Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 6: marginally above the acceptance threshold
review_content: 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.
strength_and_weaknesses: Strength:
+ 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.

Weaknesses:
+ 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)
+ 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), 

Ref: 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.

clarity,_quality,_novelty_and_reproducibility: The proposed idea is interesting and may have promising potential for further applications. However, the authors may need to improve the notations and discuss relations between the proposed method with the existing ones in the literature.

Some concerns are as follows:
+ For Section 2.4, the notations are confusing, the authors use the same notation \Delta \phi_j for Equ. (2.6) and (2.7), but with different meanings. What is the relation between them? Could the authors elaboration motivations, relations for changing the optimal map between barycenter and P_j into the one between P_0 and P_j?
---> In this case, one may not need to compute barycenter anymore for Problem 2.3, but it may simply precompute the optimal map between P_0 and P_j for every j = 1..J. Therefore, this relaxation simplify the Problem 2.3. a lot. It is better in case the authors elaborate their relation.

+ It is hard to follow the idea in Section 2.3. In Equ. (4), to define a geometric tangent cone structure at each measure P, could the authors elaborate what is the "optimal" meaning for (\pi_1, \pi_1 + \epsilon \pi_2)_# \gamma is "optimal" for some \epsilon > 0)?
--- Especially, in Equ. (2.4) it seems that the authors use optimal map to represent for each distribution (e.g., P_2, P_3) to the anchor P_0. It is better to discuss such choices with existing results in the literature.

Ref: 
[For optimal map representation] Ref: 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.

[For cone structure of W_2] Takatsu, A. and Yokota, T., 2012. Cone structure of L2-Wasserstein spaces. Journal of Topology and Analysis, 4(02), pp.237-253.

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.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: 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.
strength_and_weaknesses: Strength:
1. Paper is well written with enough references.
2. Prop 2.2 seems interesting,

Weakness:
1. 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?
clarity,_quality,_novelty_and_reproducibility: Clarity:
Overall the paper is weel written.

Quality: 
The paper presents an interesting problem and develops interesting results related to it

Novelty:
The idea in (2.7) seems interesting but I am unclear about its relationship with (2.6) 

Comments
1. In (2.6), the OT is between the barycenter and the control measures, whereas in (2.7) the OT is between the target P_0 and control measures. May be I am missing something, but is there a quantitative relation the provides the bounds the difference between these two solutions? Or is there any other justification that I am missing?
2. By the very nature, (2.7) promotes sparse optimal solutions. So will be performace be good enough when non-sparse combinations are actually involved? Even in the simulations, the application seems to have a sparse optimal solution. Can we have some simulation results on applications where non-sparse lambda is optimal?

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.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: 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.

The 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).

The 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/


strength_and_weaknesses: Strengths
(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.

Weaknesses
(2) See clarity

clarity,_quality,_novelty_and_reproducibility: 
(1) I feel that background and contribution should be split so that the contribution of the authors is much more clear. For example, Section 2.1 and the first half of Section 2.2 and Section 2.3 are backgrounds. While Eq.(2.3) and Section 2.4 onwards appear to be the author's contribution which appears to be mixed into the background. (of course, this is just the preference of the writing style, the authors may say this reads a bit better. But I tend to disagree.)

(2) Are the control measures the same size as the target? If so, how scalable is this? It doesn't appear to be scalable to high dimensions at all. This should be at least discussed as from my understanding, the projection into lower dimensional space is to make this much more computationally efficient. But the complexity of the model was not been discussed at all.

(3) Section 4 doesn't appear to be self-contained. After reading Section 4, I have no idea what you have done. I may need to read your references to understand your experiment setup.
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.
review_title: null

final_score: rating: 3: reject, not good enough
review_content: 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.

strength_and_weaknesses: **Strength**

* 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].
* Theoretical consistency results for the weights recovered from empirical samples;

**Weaknesses**

* The proposed approach to projection method is rather obvious;
* The writing of the paper is not very good in many aspects and the paper is hard to parse:
* * **(a)** the current text overcomplicates the proposed idea too much; 
* * **(b)** the actual contributions are not clearly stated;
* * **(c)** the experimental part is not transparent;

* Experimental support of the proposed ideas is not sufficient;

clarity,_quality,_novelty_and_reproducibility: The proposed idea is novel but I would argue that it is obvious and lies on the surface. Indeed, the theory and the entire idea presented here might seem non trivial, but actually it is just the 
* linearization of the Wasserstein-2 space based on the Monge embedding w.r.t. P0 (which maps measures P to transport maps P0->P which are elements of the functional L2(P) space), see definition 1.1 in [1]
* Projection of the identity function “x” in L2(P) to the convex Hull of Monge embeddings of P1, P2,...,Pn which can be reduced to n-dimensional convex problem.

