJECTIONS
Anonymous authors Paper under double-blind review
ABSTRACT
We develop notion of projections between sets of probability measures using the geometric properties of the 2- Wasserstein space. In contrast to existing methods, it is designed for multivariate probability measures that need not be regular, is com- putationally efficient to implement via a linear regression, and provides a unique solution in general. The idea is work tangent cones of the Wasserstein space using generalized geodesics. Its structure and computational properties make the method applicable
variety of settings where probability measures need not be regular, from causal inference to the analysis of object data. An application to estimating causal effects yields a generalization of the synthetic controls method for systems with general heterogeneity described via multivariate probability mea- sures, something that has been out of reach of existing approaches.
INTRODUCTION
The concept of projections, that is, approximating a target quantity of interest by an optimally weighted combination of other quantities, is of fundamental relevance in learning theory and statis- ties. Projections are generally defined between random variables in appropriately defined linear spaces (e.g. |van der Vaart 2000 chapter 11). In modern statistics and machine learning applications, the objects of interest are often probability measures themselves. Examples range from object- and functional data (e.g. Marron Alonso 2014) to causal inference with individual heterogeneity (e.g.
Athey & Imbens, 2013).
A notion of projection between sets of probability measures should be applicable between any set of general probability measures, replicate geometric properties of the target measure, and possess good computational and statistical properties. We introduce such notion of projection between sets of general probability measures supported on Fuclidean spaces. It provides unique solution to the projection problem under mild conditions. To achieve this, we work in the Wasserstein space, that is, the set of all probability measures with finite second moments equipped with the 2- 2-Wasserstein distance.
Importantly, we focus on the multivariate setting, i.e. we consider the Wasserstein space over some Euclidean space IRd, denoted by Wz, where the dimension can be high. The multivariate setting poses challenges from mathematical, computational, and statistical perspective. In particular, W2 a positively curved metric space for over, the 2- 2-sssserstein distance between two probability measures is defined as the value function of the Kante--intooovic optimal transportation problem [Villani 2003, chapter 2), which does not have a closed-form solution in multivariate settings. This coupled with well-known statistical curse of dimensionality for general measures (Ajtai et al. 1984 Dudley 1969 Fournier & Guillin
(c.g. Ambrosio et al. 2008 Kloeckner 2010). More-
2015
Talagrand 1992 1994 Weed Bach 2019).
1.1
EXISTING APPROACHES
These challenges have impeded the development of a method of projections between general and potentially high-dimensional probability measures. A focus so far has been on the univariate and low-dimensional setting. In particular, Chen et al.] (2021), Ghodrati & Panaretos (2022), and PP- goraro & Beraha (201)) introduced frameworks for
regressions in the
univariate setting for object data. Bigot et al. (014)), [azzllls et al. (201)) developed principal component analyses on the space of univariate probability measures using geodesics on the Wasser- stein space.

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The most closely related works to ours are Bonneel et al. (2016), Mérigot et al. (020)), and Werenski et al. (2022). The first develops a regression approach in barycentric coordinates with applications in computer graphics well as color and shape transport problems. Their method is defined directly on W2 and requires solving a computationally costly bilevel optimization problem, which does not necessarily yield global solutions. The second introduces linearization of the 2- 2-Wasserstein space by lifting it to a L -ppcce anchored at measure that is absolutely continuous with respect to Lebesgue measure. This approach relies on the existence of optimal transport maps between this absolutely continuous "anchor" distribution and other distributions and hence only defines tangent spaces at absolutely continuous measures. The third works on a tangential structure based on "Karcher means" (Karcher, 2014 Zemel& & Panaretos] 2019). which is more restrictive still. This implies that their method requires all involved measures to be absolutely continuous measures with densities that are bounded away from zero, with the target measure lying in the convex hull of the control measures.
1.2 OUR CONTRIBUTION
In contrast to the existing approaches, our method is applicable for general probability measures, allows for the target measure to be outside the generalized geodesic convex hull of the control mea- sures, can be implemented by standard constrained linear regression, and provides a global-and and in many cases unique-soluiee
The proposed method transforms the projection problem on the pos- itively curved Wasserstein space into a linear optimization problem in the geometric tangent cone, which can be implemented via a linear regression. This problem takes the form of a deformable template (Boissard et al., 2015 Yuille 1991). which connects our approach to this literature. Our method can be implemented get measure, (ii) construct tangent space from the tangent cone via barycentric projections if it does not exist, and (iii) perform a linear regression to carry out the projection in the tangent space. This implementation of the projection approach via linear regression is computationally efficient, in particular compared to the existing methods in Bonneel et al. (2016) and Werenski et al. (002)..
three steps: (i) obtain the general tangent cone structure at the tar-
The challenging part of the implementation is lifting the problem to the tangential structure: this requires computing the corresponding optimal transport plans between the target and each measure used in the projection. Many methods have been developed for this, see for instance [Benamou & [Jacobs & Léger 2020), Makkuva et al. (2020). Peyré & Cuturi 2019). |Ruthotto et al. (2020) and references therein. Other alternatives compute approximations of the optimal transport plans via regularized optimal transport problems (Peyré & Cuturi 2019), such as entropy regularized optimal transport (Galichon & Salanie, 20100 Cuturi 2013). The proposed projection approach is compatible with any such method, therefore its complexity scales with that of estimating optimal transport plans. We provide results for the statistical consistency when estimating the measures via their empirical counterparts in practice.
Brenier jJaoossaacbbs
To demonstrate the efficiency and utility of the proposed method, we apply our method in different settings and compare it to existing benchmarks such as Werenski et al. (2022). Furthermore, we extend the classical synthetic control estimator (Abadic & Gardeazabal] 2003 Abadie et al.) 2010) to settings with observed individual heterogeneity in multivariate outcomes. The synthetic controls estimator is a projection approach, where one tries to predict an aggregate outcome of a treated unit by an optimal convex combination of control units and use the weights of this optimal combina- tion to construct the counterfactual state of the treated unit had not received treatment. The novelty of our application that it lets us perform the synthetic control method on the joint distribution of several outcomes, which complements the recently introduced method in Gunsilius (2022) designed for univariate outcomes. The possibility to project entire probability measures allows us to disen- tangle treatment heterogeneity at the treatment unit level. The possibility of working with general probability measures key in this setting, as many outcomes of interest are not regular. We illus- trate this by applying our method to estimate the effects of a Medicaid expansion policy in Montana, where we consider- as outcome-noo non-rellar probability measure in d' = 28 dimensions.
