Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This paper proposes to embed two sets of data into a common low dimensional space and at the same time estimate the correspondences between the data points of two sets.  The proposed method combines Multidimensional Scaling (MDS) and Wasserstein Procrustes analysis into a joint optimization problem, which is solved using alternating updates. The authors demonstrate the effectiveness of the proposed method on a variety of datasets.
strength_and_weaknesses: Strength:

1. The paper is very well-written and technically sound. The idea of the paper is natural and clearly motivated.

2. The proposed method is effective on a variety of datasets.

Weakness and questions:

1. The novelty of the paper seems limited to me. It looks to me that this work simply combines the methods in two regimes and implement a joint optimization. 

2. From the optimization perspective, the proposed method is essentially solving the two problems (MDS and Wasserstein Procrustes) in an alternating way. It is hard to tell whether the resulting heuristic algorithm will lead to convergence to a local/global min.


clarity,_quality,_novelty_and_reproducibility: Clarity: The paper is clearly written and easy to follow.

Novelty: As I commented, the novelty of this work seems limited.
summary_of_the_review: In summary, the idea of the paper is very natural for solving the proposed task, although its novelty looks limited. In general, I enjoyed reading the paper. Given both of its limitations and advantages, this work has a potential to be published at ICLR.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This work proposes a practical method to jointly estimate the low dimension embedding in Euclidean space and the correspondences between data sets, which can be viewed as the alignment extension of classical embedding work by Multidimensional Scaling (MDS). Unlike some previous works in this domain, the proposed method is unsupervised by nature without given any known subset of alignment.  

This joint MDS framework is formulated as a combined stress objective function plus the additional matching penalty term from Wasserstein Procrustes analysis. Given this proposed join objective function is non-convex (same as similar problems), this paper proposed an alternative optimization algorithm (similar to EM type), which can be viewed as the combination of weighted MDS problem and Wasserstein Procrustes problem in each iteration. 

Experimental results on both synthetic data and real-world data (e.g., cell, MNIST, human body data, protein, etc) on a number of tasks from joint 2D visualization, domain adaptation, graph matching and protein alignment. 
strength_and_weaknesses: Strengths:

The problem itself is interesting and the article is pleasant to read too, i.e., join alignment and embedding from 2 different data sets without supervision can be find a lot of application across NLP, computer vision, healthcare, biology, etc. 

The optimization part is also solid as with enough considerations of previous works to joint optimize this non-convex problem by SMACOF algorithm and Wasserstein optimizations from (Alvarez-Melis et al., 2019), and with detailed analysis of computational complexity (Section 4.3), which is quite helpful to see. 

Experimental results also can be considered solid as across a number of synthetic or real data coupled with different tasks (with more results in Appendix B), and the proposed method often show (not always) the relative best performance among existing methods. 

Weaknesses:

One of the key concerns is, the proposed joint MDS method is based on MDS as the starting point, which utilize the (dis) similarities from Euclidean space and this may conflict with the nature of manifold embedding when data with strong nonlinear intrinsic geometry structure. Indeed, there are some discussions provided in Section 4.2, which is good to see, but still some unclear remains. As ISOMAP is essentially MDS coupled with geodesic distance, and not sure this is applied in which part of results in Section 5? (Maybe I missed).  Also, outside MDS and ISOMAP, there are other graph spectral works in this direction such as Laplacian Eigenmaps (and others), should be helpful to have some judgment for the choice in this work. 

Additionally, it seems using GW as the initialization step (mentioned in Section 4.3) to provide a better coupling matrix P in the algorithm, make the contribution from this proposed iterative optimization less clear as maybe the non-convex formulation is good in theory but less practical in the numerical way. 

Reference:
1. D. Alvarez-Melis, S. Jegelka, and T. S Jaakkola. Towards optimal transport with global invariances. In International Conference on Artificial Intelligence and Statistics (AISTATS), 2019
clarity,_quality,_novelty_and_reproducibility: This work can be considered as marginal contribution on existing work from MDS optimization and Wasserstein Procrustes analysis.

Author provides the code and data in the supplementary material; this is good to see.

Also, given sequence alignment has been considered in this work, it seems that the reference to the previous spatial-temporal alignment domain (i.e., which is also join alignment between 2 different data sets with some intrinsic structure, see reference 2 below) can be added, for the common and differences in the framework. 

