__ Summary and Contributions__: The paper investigates multi-object matching via robust permutation synchronization. The authors propose an iterative reweighting strategy, where the weights are initialized using CEMP (a technique borrowed from related work, but with a novel interpretation) and then an iterative approach alternates weight update and permutation estimation. Performance guarantees under a specific corruption model are proven. The approach is also validated in simulated and real datasets.

__ Strengths__: - The paper tackles an interesting problem.
- The discussion of the limitations of IRLS methods is reasonable and highlights shortcomings of existing methods.
- The theorems, drawing connections with CEMP, are interesting and sound. Moreover, the performance guarantees in Theorem 5.2 were very well received. It is indeed important to understand the theoretical limits of robust synchronization algorithms.
- The approach works very well in practice and dominates baseline methods, as can be seen in Fig. 1 and Table I.

__ Weaknesses__: - The presentation can be largely improved: some of the sentences do not help the reader and currently the draft is not easy to read. For instance, the transition to the second paragraph of the introduction is quite abrupt: while the first paragraph talks about keypoints and images, the keypoints disappear when talking about the permutation synchronization. Moreover, the sigma is not properly defined. Similarly, Section 4.1 is hard to read: while the overall idea is clear, the reader gets trapped in too many details, such as those in lines 196-199. Similar comments hold for Section 4.2: why presenting two techniques when you recommend using only one? What about moving the other to the appendix? While I understand the desire of the authors to be comprehensive, mentioning too many details may compromise clarity.
- The literature review misses several related papers on permutation synchronization:
[1] Bernard, Thunberg, Goncalves, Theobalt, Synchronisation of partial multi-matchings via non-negative factorisations, Pattern Recognition, 2019.
[2] Serlin, Sookraj, Belta, Tron. Consistent Multi-Robot Object Matching via QuickMatch. In International Symposium on Experimental Robotics, 2018.
[3] Fathian, Khosoussi, Tian, Lusk, How, CLEAR: A Consistent Lifting, Embedding, and Alignment Rectification Algorithm for Multi-View Data Association, Arxiv, 2019.
As well as other papers on synchronization on Lie groups:
[4] Rosen, Carlone, Bandeira, Leonard, A certifiably correct algorithm for synchronization over the special Euclidean group, IJRR, 2019.
[5] Arrigoni, Fusiello, Synchronization problems in computer vision with closed-form solutions, IJCV, 2020.
- Section 5: It is unclear why the superspreader model (in particular, in conjunction with assumption (12)) is more realistic or more challenging than the uniform corruption. It might be good to add more comments. Also: why restricting to a complete graph? It seems unrealistic for the SfM (and other) applications mentioned in the intro.
- The approach (and Algorithm 2) is technically sound but looks overly complicated. If CEMP can reliably estimate the corruption levels, isn’t it enough to do a single weighted least squares optimization afterwards (ie stopping before the "for" loop in Algorithm 2)? am I missing something? Indeed Theorem 5.2 seems to guarantee the performance of CEMP rather than the overall IRGCL algorithm.
- Related to the previous point: it would be good to assess the performance of CEMP in the experiments. In other words, how good is P_(1) in Algorithm 2?
- Section 6.2 preprocess the real dataset to make it usable by the proposed algorithm (e.g., by removing some of the corrupted measurements): how would that work in practice?

__ Correctness__: The technical results seem correct. The empirical evaluation can be improved since it requires pre-processing the real datasets, which seems unrealistic.

__ Clarity__: The presentation is the main issue of the paper. I think the technical contribution is nice, but the presentation is hard to follow.

__ Relation to Prior Work__: The literature review can be improved by adding missing references (see previous comments) and better discussing the relations with CEMP.