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2 METHODOLOGY
2.1 THE 2- WASSERSSTNN SPACE W2(R2)
The 2-Wasserstein Distance For probability measures Px, Py E PR with supports X, _
Rd respectively, the 2- W-sssersteinn distance W2(Px, Py) is defined as
Wx(Px,Py)
min (x)
1z vl vfd77
(2.1)
Here, |. denotes the Euclidean norm F and
T(Px, Pv) - {rr PRR RR) : (7 Px, (T2)#7= = pr}
the of all couplings of Px and Py. The maps тI and m2 are the projections onto the first and second coordinate, respectively, and Tx denotes the pushforward measure of P via T' i.e. for any measurable A C V. Ty P(A) =\t PAT-) An optimal coupling E [(Px. Py) solving the optimal transport problem equation 2. optimal transport plan. By Prokhorov's theorem, a solution always exists in our setting. When Px regular, i.e. when it does not give mass to sets of lower Hausdorff dimension in its support, then the optimal transport plan 7 solving equation 2.1 is unique and takes the form = (Id x V lxxx where Id is the identity map on Rd and V the gradient of some convex funetion. This result known as Brenier's theorem (Brenier 1991 McCann 1997 Villani 2003 Theorem 22) By definition, all measures that possess density with respect to Lebesgue measure are regular. Our main contribution is to allow for general probability measures, where only optimal transport plans but no maps exist.
The 2-Wasserstein Space The 2-Wasserstein space W2 (92(())) W22) is the metric space defined on the set (RR)) of all probability measures with finite second moments supported on Rd, with the 2- Wasserstein distance the metric. It is geodesically complete space in the sense that between any two measures P, P E W2, one can define a geodesic P. [0, [] + W2 via the interpolation Pt - (TT)47, where y is an optimal transport plan and IRd x IRd + IR is defined through (ar, y) A (1-1)x++ (Ambrosio et al. 2008 McCann 1997). ). Using this, it can be shown (Ambrosio al.J 2008 Theorem 7.3.2) and flat for d (Kloeckner,) 2010), where curvature is defined in the sense of Aleksandrov (195I). This difference in the curvature properties is the main reason for why the multivariate setting requires different approaches compared to the established results for measures the real line.
that W2 is a positively curved metric space
2.2 TANGENT CONE STRUCTURE ON W₂
We exploit a tangential structure that can be defined for general measures on W2 (Ambrosio et al., 2008 Otto, 2001 Takatsu Yokota 2012). In particular, it allows us to circumvent solving bilevel optimization problem as the one considered in Bonneel et al. (2016), whose statement we have included in the appendix.
The tangential structure relies on the fact that geodesics Pt in W2 are linear in the transport plans (#r) This implies a geometric tangent cone structure at each measure W that can be defined as the closure in P2 of the set
G(P) {rE L (R x RR (#1)#7 P. (MI, + ex2) #7 y is optimal for some > o}
with respect to the local distance
{
W3 m1s) A min
[n0 sn5l d7123 : 7123 T1(71 713)
(2.2)
(R:43
where 712 and 7i3 are couplings between P and some other measures P3 and Pa, respectively, and T1(722 Tha) ) is the set of all 3-couplings 7123 such that the projection of 7/23 onto the first two elements is 722 and the projection onto the first and third element is 713 (Ambrosio et al. 2008
3

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Appendix 12). We can then define the exponential map P with respect to tangent element y E G(P) by
expr(7) (T) #2)
This tangent cone can be constructed every P E W. irrespective of its support; in particular, we do not assume that the corresponding measures are regular, i.e., give mass subsets of TRd of lower Hausdorff dimension. In the where P is regular the tangent cone structure reduces space (Ambrosio et al. 2008 Theorem 8.5.1). This tangent space structure has been exploited in Mérigot al. (2020) and Werenski al. (2022), and we include the results for our projection approach in this special case in Appendix A
tangent
2.3 TANGENTIAL WASSERSTEIN PROJECTIONS
Our main contribution to define projection approach between general probability measures, where the target need not be regular. To define this notion of projection, we need to first define an appropriate notion geodesic convex hull. The novelty here that we define this notion via generalized geodesics (Ambrosio et al. 20088 section 9.2) centered at the target measure Po- For this, we extend the definition Wp to the multimarginal setting, by defining, for given couplings Juy jy). [
W Po:A(701, (701, 702..... (701;702,....... 70.) A min
ΣdE [. mol2 dr (1o1 7057) j
(2.3)
where (7..,....., 70J) S. I(Po, Pi,...,PP) is the set of all (J cooupling y such that the projection of y onto the first- and j-th element is 70j. Note that this definition is similar to the mul- timarginal definition of the 2-Wasserstein barycenter (Aguch & Carlier 2011 Gangbo & Swięch, 1998). but "centered" at Po. Based on this, we define the generalized geodesic convex hull of mea- sures [Pa)e with respect to the measure Po as
Coro ((B))z=)) & 2 P()) E 3xRR))
Σ XyTs+11 #7,
AEAr
7 solves Po:a Wo (701........ Yo.); Voj is optimal in I'Po, PJ) Vj [jE
(2.4)
direct application of our tangential projection idea would lead us to solving
X arg min Po: (7.. XEA
(2.5)
which would be computationally prohibitive bilevel optimization problem similar to the in Bonneel et al.] (2016). We therefore rely on barycentric projections to reduce the general cone structure to a regular tangent space which denote by Tr, Wa (Ambrosio al. 2008). In this structure the projection problem equation 2.5 is replaced by
with byo, (xI) & / 22 dr0f.m (x2)
x é arg min XEA
Σ Ay (bros Id)
(2.6)
(.3(P))
denoting the barycentric projections of optimal transport plans Yoy between Po and Ps. Here, 7a denotes the disintegration of the optimal transport plan 7 with respect to Po.
This approach is a natural extension of the regular setting to general probability measures for two reasons. First, if the optimal transport plans To are actually induced by some optimal transport map, then byo, reduces to this optimal transport map; in this case the general tangent cone (PP) reduces to the regular tangent cone Tra W2 (Ambrosio et 2008 Theorem 12.4.4). Second, by the

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Tp Wa
1d
Po
Coru
W:
Figure 1: Tangential Wasserstein projection for general target Po-
Tr, Wz is the regular tangent space constructed by applying barycentric projection (Po)). the general tangent anchored Po. Thick dashed lines are tangent vectors (b, Id) generated by the respective barycentric projections. The gray shaded region their convex hull this constructed tangent space and the projection of Id onto this convex hull. Pa & expa, (x) is the projection of Po onto the generalized geodesic convex hull Cor, ((Pi, P2, Ps)) C W2 (blue).
