__ Summary and Contributions__: This paper considers the recommendation problem, where a goal is represented by an ideal point in some multi-dimensional space, and items are ranked based on their distance from this point. Rather than using a standard Euclidean distance, this paper proposes learning a Mahalanobis distance while estimating the ideal point based on pairwise preferences.

__ Strengths__: - This work presents a very elegant approach for an extremely popular problem with high novelty and broad relevance.
- The presentation is very clear and easy to follow.
- By relying on pairwise comparisons, this model also depends on easier to obtain data.
- The algorithm is evaluated on synthetic and real dataset.
- The motivation and broader impact is thoughtfully discussed

__ Weaknesses__: - The application of recommendation to candidate admissions is, while very important as an example of fairness in AI, a little uncompelling from a recommendation evaluation perspective. The paper would be stronger if the method were also tested on a classic recommendation problem. This also means that future comparisons with this method as a baseline will be challenging (it will require re-implementation of the method as the authors have not indicated a willingness to share their code. While addressed in the author feedback, it is still a comparison I would like to see.
- The authors note a lack of theoretical guarantees, which is a little disappointing especially given the iterative EM nature of the solution.

__ Correctness__: This reviewer did not check the algorithmic derivations in detail.
However, the experimental validation is appropriate and correct. The claims of how the method works are well supported.

__ Clarity__: The paper is very clearly written, thoughtfully introducing concepts in a way that does not feel rushed. This is an exceptional presentation for a method that needs to introduce many concepts and steps within the tight limits of NeurIPS.

__ Relation to Prior Work__: Prior work is thoroughly and clearly discussed, with contrasts drawn as required.

__ Reproducibility__: Yes

__ Additional Feedback__: In terms of reproducibility, the real world dataset may or may not be available publicly, we do not know. This may limit the possibility to reproduce the specific results, or for future work to compare directly to this approach.
How were the regularization parameters set?
And, trivially, Figures 1 and 5 could be shown on a logarithmic scale.

__ Summary and Contributions__: This paper proposes two algorithms that use pairwise comparisons of items to learn both user preference vectors and a Mahalanobis distance over the space of preference vectors (which are in the same space as the items) under the ideal point model. The paper presents experimental results on synthetic data as well as graduate admissions data, showing that the proposed methods can rank graduate school candidates and find interesting feature interactions.

__ Strengths__: This paper is very well-written and the proposed algorithms are very cleanly explained. Researchers working on pairwise comparisons and preference learning should find this paper to be interesting and valuable.

__ Weaknesses__: Update: The authors provided thoughtful feedback on noise considerations that addresses the main weakness mentioned below from my original review. Provided that this discussion on the noise model is incorporated into the final text, I think the paper will be substantially stronger. Additionally, the authors provide detailed feedback for other weaknesses that other reviewers brought up. Altogether, the additional discussion makes the paper much more thorough. In light of this, I am increasing my score of the paper from 7 to 8.
One unresolved issue though: one of the reviewers pointed out that the ideal GPA of 4.06 is greater than 4 despite the GPA being normalized to a 4.0 scale. It would be helpful to clarify why this happens (e.g., I know some universities have a 4.0 scale but allow A+'s to be greater than 4.0, etc), and what exactly you mean by normalizing (e.g., dividing by the max which would undo the A+'s being greater than 4.0 for some schools, or changing some other school's scaling to be up to 4.0 in the case of, say, MIT which has a scale out of 5).
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The main weakness I see in this paper is that the algorithms are motivated in a slightly heuristic fashion based on a noiseless model, and then the authors allow the solution form to be slightly different to tolerate deviations from the noiseless model. This leaves the question of whether there's a cleaner way to model the Bernoulli noise (simplest case would be where the noise is the i.i.d. across all comparisons) where learning preferences and the distance just amount to maximum likelihood? If so, in what way does this resemble the algorithms proposed?

__ Correctness__: Yes, the claims and experiments look correct.

__ Clarity__: Yes, the paper is very well-written and the exposition is quite clean.

__ Relation to Prior Work__: Yes, the authors relate their work to existing literature.

__ Reproducibility__: Yes

__ Additional Feedback__: After reading the author feedback/other reviews: please incorporate the discussion points from the author feedback into the main paper. Also, see the GPA comment (originally from reviewer #3) and now also stated in the "weaknesses" question above.

__ Summary and Contributions__: The authors develop a method to jointly estimate ideal points and a Mahalanobis metric from noiseless pairwise comparisons and apply it to some synthetic and empirical datasets.