Reference:
2. F. Zhou and F. De la Torre, Canonical Time Warping for Alignment of Human Behavior, Advances in Neural Information Processing Systems (NeurIPS), 2009 




summary_of_the_review: The overall contribution seems marginal but given the solid technical work behind this alternative optimization algorithm, and the rich empirical evaluation to this joint alignment problem, my recommendation for this paper is 6 "marginally above the acceptance threshold".
review_title: null

final_score: rating: 3: reject, not good enough
review_content: summary_of_the_paper: This paper generalizes the classical  multidimensional scaling approach for unsupervised manifold alignment. This enables the mapping of , datasets from two different domains, without requiring correspondences across the datasets, to a common low-dimensional Euclidean space. This approach generalizes approaches based on dictionaries that map the source and target data onto a common latent space. Results on a few benchmark datasets are proposed.
strength_and_weaknesses: Strengths: A useful extension of MDS with applications in unsupervised domain adaptation (UDA) is proposed. This approach enables the mapping of datasets from two different domains onto a latent space.
Weaknesses: I am not impressed with papers that have reasonable theory but demonstrate results on synthetic data, MNIST etc. Results in table 3 are not impressive either.

clarity,_quality,_novelty_and_reproducibility: The paper is clearly written. While the proposed extension of the classical MDS technique enables new possibilities for UDA, the authors have chosen rather simple datasets to demonstrate the results of their approach.
summary_of_the_review: I will encourage the authors to do more experiments with real and challenging datasets.
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: This paper proposes a new framework for unsupervised manifold alignment with joint multidimensional scaling. It combines a MDS method with a optimal transport to join the embedding. The main contribution is the combination of these two methods forming a general framework and a optimization scheme to solve both problems jointly. The experiments shows interesting results on two different tasks (visual exploration and graph matching).  
strength_and_weaknesses: ## Strength

The proposed framework is interesting on several points,
1. It relies on tools with good mathematical properties. Optimal transport has now effective methods for solving the problem with appropriate regularization and MDS can be approximated with the SMACOF algorithm.Thus, both sub-optimization problems are easily solved.
2. The experiments are strong with very good results for graph matching leading to potentially new methods.
3. The generality of the framework opens it to many problems that occurs in different community (especially all the problematic around matching).
4. The method seems easy to code and then enjoy a good reproducibility.

## Weaknesses

However I see several issues in the current work,
1. While the state-of-art is mostly complete, the authors forget some recent methods that may be competititve [1].
2. For graph matching, a comparison against KerGM [2] would be welcomed.
3. The method does not scale to large dataset.
4. There is no analysis of the convergence of Algorithm 1. Since it is a classical two stage optimization scheme, with appropriate hypothesis, convergence toward a stationary point should be possible [3].
5. There is no comments about the initialization. How dependent are the results to the initialization? Is it stable enough?

[1] Jin, K., Liu, C., & Xia, C. (2021). Two-Sided Wasserstein Procrustes Analysis. In IJCAI (pp. 3515-3521).
[2] Zhang, Z., Xiang, Y., Wu, L., Xue, B., & Nehorai, A. (2019). KerGM: Kernelized graph matching. Advances in Neural Information Processing Systems, 32.
[3] Bezdek, J. C., & Hathaway, R. J. (2003). Convergence of alternating optimization. Neural, Parallel & Scientific Computations, 11(4), 351-368.
clarity,_quality,_novelty_and_reproducibility: ## Clarity

The article is pleasant to read and easy to understand.

## Quality

The proposed work is good.

## Novelty

While the problematic itself is not new, the proposed joint optimization is novel and may lead to new methods in different area.

## Reproducibility

The code by itself is not available, but the algorithm is well described and with the appendix a direct implementation should be possible. All the experiments are clearly described with details in appendix.
summary_of_the_review: This article proposes a new joint optimization problem for both multidimensional scaling with unsupervised manifold alignment. It relies on two well founded methods: multidimensional scaling  with SMACOF as solver and optimal transport with Sinkhorn's solver. The complete framework is general enough to be applied on different kind of tasks and the experiments shows promising results. Despite some weakness, this framework may interest a large part of the community.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: In this paper, the authors propose a new manifold alignment, which jointly estimate the low dimension embedding in Euclidean space and the correspondences between data. More specifically, the proposed method consists of Multidimensional Scaling (MDS) and Wasserstein Procrustes analysis, and the model parameters are updated in an alternating manner. Through experiments, the authors demonstrated that the proposed method compares favorably with existing methods.

The proposed method is simple yet effective. However, the technical contribution is a bit limited; this is a borderline paper. 
justification_for_why_not_higher_score: This idea is simple and it is good to have a method that combines MDS and Wasserstein distance. Many reviewers are happy to accept. However, they are not that excited about the results. At this point, I lean towards acceptance, however, it is completely fine that it can be rejected.
justification_for_why_not_lower_score: This idea is simple and it is good to have a method that combines MDS and Wasserstein distance. Many reviewers are happy to accept.
Decision: Accept: poster