definition of by and disintegrations in conjunction with Jensen's 's inequality it holds for all X E A that
(br
Id)
W PosA (101 7oJ)
(2.7)
(P(P))
This implies that for general Po we can also define a convex hull based on barycentric projections, which is of the form
Copo (1B)))-)) & P(N) (((")
Σ Xyboos
# Po.
XEAS
(2.8)
Furthermore, the contraction property equation 2.7 implies that Cop, Cop,, with equality when all transport plans are achieved via maps Viy, Using these definitions, the following defines our notion of projection for general Po and shows that it projects onto Cop--
Proposition 2.1. Consider a general target measure Po and a set (Pyliet of general control measures. Construct the measure Pa as
P exP
Σ X7bm0: Id
where the optimal weights X* E A are obtained by solving equation 2.6 and "Yoj are optimal plans transporting Po to P,, respectively. Then given optimal plans "ooy- Pa is the unique metric projection of Po onto Cor, ((P))/
The optimal plans Tuy transporting Po to P, need not be unique if P3 lies outside the cut locus of Po. i.e., when there is more than optimal way to transport Po onto Py. However, the projection for fixed Yn is always unique by virtue of the linear regression.
3 STATISTICAL PROPERTIES OF THE WEIGHTS AND PROJECTION
We now provide statistical consistency results for our method when the corresponding measures (P) Je[J estimated from data. We consider the case where the measures P3 replaced by

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their empirical counterparts
N₂ 6x, (A)
PN,, (A) - N
for every measurable set A in the Borel o-algebra on IRd where (A) the Dirac measure and (Xij...., XN,.j) is an independent and identically distributed set of random variables whose distri- bution is Pj. We explicitly allow for different sample sizes Uن= Nj N for the different measures. To save notation write CN, = Gj. bos b,,,,,,, and Poz NNo in the following.
Proposition 3.1 (Consistency of the optimal weights). Ler 1P, } be the empirical measures which are independent and identical draws from P3, respectively. and are supported on some common latent probability space (9,, s ,)). Assume all Pj have finite second moments. As N₂- - oo for all E [J], the corresponding optimal weights Xxx = (Xi.....,) .Nx,) E A obtained via
corresponding to the data
Xiy
A arg min
Σ X3 Ongj - Id]
(3.1)
NEA
LP((P
satisfy
P XN 12
x² -5) +
+0
for all 0,
where X solve equation 2.6
This consistency result directly implies consistency of the optimal weights case the optimal trans- port problems between No and each NN,, are achieved by optimal transport maps V We also have a consistency result for the empirical counterparts iP..N of the optimal projection Px.
Corollary 3.1 (Consistency of the optimal projections). In the setting of Proposition 3.1 the esti- mated projections PT,, converge weakly in probability to the projection P. as N₂ + 00 for all j E []].
Proposition 3.1 and Corollary 3.1 hold in all generality and without any assumptions the corre- sponding measures Pj. except that they possess finite second moments. To get stronger results, for instance parametric rates of convergences, one needs make strong regularity assumptions on the measures Pj. Without these, the rate of convergence of optimal transport maps in terms of expected square loss is as slow n 22d (Hütter & Rigollet, 2021). Under such additional regularity condi- tions, the results for the asymptotic properties are standard, because the proposed method reduces to classical semiparametric estimation problem, as the weights A are finite-dimensional.
4 ILLUSTRATIONS
4.1 MIXTURES OF GAUSSIANS
We consider mixtures of Gaussian dimension d = 10. We draw from the following Gaussians:
X, ~ N (13, Σ), j == 0,2,,33
where 140 = 10, 10, 10). #1 = 50, 50, ,0,....,,0)) 50). 122 2000, 200, ..... 200). μ4 [-50, -50, 50) and Σ Idio +0.8 id with Id the 10 10 matrix with zeros on the main diagonal and ones all off-diagonal terms. We then define the following mixtures: Yo as target, and Y,. Y2, and Y3 as controls, where
Y, 0.7X. + 0.15X; + 0.15X2 Y, = 0.7X, + 0.2X, + 0.1X3,
Y, 0.6X, 0.3X, + 0.1X2
Y, = + 0.1,, 0.6X, 0.6X;

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Each sample is 10000 points. The estimated optimal weights are X [0.4329, 0.4002, 0.1669 with corresponding projection Yo. With only control units, it not possible to perfectly replicate the entire target distribution. Still, in Figure 2 the optimal projection approximates Y, reasonably well. Moreover, the weights are non-sparse in this case, indicating that the target Py, lies inside the observe sparse
geodesic convex hull of the control measures. In many real-world applications optimal weights; see, for instance, Section 4.3 and our application to synthetic controls Section 5 This implies that in these settings the target lies outside the geodesic convex hull of the controls and is projected onto one of the faces.
0016-
0.04
0012
0.03
Figure 2: Kernel density estimates of the average of all dimensions comparing target Pro (blue) and its projection Ps (orange) onto the generalized geodesic convex hull [,y,, PY,, Pr.).
4.2 IMAGE EXPERIMENT: MNIST
We compare our results to those from the experiment in Section 4.3 of Werenski et al. (2022). We follow the experimental procedure described therein, taking experimental data the MNIST dataset of 28 x 28 pixel images of hand-written digits (LeCun) 1998). We show comparison to the test case with image occlusion. We treat the normalized matrix probability measures supported on a 28 x 28 grid. Figure 3 shows our results. We are able to more clearly replicate the edges and contours of the target image. compared both the Euclidean projection and the method described in Werenski et al. (022)). Moreover, our method manages to replicate the overall shape of the specific handwritten number closer than the other methods; in particular, it is the only method that correctly replicates the horizontal bar at the bottom of this particular handwritten "4".
Figure 3: Left to right: occluded image; Euclidean projection; result from Werenski et al. (2022), using optimal weights obtained from their method; result from approach, using optimal weights obtained from equation 2.6 target image.
4.3 IMAGE EXPERIMENT: LEGO BRICKS
To examine the general properties of how our method obtains the optimal weights, we provide application replicating target image of an object using images of the same object taken from different angles. We use the Lego Bricks dataset available from Kaggle, which contains approximately 12,700 images of 16 different Lego bricks in RGBA format. Our method manages to replicate the target block rather well, while only using the information of control units that look sufficiently like the target. In particular, in replication, our method does not use information from any image of the underside of the Lego brick. In contrast, the Fuclidean projection does not provide the correct rotation in the replication, and suffers from the standard blur induced by using a mixture of images.

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Target image
(b) Euclidean projection
Our method
(0.059, 0.791)
(0.313, 0.108)
(0.077, 0)
(0.083, 0.008)
(0.468, O)
(0.