__ Strengths__: The paper has a great treatment of the problem and analyzes the estimation problem at length. There are some interesting applications to synthetic and empirical data. The paper claims that the joint estimation problem of ideal points and Mahalanobis metric hasn't been solved before, which may be of independent interest.

__ Weaknesses__: The over arching issue with the paper is a lack of prove theoretical claims or empirical wins over existing work, so this reads more like a deep dive on how to estimate a model along with some examples, which is interesting and possibly publishable in its own right, but doesn't stack up against contributions with empirical and theoretical wins.
Other notes:
-Focus on the noiseless case, which has less relevance in practice.
-Lacks discussion on runtime (beyond pointing out that things are linear or convex) or how much data is needed for estimation
-Seemed weird to me to include that the ideal GPA learned was 4.06, greater than the max GPA of 4.

__ Correctness__: Everything is correct as far as I can tell.

__ Clarity__: The paper is generally well written but I found some of the style in the related work to be dismissive of prior work. Claims like "The ideal point model is an intuitive and interpretable way to model preferences and has been empirically shown to exhibit superior performance compared to other models of preference" are overblown relative to the two citations that follow, you'd need some recent authoritative work on preference learning to justify a claim like that, especially when your paper doesn't include a comparison of performance against other methods.

__ Relation to Prior Work__: The relation to prior work from the metric learning side is well discussed, but I think a lot of the existing work on estimating pairwise comparisons were swept over a bit lightly. The biggest need for prior work comparison is in the empirics section in my opinion, it's hard to know how useful this is when it's only compared against itself on a school choice dataset.

__ Reproducibility__: Yes

__ Additional Feedback__: I think this is good work but the crux of my score is on its relative theoretical and empirical strength to other work. Perhaps the estimation problem and formulation are of sufficient strength to justify publication from the metric learning perspective, which I have less expertise in and will defer to the other reviewers on if my review is an outlier.
From the perspective of estimating models of comparison I think more work is needed to compare this method to existing ones and/or derive theoretical bounds on estimation and performance.

__ Summary and Contributions__: Authors propose method for simultaneously learning user preference vectors and an associated mahalanobis distance under the "ideal point" model of user choices and preference representation. They provide a convex program and two related algorithms for their learning problem. And they run experiments demonstrating the soundness of their algorithms on both synthetic data and datasets of graduate school applicants who have been accepted / rejected from academic programs.

__ Strengths__: What is interesting, different, and perhaps surprising about this work is that given a set of item vectors { x_i } and a data set of user choices of the form "the user preferred item x_i over x_j", one can learn *both* the user's "ideal item" vector and her associated importance weighting of the item attributes (in the form of a mahalanobis distance) and combinations thereof, and as a convex program. Prior work has only focused on either learning the user's ideal point vector, or a mahalanobis distance, but not both simultaneously.
The experiments demonstrate the authors' method works better than an alternative approach from the literature on small to modestly sized datasets when the underlying preferences are described by a mahalanobis distance that is not Euclidean.

__ Weaknesses__: It should be pointed out that the ability to learn both a mahalanobis distance and user's ideal point is not totally novel since performing user/item matrix factorization from user pairwise preferences could be viewed as a case in which both the distance metric and the user representations are learned. But in matrix factorization, the mahalanobis distance gets pushed into the learned item embeddings rather than being explicitly captured in a separate parameter). Consequently, it is not clear that this method provides the community with truly new capabilities. And theoretically speaking, the existence of a convex program for this problem might have been expected, though it has not been published before.
Second, by relying on semidefinite programming, the proposed method is not scalable to industrial recommender systems, at least "out of the box". But there are likely custom modifications that would make the proposed algorithm(s) useable in these settings. The authors might briefly discuss these since recommender systems are listed as a use case in the introduction.

__ Correctness__: the claims appear correct

__ Clarity__: yes, the paper is very well written and easy to follow

__ Relation to Prior Work__: relationship to prior work is discussed and is clear.

__ Reproducibility__: Yes

__ Additional Feedback__: While the authors discuss identifiability, they might also want to discuss the sample complexity of learning ideal points and distances. How many comparisons are needed to estimate "u" and "M" well (for some definition of 'well').
Also, the authors consider learning "u" and "M" for a single user. But what if we have preference choices obtained from multiple users? If we assume all users have the same underlying "M", then we would need fewer choice observations per user to learn "M" well, would this in turn help us more quickly learn the ideal points for each of users?