Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 1: strong reject
review_content: summary_of_the_paper: The paper proposes to solve the high-rank matrix completion problem by learning a fusion (or a collection) of subspaces. Since the subspaces form the Grassmann manifold, the paper poses the problem as a learning problem on the Grassmann manifold. It proposes a particular cost function that minimizes the distances between all the n subspaces, which correspond to n data points. No technical results are presented. Convergence of the standard stochastic gradient method on manifolds is discussed (which is a known result). Empirical results show some benefits but they are not clear to me. 
strength_and_weaknesses: Pros:
This is an interesting problem.

Cons:
I found a number of arguments in the paper very problematic. This is apart from the lack of theoretical basis of the arguments presented in the paper (which by the way appear very hand wavy at best).

In Page 1, it is mentioned “the geodesics between subspaces of all data, to encourage the subspaces from points that belong together to fuse (i.e, represent the same space)…” as a contribution. However, no results are shown to see whether the modeling actually allows all n points to converge to the same. There seems to be a mis-match between the text statement and the actual work.

No motivation for the loss function, especially the fusion cost term. Why is it a good choice? Ablation studies are missing.

As the paper alludes to “high-rank” completion problems, literature survey should be done on what are precise difficulties and what works already exist. 

The baselines are not properly explained. Furthermore, the legend of those are mismatching to those in the text. Also, it would make sense to include [39] as a baseline by considering all the pairs as the paper considers [39] it to be “tightly related”. 



clarity,_quality,_novelty_and_reproducibility: The paper lacks quality and novelty. The paper clearly does not give due consideration to the literature on high rank matrix completion problem and the distinction that it tries to make. 
summary_of_the_review: I consider the paper requires much improvements in order to be considered a viable submission.
review_title: null

final_score: rating: 8: accept, good paper
review_content: summary_of_the_paper: The paper tackles the high rank matrix completion problem in which only partial observations are present and the underlying data comes from the union of subspaces.  

The fundamental difficulty with all existing approaches is that they rely on assessing distances (e.g., euclidean, or in the form of inner products) between partially observed vectors To circumvent this problem, this paper introduce a new paradigm to cluster incomplete vectors, using subspaces as proxies, thus avoiding the need to calculate distances or inner products or other notions of similarity between incomplete vectors, as other methods require. In this approach, each (incomplete-data) point is assigned its own (full data) subspace, and the objective is to simultaneously minimize over the Grassmann manifold two terms: (1) the chordal distance between each point and its assigned subspace, to guarantee that the subspace stays near the observed entries, and (2) the geodesics between subspaces of all data, to encourage the subspaces from points that belong together to fuse (i.e, represent the same space). These two terms (1) and (2) constitute the 'fusion' between two different types of distances in the non-convex optimizaiton problem.

In a nutshell, this work introduces the idea of fusion penalties in the Grassmann manifold, for the problem of high-rank matrix completion.
strength_and_weaknesses: The paper is quite interesting and well written. The idea of fusion (specifically, in this example combining the chordal and geodesic terms in the optimization function) is interesting.



clarity,_quality,_novelty_and_reproducibility: * Note that the template doesn't seem to be the original ICLR one

Clarity: the paper is very clearly and well written. I enjoyed reading it! It's style is somewhat a atypical (in a good way!) in terms of its structure and presentation. I appreciate that 1) the authors are upfront about the regimes in which their algorithm doesn't perform as well; and 2) they provide quite a bit of technical details on possible future work. 

Quality: the paper is of high quality, it has very good articulation of the underlying intuition as well as provide very thorough and self contained review of the related literature 

Novelty: the idea of fusion gradients on the Grassmanian is novel and interesting 

Reproducibility: the authors provide some details and practical details on running the algorithm on real data (e.g. sketching) but I would have liked to see concrete details such as code. Hopefully this can be addressed during the review process. 
 
summary_of_the_review: In summary, I find the paper interesting and novel, hence  worthy of a conference publication.
review_title: null

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This paper proposes an approach for "high-rank matrix completion", a combination of matrix completion with clustering on a union of subspaces. It designs an optimization problem that pushes the (incomplete) observation's subspace to contain a possible completion of the observation, while pushing for different subspaces to get close to each other. Optimization is done by gradient descent on the product of grassmanian manifold, and there is a local convergence theorem. Several numerical experiments are conducted with promising results.
strength_and_weaknesses: Strength: This paper builds on well-grounded results in Riemannian manifold optimization. Subspace clustering and matrix completion is solved together, and it doesn't require the number of clusters as an input. It doesn't have low-rank assumption on matrix completion. Experimental results in the low sampling regime are promising.