(0. 0)
(0, 0)
(0. 0)
(0. 0.093)
Figure 4: Top row: target block, Euclidean projection, and projection from our method. Middle and bottom rows: Control units used in simulation. Left entry in parentheses is optimal weights from our method, right entry are optimal weights from Euclidean projection. Weights are denoted as zero they are less than 1e-6.
APPLICATION TO CAUSAL INFERENCE VIA SYNTHETIC CONTROLS
When analyzing the causal effect of treatment on unit, such as that of public policies or medical interventions, there is often no comparable control unit that can capture the treated unit's underlying characteristics. The classical synthetic controls method (Abadie & Gardeazabal] [2003 Abadie al. 2010) aims to create suitable control unit by replicating the pre-treatment outcome trends of the treated unit, using some optimally chosen set control units. This achieved by projecting the observed characteristics of the target unit onto the convex hull defined by the characteristies of control units in the pre- treatment periods. The optimal weights obtained by this projection, therefore, describe how much each control unit contributes to the target unit's 's counterfactual outcome in the post-treatment period (Abadie 2021).
We apply notion of projections extend the classical synthetic control method to work on joint measures of several outcomes, which allows to disentangle heterogeneous treatment effects and complements the univariate method introduced in Gunsilius (2022). As demonstration, the effect of health insurance coverage following state-level Medicaid expansion in Montana in 2016. The variables of interest are Medicaid coverage, employment status, log wages, and log hours worked. For control units, we use the twelve states for which such expansion has never occurred; these are: Alabama, Florida, Georgia, Kansas, Mississippi, North Carolina, South Carolina, South Dakota, Tennessee, Texas, Wisconsin, Wyoming. Additional information can be found in Appendix
study
We estimate "synthetic Montana", i.e. Montana had it not adopted Medicaid expansion, by estimat- ing optimal weights X using data from 2010 to 2016, and solving equation 2.66 [2.] over joint distribution of the four outcomes over the time period from 2010 to 2016, which generates measures in = 28 dimensions. We note that we estimate one set of optimal weights weights-Speciffeenee specifically, one for each control state- over the entire time period. We then estimate the counterfactual joint distribu- tion using data from 2017 to 2020, by using the optimal weights X and computing the weighted
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barycenter (Agueh & Carlier 2011) of the control states using these weights. Details of sample
lection and estimating "synthetic Montana" are described in Appendix [ The results of the general causal effect of the Medicaid expansion policy in Montana averaged over the years 2017 2020 are illustrated Figure 5
Consistent with findings in Courtemmanche al. (2017). Mazurenko et al. (2018). find significant first- and second order effects of Medicaid expansion, which are summarized the top row and the bottom row of Figure respectively. "Synthetic Montana" has much lower proportion of individuals insured under Medicaid, suggesting that expanding Medicaid eligibility directly affects the extensive margin of Medicaid enrollment. The disemployment effect is less pronounced in comparison the enrollment effect we estimated, but nonetheless positive and nontrivial, consistent with the findings
in, e.g., |eng et al. (2020p, but inconsistent with those in, e.g., Gooptu et al. (2016). We also find positive second-order effects, summarized in the bottom row of Figure Additional details are in Appendix C
20
18 mM
Inmoned
02
Tepoont
Emninees
(a) Covered by Medicaid
(b) Employment Status
(c) Log Wage
(d) Log Labor Hours Supplied
Figure 5: Counterfactual (blue) vs actual (orange) Montana from 2017 to 2020. In the bottom row, histograms of data distributions are shown on the left, and cumulative distribution functions are shown on the right.
6 CONCLUSION
We have developed projection method between sets of probability measures supported on Rd based on the tangent cone structure of the 2- Wasserstein space. Our method seeks best approximate some general target measure using some chosen set of control measures. In particular, it provides a global (and in most cases unique) optimal solution. Our application to evaluating second-order effects of Medicaid expansion in Montana via an extension of the synthetic controls estimator (Abadic & Gardeazabal 2003 Abadie et al., 2010) demonstrates the method's utility in allowing for method that is applicable for general probability measures. The method still works without restricting optimal weights to be in the unit simplex, which would allow for extrapolation beyond the convex hull of the control units, providing a notion of tangential regression. It can also be extended to a continuum of measures, using established consistency results of barycenters (e.g.
first- and
Le Gouic Loubes 2017).
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APPENDIX
A WASSERSTEIN BARYCENTERS AND THE SPECIAL CASE REGULAR TARGET MEASURE
The natural approach to defining projections on W2 is to work on the manifold directly. As men- tioned in the main text, this leads to bilevel optimization problem, based on the notion of barycen- ters in Wasserstein space (Agueh & Carlier 2011 Carlier & Ekeland 2010):
P(A) = arg min (E)
(2))))))
2
With this definition, and assuming that the barycenter P(A) is unique for given X, the bilevel pro- jection problem reads:
X E arg min W2,Pu, P())), where P(A) = arg min (
(PP.P)
(A.1)
YEA
A version of this approach is used in Bonneel et al. (2016) to define notion of regression between probability measures low dimensions. The challenges here are mathematical and computational. Importantly, the optimal weights X need not be unique. This is not an issue for the applications con- sidered in BBnneel et al. (2016). like color transport; however, when the weights convey information used in further procedures, like causal inference via synthetic controls, where the optimal weights are used to introduced a counterfactual outcome of a treated unit had it not been treated (Abadie & Gardeazabal 20033 Abadie al. 2010 Abadie (021). Moreover, the bi-level optimization structure makes solving the problem prohibitively costly in higher dimen- sions. Bonneel et al. (2016) introduce gradient descent approach based on antropy-regularinn analogue of W2 (Cutuni 2013 settings.
important in statistical settings
Peyré & Cuturi 2019) that can be implemented in -ww-dimensional
introduce a tangential approach, but under strong as- sumptions on the involved measures: they need to be absolutely continuous with densities bounded away from zero on their support, and in particular the target measure must be known to lie inside the convex hull of the other measures. A starting point for this the barycenter P(A) for fixed weights of a set (Py (Ps)eeL in regular tangent spaces. (Agueh & Car- lier (201T) Equation (3.10)) show that if least one of the measures absolutely continuous with respect to Lebesgue measure, then P()) can be characterized via
Other approaches like Werenski et al. 2022
consider characterization of
A (V
Id) 0,
(A.2)
where { are the optimal transport maps from the barycenter to the respective measure Ps. ie. (5,) = P,. Each term of the summand in equation A.2 is an element in T P()) (2RR)
by construction.