I do no understand why do the authors overcomplicate the entire exposition so much and why the theory of the tangential spaces is needed in this paper. It does not provide any useful information. In my view, everything what is described in pages 4-6 could and should be compressed to at most 1 page.

The experimental part is not sufficient to demonstrate that the proposed methodology is practically useful. The main issue here that it is not possible to understand the experimental setup of section 4. What are the distributions P_0, P_1,.., P_n here, how exactly are they obtained? It is not even written what is the dimension of the considered experiment. I bet some details are in Appendix, but this is not satisfactory as all the major details should be explicitly clarified in the main text. I think that it would be several times better if the authors include more experiments and experimental details to the main text instead of too long exposition of the tangential projection (see my comment above).

Finally, as far as I understand, no baselines are considered in the paper. I wonder why the authors do not test their method at least in some setups of [2] and do not compare with [2]. Specifically, it would be interesting to see how the proposed method works in the case of Figure 10 of [2].

**Replicability:** moderate. On the one hand, there are not a lot of clear technical details of the implementation in the paper. On the other hand, the code for all the experiments seems to be present in the provided supplementary.

**References**

[1] Mérigot, Q., Delalande, A., & 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.

[2] Bonneel, N., Peyré, G., & Cuturi, M. (2016). Wasserstein barycentric coordinates: histogram regression using optimal transport. ACM Trans. Graph., 35(4), 71-1.

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.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: The paper proposes a new approach to project a measures onto a "linear" combination of probability measure, in the W2 space, a task which can be related to a "regression" problem in the Wasserstein space. Two approaches have been proposed for that task, either a bilevel optimization problem (where the barycenter of a set of measures using a family of weights is the argmin of a given energy, and its distance to the query point is further differentiated) or using some form of Linearization on the tangent cone of a reference measure to fall back on a Euclidean parameterization in which the barycentric problem is simply a (Euclidean) regression problem. Both approaches have been proposed. The novelty here, as underlined by the authors, is that the method uses, instead of building on Monge maps to a reference measure (which are difficult to access), the barycentric projections from similar plans as an approximation.

Reviewers have had split opinions on this paper. Ultimately, and after careful reading of their opinions and the author rebuttals, I propose to reject the paper based on a few issues. The first is that the paper has undergone significant changes during the review process, correcting for some flaws, which, as even positive reviewers aknowledge, should warrant a new round of fresh reviews due to their breadth. The second is that the experimental section remains weak, and one reviewer in particular has raised numerous concerns. Finally, the authors mention that

"In contrast, approaches that depend on the classical Wasserstein barycenter consider optimal transport maps between the barycenter and the control measures (e.g. Bonneel et al. (2016); Werenski et al. (2022))."

but as presented in the paper that distinction is only "temporary", since the authors naturally end up using a Monge map approximator, the barycentric projection operator

"To your second point on the sizes of optimal transport plans, we note that we apply the barycentric projection to the optimal transport plans, i.e.  in our notation. In particular, this means we always work in the same dimension as our target probability measure ."

However, https://arxiv.org/pdf/2201.12195.pdf (Sec.3.1) follow exactly the same route (including entropic regularization) since they rely on an entropic map estimator (identity - 1/2 gradient of dual potential) which is conceptually equivalent to what is proposed here.

As a result, it feels that the gap between the method proposed here and the method by Werenski (2022) needs to be clarified. That paper was published on arxiv close a year ago (ICML submission) and so more care must be used to differentiate this submission from that paper which looks very similar.


justification_for_why_not_higher_score: Novelty, experiments.
justification_for_why_not_lower_score: na
Decision: Reject