Weakness: The paper seems to have crammed a lot of material to fit the page limit. It may have been a better fit for a journal where they can define, derive, and present the results much more rigorously and carefully.
clarity,_quality,_novelty_and_reproducibility: Clarity can be improved, especially in section 4. Even though the idea of regularization using geodesic distances defined on the Grassmannian manifold is not new, the choice of these particular metrics applied to this particular problem seems novel and original.
summary_of_the_review: While I still think this paper might be a better fit for a journal, the problem formulation is novel and it has interesting theoretical and experimental content that tackles a difficult problem with many applications.
review_title: null

final_score: rating: 5: marginally below the acceptance threshold
review_content: summary_of_the_paper: This paper introduces a new paradigm to cluster incomplete vectors using subspaces as proxies to exploit the geometry of the Grassmannian. The authors leverage this new perspective to develop an algorithm to cluster and complete data in a union of subspaces via a fusion penalty formulation. The analysis with synthetic and real-data experiments is conducted to serve as proof of concept.
strength_and_weaknesses: Strength:

1. The core idea that to cluster incomplete data using subspaces as proxies is novel. 

2. This paper is well-written and clearly organized.

3. Detailed theoretical derivations and analyses are provided.

Weaknesses

1. The scenario to which the article applies, and the problem it solves, are unclear. What is incomplete data, and what is the relationship between incomplete data and HRMC?

2. The article provides detailed mathematical proofs, but I think they are too much at the expense of readability. I carefully read the article twice but still had difficulties following it.

3. There are not enough experiments to prove the efficiency of the proposed method, and the paper did not provide codes or other supplementary materials for replication.

clarity,_quality,_novelty_and_reproducibility: This paper is well written and clearly organized. The clarity, quality, novelty is good.
But the paper did not provide codes or other supplementary materials for replication. The Reproducibility is not good.

summary_of_the_review: This paper introduces a novel paradigm that optimizes on the Grassmannian to complete and cluster incomplete data in a union of subspaces. The core idea is novel and neat, and the paper is well-written. However, there is some unnecessarily technical writeup, some parts appear redundant. And the experimental results section is not sufficient.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: There are four reviews for this paper: two with rather neutral scores, one with a very negative score and one with a clearly positive score.
The two "neutral" reviewers mentioned some strong points, such as the underlying idea of treating subspace clustering and matrix completion jointly, or promising experimental results in small sample-size regimes.
On the other hand, these reviewers also  pointed to some weaknesses, such as unclear application scenarios and lack of conceptual clarity an mathematical rigor. During the rebuttal and discussion phase, it seems that none of these two reviewers were fully satisfied with the authors' reply. They both wanted to stick with their borderline scores, and none of them wanted to champion the paper.

Then there is a third and clearly negative review. This reviewer raised many points of criticism, such as
- missing ablation studies that would help to understand the use of the cost term (for example, why use the chordal distance?)
- missing formal basis: 
  (i)  the paragraph "Why this should work?" is more subjective in nature, without any precise and formal arguments.
  (ii) unclear use of multiple sub-spaces, given that the minimizer is a single point.
- missing comparisons with numerous LRMC  approaches in the literature
- unclear novelty / high similarity to  [39] ([47] in updated version)
At the end of the discussion phase, this reviewer clearly expressed that basically none of these concerns have been addressed in a clear and transparent manner (and honestly, to some degree I also share this impression).

Finally, there is a fourth review with a clearly positive score. Unfortunately, this review is very vague in terms of actual strengths/weaknesses, containing only relatively general statements like "interesting and well written" or "interesting idea". During the discussions, no further positive arguments have been placed, and this reviewer further mentioned the possibility of lowering the score. 

In summary, I strongly belief that for this paper the negative arguments clearly dominate the more positive aspects, and therefore I vote for rejection. 






justification_for_why_not_higher_score: There are too many open questions that could not be addressed during rebuttal / discussions. Some of these questions refer to severe conceptual problems.
Decision: Reject