More generally, the condition equation A.2 is a sufficient condition for P(A) to be a "Karcher mean" (Karcher 2014) in We (Zemel & Panarctos, 2019). In fact, a "Karcher mean" of a set of measures [Pybe[J] is defined as the gradient of the Fréchet functional in W2 and is characterized through equation A.2 holding P(A)-almost everywhere. equation A.2 is a stronger condition because it is assumed to hold at every point in the support of P(A), not just almost every point. Alvarez--Esteban introduce fixed-point approach compute Wasserstein barycenters, and Werenski et al. (2022) use this structure to introduce a replication approach for absolutely continuous measures whose densities are bounded away from zero and whose target measure lies inside the convex hull of the control measures. Related is the recent definition of weak barycenters in Cazelles et al. (202T), where the authors replace the optimal transport maps from the classical optimal transport problem by the weak optimal transport problem introduced in Gozlan jet al. (2017). Heuristically, this characterization is that of a deformable template. A measure P deformable template if there exists set of deformations { j=1.... such that 3/ P P,. in
al. (2016) use this characterization
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way that their weighted average is "as close the identity" as possible. In our setting by Vip 19y-Id Id (Anderes et al., 2013 Boissard et al. 2015 Yuille 1991).
In our setting of interest, tangential projection reduces to
X arg min xEA
A 9j Id)
(A.3)
LP(P))
where Vij are the optimal transport maps between the target Po and the control measures Pj. j [J] In contrast to Werenski et al. (2022) the target measure does not need to lie inside the convex hull of the other measures.
Based these definitions Copo (P)})=) in the case where Po is regular.
show that approach is projection of the target Po onto
Proposition A.1. Consider regular target measure Po and set (Po)eel of general control measures. Construct the measure P3 as
J Σ
Px - expro
x - Id)
where the optimal weights X E A are obtained by solving equation A.3 and Vij are the opti- mal maps transporting Po to Pj. respectively. Then Px the unique metric projection of Po onto
Con, (1P))
B PROOFS
Proof of Proposition A.7 Define the following closed and convex subset C C L2(P)) for fixed optimal transportation maps between Po and Pj. denoted Vyyy
My Vi for some X E A
c
(P f=
Recall that the transport maps Viy exist since Po is regular. Using C, we can rewrite equation A.3
arg min XEA
ld
arg min| f - -d (P()))
JEC
LP(P))
which by definition is the metric projection of Id onto C. Since C is non-empty closed and convex subset of the Hilbert space (P), this metric projection exists and is unique (Aliprantis Border 1999 Theorem 6.53). Moreover, if ld C, then Te Id; otherwise, se OC, where OC is the boundary of C (Aliprantis & Border 1999 Lemma 6.54).
Since Po is regular, the exponential map is continuous. In fact, for every j j k,
(P Ps) -=)= = W (9)) Po. (V (x) Po) <
V V px dPo(x).
In other words, the distance hetween P3 and Pk in (TR") is smaller than that between correspond- ing elements Vipj, V in the tangent space. This implies continuity of the exponential map.
Furthermore, in this regular setting, the exponential map sends convex sets in T Tr. W2 to generalized geodesically convex in W2. Mechanically, for any two (scaled) elements ((Vi, Id) and s( (V Id) in Tr, W2. and any E [0, 1].
expr, (pt V pj Id) (1- p)s(p)svvv po -dd)
=expp, (pl V ( (1 -pss V (p)) (pt p)s) Id)
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pt pi + (1-p)s
exppr
p
V 4p:: - ld]
= -( [[ptVp; Ps + +(1 (1- p)s pi] (1 (1- Id] Po
(lot(V 92 -dd + (1 p)s Sx Id)] + ld Po
#
where 7 pt + (1 -11--)ss p)s. This is a generalized geodesic connecting P3 and Pa. via the optimal transport map between them and Po (Ambrosio al. 2008 section 9.2). The same argument holds when extending generalized geodesics to generalized barycenters by taking convex combination of more measures than binary interpolation with respect to p. Mechanically, for any E A and t > 0 for all j [J]
1 V Pj
exPp.
Xg4(V 9j - -ld) = =exppo
N
Σ 1=1
2gly ld
/=1
expp%
PJ
Σ 5=1
Vo - ld
411
Po
#
(V - Id)
+ld
Po
where P, a Ej=1 Aztz. This proves the exponential map is generalized geodesically convex.
From above it follows that Pa expp, (me) is either in the interior of C, which the case if Id C. or on its boundary: since TC E aC, exp n.. (Tc) E expr, (0C). By continuity of the exponential map follows that expr, (DC') r (C). Combining all steps above show that Ps geodesic
metric projection of Po onto the geodesic convex hull of {P)
Proof of Proposition 2.7 The result follows from the same argument as the proof of Proposition A.1 Theorem 12.4.4 in Ambrosio et al. (2008) shows that TP, W2 is the image of the barycentric projection of measures in the general tangent cone: (x(x) is an optimal transport map if y is an optimal transport plan. But the exponential map satisfies
expp, (v) = (v + Id) Po for all v E Tp, W2.
This implies that
J Σ
P
exPpo
Σ Xgbro; - Id
Ajbnos
Po e Coro (1B3)/))
since the convex combination of elements in the subgradients of convex functions lie in the subgra- dient of a convex function (provided the subgradient of each convex function is nonempty, which is the case here). Then the continuity and generalized convexity of the exponential map for elements in the regular tangent space Tp, W2 implies the result.
We split the proof into two parts. In the first part we prove the con- vergence in probability of the family of objective functions equation 3.1 to their population counterparts equation 2.6 the empirical measures IN,, converge weakly probability to the
Proof of Proposition 3.7
population measures P, In the second step the fact that X is classical semiparametric
estimator (Andrews 1994 Newey McFadden, 1994) to derive the convergence of the weights.
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Step I: Convergence of the objective functions functions for obtaining the weights X², we write
To show the convergence of the of the objective
Xzb0 ld
Xybo ld
(2(())
1.2(PNo)
/
Xybo, (x) - dPo - -1
Σ
Ay (2)
dlP No
1=1
We hence want to show that
2
2
lim Ag 0000
E (x))
dPo(z) - Σ (2)))(())
dIp No (x) =0,
where Ag Ni = min Ny).
We split the result into two parts. The first part shows that
2
lim inf Ag +00
2, (zo) - 20 NNN,
Azb0)(z0)) - xo dPo(zo).
In the second part we use the (Po) convergence of the barycentric projections to prove that the limit exists and coincides with the limit inferior.
For the first part, we have
lim inf Ag No00
[
Σ
(x00)(20) 2o 1=1
dIP No (1r)
lim inf Ng -00 (RI
Σ 1=1
Njæz To
where (xo, I1, ;1,...,;.) a measure that solves
min ( - zol2 dy: T ... Fiffo......... F0o)
Toy the optimal couplings between IP No and IN,, (Tos) No- Since all measures are defined on the complete and separable space Rd, and by assumption of finite second moments, i.e.