__ Reproducibility__: Yes

__ Additional Feedback__: - line 39: “the the”
- line 213: “it faster” -> “it is faster”
- Line 175: “CEMP is provably robust to adversarial corruption, but is more difficult to motivate”: this sentence is confusing and CEMP is not well reviewed in the related work section.
- Line 99: sigma_ij should have a tilde I think - It’s the measured quantity.
POST REBUTTAL:
I found the rebuttal very convincing. I agree with R2 that adding more experiments (and discussing computational aspects) could further improve this paper. R3's concern is mainly about discussing connections with IRLS (which seems fixable) and comments on convergence -- as far as I see, the authors' rebuttal provides fairly convincing comments. In general, having a IRLS-like algorithm that can tackle permutation synchronization seems interesting and novel to me. I also agree with R4 about the lack of clarity: that seems to be the main issue of the currrent paper. In summary, I like the paper and I'm convinced about the technical quality, but the presentation is not very accessible and I hope the authors can improve in the final submission.

__ Summary and Contributions__: The main contribution of this work is an iterative reweighting algorithm for the permutation synchronization problem. The authors do a good job at placing the work in the context of the group synchronization problem, of recovering (compact) group elements from a small noisy subset of pairwise measurements. The link between the two is given by the standard representation of the permutation group elements as doubly stochastic binary matrices.

__ Strengths__: The paper addresses a rather niche regime (of heterogeneity) in which other methods do not seem to perform well. The obvious previous work on the topic of this paper is [22]
D. Pachauri, R. Kondor, and V. Singh. Solving the multi-way matching problem by permutation synchronization. NIPS 2013
which essentially leveraged the group synchronization existing literature at the time to propose a spectral methods for permutation synchronization.
The popular iteratively reweighted least squares (IRLS) does not trivially extend itself applicable to the permutation synchronization problem due to the discrete nature of the permutations which may lead to zero residuals, which in turn will lead to overweighted edges in the iterative scheme.
The authors proposed an interesting scheme that involved iteratively reweighing
edges in the graph Connection Laplacian, and draws similarities with the popular VDM framework of Singer and Wu, that extends the classical diffusion maps framework to incorporate not only scalar similarities between the nodes, but also the orthogonal transformations between bases for the tangent places at points close enough on the manifold. The same matrix operator is leveraged here, with additional normalizations and weighting schemes.
The authors provide performance guarantees under the adversarial effect of nonuniform corruption, with bad/corrupted edges being attached to a single node in a star shaped topology. I think this model is very realistic one and often encountered but understudied in the literature, and a nice contribution overall. (For example in certain graph embedding problems, where nodes represent embedding of overlapping subgraphs, and edges hold pairwise ratios of the group element at the endpoints of each edge. If a subgraph embedding is grossly corrupted, so will be the information contained on the incident edges).
I think this is an interesting paper. I could see one argue that the novelty is slightly incremental compared with the approaches in works such as [22] and the given VDM framework, but I think the idea of reweighing in this context is neat and well motivated by the fact that existing methods such as [22] only cope well with uniform corruption which is less realistic in real applications. It would be very interesting to extend this to the setting of group synchronization over other groups, if this is something that has not been looked at yet.
Overall, I think this paper could be of impact in the computer vision literature.

__ Weaknesses__: The authors should perhaps also show simulations when the measurement graph is not complete, but also vary the sparsity level, in addition to the noise levels. It would be interesting to see how the different methods compare in this regime (below (log n)/n), especially since spectral methods often require regularization to handle such sparsity.
Can the authors also comment on the computational aspects? Especially given the data sets considered, in both synthetic and real data, are fairly limited in size.

__ Correctness__: The paper appears to be technically sound.

__ Clarity__: Yes, the paper is well written and motivated. The authors could spend some in the intro to briefly explain the group synchronization problem to readers less familiar with this literature.

__ Relation to Prior Work__: Yes, the authors explain how this work relates to existing literature and what gaps it aims to fill.

__ Reproducibility__: No

__ Additional Feedback__: ----- Post Rebuttal -----
After reading the rebuttal, I am happy to maintain my initial score and think this is a solid submission. The authors should address the few loose ends for the final version.

__ Summary and Contributions__: The paper presents a method for multi graph matching that exploits detailed more detailed problem constraints than existing methods.

__ Strengths__: Advances the state-of-the art showing a way to solve a relevant and complex problem with a more sophisticated approach than existing methods.