/ |1r, mo/ drig +00,
max sup jE[j] N₂
holds that each sequence Por is tight by Ulam's theorem (Dudley 2018 Theorem 7.1.4). Using the fact that A and 7N r1 (fo1...., PoJ). applying Jensen's 's inequality gives us
2
max sup JED N2 ( 243
Σ A5æ To d
max sup JE N j=1 Jgd [x
xol² dios < t00
which implies that 7N is tight. By Prokhorov's theorem, there exists subsequence 7N, that weakly converges to a limit measure 7. Therefore, by the continuity of the map (No, I.))) + Σ, A5æı Vo.
follows from classical convergence results (Ambrosio et al. 2008
Lemma 5.1.12(d) that
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2
lim Ag 7,0 (Ra)s+1
2 2
20
din (1,...,
Ay To dy(ro......)
(Re) (
Furthermore, by the same argument via Jensen's 's inequality, i.e.,
2
J
dy( ........................ ...)) & Σ / Paaa zul" d7o +00,
20 و5Jリ
j=1 (3072
j=1
follows that the limit E T (701,.... :o.))..
Now note that by the definition of disintegration it follows that (Ambrosio et al. 2008
Lemma .)3.2
7 E Ii "1
Yso ( 72110)
where
y= / Yao (x))
Yoj / /rjioo d.Po
and
the disintegrations of y and ma with respect Po. respectively. Therefore, we have
2
20
dy(=o..... ...;)...;)) ..)
j=1
1
(cxery
dy (2 (5) d.)(((.))))PP(
3o
(BOO))"
Σ 25zg xo
dr (#1,..., xz) dPo(xo)
Rd
/eecy Tiy (21,,..)
2o dPo(ao)
Rd
1=1
2
Jad Ij (7351a0
20
dPo(zo)
pd
j=l
(xoo)(zo) xo dPo(zo),
Rd
where the third lines
follows from Jensen's inequality and the fifth line from Yeo
(T1]aas.....7J)))))) This shows the first part.
For the second part we use the fact that each barycentric projection buy (x1) is an optimal transport map between IP NNo and IP N, if Top is an optimal transport plan between IP No and IP PN,,, which follows from Theorem 12.4.4 in Ambrosio et al. (2008). As before, we know that Dos
IP IN,, is tight #
sequence that converges to some P,. By definition and the fact that buy is the gradient of convex function between IP No and IP N, bo; is the unique optimal transport map between IP No and IP NN,, for all N, and all Since the measures P3 have finite second moments by assumption, we have
lim NoA,, -,,N 3d
12,12 dip,, lim sup
|o (2o)
dP No
NoAN, - JEd
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NoAN -00 Rd 1/e xi diyl (17z)
sup [
12 dlP No
simsup NoANy -100 (RE)2 |=ss (Foo)(00,3)
lim sup
[xze |oy| dros(oo, +00,
where the last equality follows from the tightness of Toi, as shown earlier. Therefore, by standard stability results for optimal transport maps (Segers) 2022 Panaretos & Zemel 2020). it holds that
boy converges uniformly on every compact subset K S R2 the support of the limit measure Pj. that is
lim sup NoAN +00 me
(1)()))) - Uy (20) = 0,
where by is the optimal transport map between Po and Pj.
We now show that Uj = boj Po-almost everywhere. From the local uniform convergence, we can
then derive "strong L L'-convergence" (Ambrosio et al.
2008 Definition 5.4.3) of the potentials:
lim sup NoA
[oo]
|
L2(PN))
lim sup
[boj
|/ (
lim sup No
limssp
[|0
|| 110/ll
NoAN5-200
LP(PNo)
lim sup N00
Thoj
U/y (2(
+limsup
| | No0t00
||05 (e(p))
Now the first term converges to zero by Hölder's inequality and the local uniform convergence of the optimal transport maps from above. The second term satisfies
lim sup No
| | - -118s11
(Le (15(z00) dIP dIP
1/2
(1 (es(Eu))
1/2
lim sup No-+00
[2) dIP No L (z0)) dPo
1/2
lim sup
No
But since Po has finite second moments, it holds that this term also converges to zero.
Based on this we can show that Poz = Id. Boy) a limit point of the sequence Poy. holds that
converge weakly to Yoj = (Id, Uz). Indeed, if you
Stery11 |05 dros (20, 2z))
lim /
No (Re)² [z) "Fios (1o, 2r3)
NoAN; (RU)2 |z; drios (xo xz))
lim
Ponylao) ? aIP No (2))
lim sup NoAN
/ (0,(20) dlo(2o).
Disintegrating the left-hand side with respect Po. and applying Jensen's inequality, gives
1eey |2, ddyo
/
1z dr (I₂) dPo()
=
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/ Rd 1pc By 100 (1z) dPo(sn)
(o,((zo)| dPo(
that is,
J |oo5 (cro) dPo(xo)) [ [uy(zro) dPo(su).
Ra
But since is an optimal transport map between Po and Pj by definition, holds that
Jec [Dby(Zz)) d > / |u5(xo))) dzo(
which implies that equality holds and we have that
J [ (05(20) dPo(Fo)) 0,
which implies that boy By Po-almost everywhere. We have hence shown that (Id, 8os) IP No converges weakly to (Id, boj) # transport map between Po and Pj (e.g. Villani 2003 Theorem (ii)).
#
Po for all j. where the barycentric projection boj is the optimal
Moreover, have shown "strong 2 P-convergeece" of the barycentric projections in terms of Def- inition 5.4.3 in Ambrosio et al. 2008). Since this holds for all j. it also holds for their convex combination for fixed weights À A Putting everything together, then have that
lim Ag Ny
Σ Uos ld
Σ Xybos Id
LP(P))
Since all observable measures Py are empirical measures, they converge weakly in probability Varadarajan 1958), which implies that
J
J
lim Ag N2 +000
Σ
Xoboi
X bo;
ld
> s
=0
for all s > 0.
(.2 No.
(PP)))
This shows convergence in probability of the objective function for fixed X.
Step 2: Convergence of the optimal weights X The convergence of the optimal weights now follows from standard consistency results in semiparametric estimation. In particular, the objective functions are all convex for any À E RJ, which implies that they converge uniformly on any compact set (Rockafellar) 1970, Theorem 10.8), so the objective function converges uniformly on A Now standard consistency result like Theorem 2.1 in Newey & McFadden 1994) then implies that
P P(X
- x² - =
lim Ny N,000
for all s >>>0 0
which what we wanted to show. Note that the result can also be shown if we allow the weights to be negative, i.e., if we only require that E/=1 AÚ = In this case, the fact that the objective functions are convex and coercive implics that an optimal > will be achieved at the interior of the extended Euclidean space, from which consistency follows by Theorem 2.7 Newey & McFadden 1994).
Proof of Corollary 3.7 We want to show that EEe= Sis, Bos
IP P. converges weakly in proba- #
bility (Σ/=1 13b0
Po. where A A (irr,..., 2zx)) the optimal weights obtained in
#
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equation 3.1 and equation 2.6 respectively. The result follows by applying the extended continuous mapping theorem (van der Vaart & Wellner 2013] Theorem 1.11.1) as follows.
As shown in the proof of Proposition 3.1 have "strong L c'=convergence" of the maps Σ' Xix do; Id Σn- Azbo; ld. Therefore, by Theorem 5.4.4 (iii) in Ambrosio al. 2008 holds that
lim | 7
Iü. Σ
6 (zo) 2o dIp dIP No (zo)=
| Rd
S
To:
Xzb0s(F0) xo (Fo
for any continuous function such that (ma)) & C + C2 To ro. for all so in the support of Po.
where C₁ < -oo are some constants and To in some element in the support of Po (Ambrosio al. 2008 equation (5.1.21)). particular, this holds for any bounded and continuous function f. which implies that
lim
f
Ni, boy (zo) dlP (xo))
/ee f J
Σ 27 (2r0)
Bd
for any bounded and continuous function, which implies that EEj= div, bos
NN converges
weakly to (Ei= X3bss Po if converge weakly to Pj.j [J].
Now we apply the extended continuous mapping theorem (yan der Vaart & Wellner 2013 Theorem I). Equip (RR with any metric d(.,) :) that metrizes weak convergence. We define the maps ==
P2 (R)). + (R)) by
(Po..... Py) =
Po
and analogously for their empirical counterparts gN. Note that g and gN are non-random functions if the measures P, and Nr are non-random themselves for all j [J]. Moreover, by definition,
and IN are continuous maps because Σj= Xōboş are gradients of convex functions, which are continuous Polalmost everywhere; the same thing holds for their empirical counterparts. Now from what have shown above and in Proposition 3.1 it holds that
gN [P No. IP Nx,) + g (Po...., Ps)
PN, converge weakly to Ps. Since (P,N, 1/2 here instead are the only random elements in X == (2()), a) the extended continuous mapping theorem implies that
(ā(ON ........,IPNN)) (Po... (il....,))) >E) = for all E > 0
lim Ag N,00
which is what we wanted to show.
C DETAILS OF MEDICAID EXPANSION APPLICATION
We use the ACS data with harmonized variables made available by IPUMS, unified source of Census and survey data collected around the world. The data is at the household peeehooldpeesso-yeal level. For application, continuous outcomes are adjusted using the person-level sample weights available in the data.
select the household head and the spouse
unit of analysis. The
We adopt the following sample restriction criteria: included individuals
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of working age, i.e. between ages 18 and 65
who has no missing outcomes (for those listed in the main text)
who has no top-coded responses
who are either household heads or their spouses
The sample size breakdown by states are follows:
State
Observations
Target
MT
25,173
Control
AL
106,464
FL
427,397
GA
227,659
KS
74,812
MS
61,505
NC
233,804
SC
107.905
SD
22,563
TN
152,470
TX
598,222
WI
157,410
WY
15,666
Table 1: Summary of the full data sample used to obtain X".
We randomly select N = 1500 observations from each unit for estimating X In the Python imple- mentation, we face a challenge where the entries of the target and control data are large enough, equation A.3 becomes too large for CVXPY to compute duce a stabilizing constant to prevent this. This stabilizing constant is determined by the mean value and dimensions of the target distribution, and the number of controls. The weights obtained sparse and are displayed in T
optimal solution. Therefore,
intro-
State
AL
FL
GA
KS
MS
NC
SC
SD
TN
TX
WI
WY
Weight
0.184
0.174
0.010
0.513
0.119
Table 2: Estimated Weights for Control States.
We check whether the obtained weights are fit for creating synthetic Montana by examining if they well-approximate actual Montana in the pre-treatment period. As seen in Figures 6 and projection is similar to the actual data.
7
siperion
10
Innered
Iahred
03
Tepeivel
Unimettue
Brokeed
Unrmdooed
(a) Covered by Medicaid
(b) Employment Status
Figure Replicated (blue) vs actual (orange) Montana from 2010 2016.
22

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Under review as a conference paper at ICLR 2023
200
150
(a) Log Wage
(b) Log Labor Hours Supplied
Figure 7: Replicated (blue) vs actual (orange) Montana from 2010 to 2016. In each panel, his- tograms of data distributions are shown the left, and cumulative distribution functions are shown on the right.
Once we obtain the optimal weights X", estimate the counterfactual outcomes of interest for the four years after Medicaid expansion in Montana (namely, between 2017 and 2020). This involves we
solving equation A.II with X obtained from the pre-intervention period. Implementation-winn computed the free-support barycenter, using the POT package; this does not fix the support of the barycenter a priori, and allows it to different from those of the control distributions. We plot the densities and distributions of the counterfactual outcomes in Figure(3]o of the main text.
To perform inference on the estimated causal effect, we use a placebo permutation test in analogy to Abadie et al.] (0100), Gunsilius (2022). The idea is to repeatedly apply the procedure described treated unit. Post-intervention.
above to each control unit, pretending in turn each control unit if an actual treatment effect only appears in the treatment unit (Montana, in this application), then the estimated effect for the actual treatment unit should be among the largest.
0.000
0.80
0.13
0.72
0.122
0.12
8.000
0.02
0.050
0.02
2014
2006
2018
2020
2030
2034
2018
(a) Weights Using All Years
(b) Weights Using Averaged Weights Over All Years
Figure& 8: In orange: Montana. In blue: pretending each control state listed in Table is a treated state.
We plot the 2- Wasserstein distance between the treated, joint distribution of all outcomes and the pre-/post-ing inposstinterventionn optimal projection (i.e. equation A.I A.I with X*). We present two sets of results Figure g in panel (A), the optimal projection is computed using X estimated using all years in the pre-intervention period; in panel (B), the X" used is constructed from taking simple average of weights estimated in each year of the pre-intervention period. Our results suggest that the estimated causal effect is valid in the post-intervention period, coming from Montana, especially from 2017-2019. The effect is less pronounced in 2020, however.
consistently observe the largest difference
23

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Under review as a conference paper at ICLR 2023
To accompany Figure 8 also compute p-values, which denote by and define pt r(du) where d1 is the 2-Wasserstein distance from the optimal projection to actual distribution when the target unit is Montana, r(d1:) is the rank of die amongst djiS at given time t. and is the number of control units. Results are described in Table 3 A smaller Pt value indicates larger treatment effect. We observed that r(d1e) = for 2018 and 2019, implying nontrivial effect the Medicaid expansion in Montana during these years. The values are Pt comparably higher in 2017 and 2020, which we attribute to the fact that was the first year of the policy implementation, and the COVID-19 pandemic, respectively.
J+1
Year (t)
Pt (Weights Using All Years)
pi (Averaged Weights Over All Years)
2017
0.231
0.308
2018
0,077
0,077
2019
0.077
0.077
2020
0.535
0.385
Table 3: Estimated pt
r(die)
in the post-intervention period.
J+1
24


Reviews:
[{"review_id": "caEDqtz63Bi", "reviewer": "Reviewer_PFzW", "summary": "", "questions": "", "limitations": "", "rating": 3, "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": "", "summary_of_the_paper": "The paper proposes a way to project a probability measure P0 onto the finite population P1, P2,..,Pn of the other measures. The projection idea is based on the weighted averaging of the optimal transport maps P0->Pn for the quadratic transport cost (the projection implies computing these weights for averaging). The authors prove statistical consistence results for the measures available by empirical samples and show the application to estimate causal effects via synthetic controls.\n", "summary_of_the_review": "I do not think that the contribution of this paper is significant. In my opinion, the authors explain rather straightforward things by employing non-trivial facts such as tangential spaces, etc. At the same time, this seems to be absolutely unnecessary as the space they work in is just linear. Thus, I think the paper does not present any non-trivial results or ideas. Moreover, the writing should be improved/simplified. The only potentially positive and interesting result is the consistency theorem, but it is outweighed by the above-mentioned drawbacks of this paper. Thus, I think this paper is currently far below the ICLR paper standard and I vote for rejection.", "strength_and_weaknesses": "**Strength**\n\n* The proposed projection methodology is straightforward and does not require solving a bi-level optimization problem as, for example, in the barycentric coordinates approach [2].\n* Theoretical consistency results for the weights recovered from empirical samples;\n\n**Weaknesses**\n\n* The proposed approach to projection method is rather obvious;\n* The writing of the paper is not very good in many aspects and the paper is hard to parse:\n* * **(a)** the current text overcomplicates the proposed idea too much; \n* * **(b)** the actual contributions are not clearly stated;\n* * **(c)** the experimental part is not transparent;\n\n* Experimental support of the proposed ideas is not sufficient;\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "QRZzh-F-yq1", "reviewer": "Reviewer_LCUu", "summary": "", "questions": "", "limitations": "", "rating": 6, "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": "", "summary_of_the_paper": "The idea is to define an appropriate notion of W2 projection of P_0 onto given measures P_1,..,P_J . The proposed definition avoids the standard bi-level optimization based one and hence is efficient to compute. Simulations on a problem of estimating causal effect are presented.", "summary_of_the_review": "Overall the paper is interesting, however, I currently see a couple of important gaps as summarized above. Hence I recommend accept with reservations.", "strength_and_weaknesses": "Strength:\n1. Paper is well written with enough references.\n2. Prop 2.2 seems interesting,\n\nWeakness:\n1. The abstract reads that the proposed methodology has several applications. However, in simulations or otherwise only one application is shown. Also, this application does not seem to be one where the proposed method is compelling. In the sense that estimating causal effect has many solutions and the proposed seems only one of the ways. Are there any more compelling applications where the problem of finding the projection is more natural/compelling?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "Gl7bKNTZUM", "reviewer": "Reviewer_mwRK", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "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": "", "summary_of_the_paper": "The authors propose to use the 2-Wasserstein space over some Euclidean space to approximate high-dimensional probability measures in a low-dimensional setting. The authors proposed to do this in three steps, (i) the first is to obtain the general tangent cone structure at the target measure, (ii) the second is to construct a regular tangent space if it doesn't exist, and (iii) the third is to perform a linear regression to carry out the projection in the tangent space.\n\nThe challenge is to compute the corresponding optimal transport plan between the target and each measure used in the projection. The authors propose to do this using the classical synthetic control estimator. The introduction of the control estimate creates a hierarchical optimization, where first the distance is minimized between the parameters of the model and the set of control measures. The second layer of optimization then finds the minimum between the minimum of the set of control measures and the target (Eq.(2.3).\n\nThe authors show that the hierarchical structure is well defined between the regular target measure and the second of control measures and the target measure and the set of control measures. The authors also derive the consistency of the optimal weights/\n\n", "summary_of_the_review": "Overall, the paper is well-written and easy to follow. I feel that the paper has some reasonable novelty to it. However, the main drawbacks are that there is no discussion of the computational complexity and a weak experiment section without any clarity of the experiment set up, and no benchmark to compare against to properly evaluate the performance of their proposed approach over other state-of-the-art approaches. Though, I do understand that this is a theoretical paper, but some sort of benchmark with comparably approaches should be made to show where it sits in the literature.", "strength_and_weaknesses": "Strengths\n(1) Addresses an interesting research question by approximating a high-dimensional probability measure with low-dimensional settings using a geometric average called the barycentric projections.\n\nWeaknesses\n(2) See clarity\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "4wAw2xRIaO", "reviewer": "Reviewer_pNhL", "summary": "", "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": "", "summary_of_the_paper": "The authors propose a new notion of projections of a distribution into a set of distributions with Wasserstein metric by working on regular tangent cones of the Wasserstein space using generalized geodesics. The authors evaluate the proposed method to estimate causal effects via synthetic controls.", "summary_of_the_review": "The proposed idea is interesting and has a promising potential for applications. However, the authors need to elaborate more details about the proposed approach and its motivation/relation with the ones in the literature.", "strength_and_weaknesses": "Strength:\n+ The authors give a new notions for projection of a distribution into a set of distributions by leveraging regular tangent cones of the Wasserstein space.\n\nWeaknesses:\n+ The authors need to give motivation and elaborate with more details relations between the proposed approach with the existing one (e.g., Problem 2.6 and Problem 2.7 where the authors use the same notations, \\Delta \\phi_j but with different meaning, where in (2.6), it is the optimal transport map from the barycenter to the respective measure P_j while in (2.7), it is the optimal map between the target P_0 and P_j)\n+ The authors need to discuss the relation between the propose method and the approach which use the optimal map to represent distributions (w.r.t. an anchor distribution, e.g., P_0 as in the submission), \n\nRef: Mérigot, Q., Delalande, A. and Chazal, F., 2020, June. Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space. In International Conference on Artificial Intelligence and Statistics (pp. 3186-3196). PMLR.\n", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Reject