__ Weaknesses__: One aspect of the papers that seems off-putting are the very strong wording regarding IRLS, without offering much insight in the ways the proposal differs from IRLS. This is made more confusing by the fact the proposed algorithm does seem to follow in the end a recipe similar to IRLS, with a weighted least-squares problem solved iteratively with the proposed method basically attaining itself to how the weights are calculated.
Is it really the case that the proposed method cannot be understood as IRLS, although perhaps with a weight calculation that is much more sophisticated that previous research has ever done? The proposed method seems to be carefully exploiting many relevant constraints of the problem, and achieving great results from that, what is all great. Is it really fundamentally contradicting the (very general) IRLS algorithm, though?
It's important to understand how the method relates to IRLS because it's a great theoretical framework even before being a very convenient, practical method. Two important questions show up when IRLS is applied to any problem in general:
1_ what is the corresponding non-linear error function that will be minimized as a consequence of the formula utilized to calculate the weights?
2_ Is there any probabilistic interpretation of the process, viewing it as a case of Expectation-Maximization?
Other important considerations are: how important is initialization to the process? The authors did seem to touch this issue a bit, although their understanding does not seem to be very clear.
While IRLS is not exactly a non-linear optimization method such as Gauss-Newton, Levenberg-Marquadt, etc, it's quite safe to say it falls it that family. Meaning it mostly will provide you a local minimum from the attraction basin of the initial solution. That means it's not a global optimization method, and therefore a "bad" initialization along with an objective function that contains many sub-optimal local minima will be a more challenging problem to solve.
The paper mentions this local minima "problem" with IRLS. First of all, it should be clear this is not a problem with the optimization method, really. The real problem is with the objective function and the initialization. The problem is initializing out of the optimal basin. And of course, anyone would prefer a global optimization if it were feasible.
What is the case with the proposed method? Is it a global optimization method, unlike IRLS, Gauss-Newton, LM, etc? Or is it a local optimization method, except the objective function proposed by the authors tends to have more convenient attraction basins, and the initialization proposed tends to lead to a global optimum, or at least a very good one? These are very important details about how such algorithm works, and while the authors bring some of these concepts relative to the alternatives, it's not so clear what happens in the proposal. Could it not be the case that the proposal is still pretty much IRLS, except the non-linear error being utilized has vastly advantageous properties?
The paper right now even seems to suggest that the proposed method is actually performing some form of global optimization, and it would be very nice to make it clear what is the case.

__ Correctness__: Experiments seem adequate.

__ Clarity__: It could be improved in terms of clarity and accessibility to a wider audience. The paper begins by citing very concrete problems such as SfM, and later attains to abstract and specific concepts that only someone knowledgeable of the related methods could grasp. It is hard to understand how practical concepts such as reprojection errors or descriptor distances get translated into graph weights. This is probably well-understood by anyone aware of the methods, although some pedagogy would be appreciated. This is important because these specifics might make a difference in the performance of the method, and why it can outperform simpler alternatives.

__ Relation to Prior Work__: There is plenty of comparisons with previous methods.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: This paper describes a new solution for multiple object matching (permutation synchronization) problem, based on an iteratively reweighting scheme. The solution outperforms the conventional IRLS approach thanks to the superior handling of non-uniform noises presented in the input.

__ Strengths__: The method is shown to perform well, and better than conventional IRLS or least square method for permutation synchronization. A theoretic proof is provided that assures the robustness of the method in the presence of non-uniformly distribution corruptions. The central idea of the method can be extending to solving other multi-object "group synchronization" tasks, and partial matching problems in 3D reconstruction.

__ Weaknesses__: Despite the task of multi-object permutation synchronization itself is conceptually simple and (should be) easy to understand, the paper is written in a form making it unnecessarily difficult to follow. Many of the writing styles and notational systems are odd and unintuitive. Although I consider I under stand most parts of the paper, From time to time I had to refer back and forth in the paper the notations due to their unnatural choices of symbols. In other parts, I felt at lost, e.g. why do you require the relative permutation "\sigma^*_{ij} = \sigma^{*-1}_{ij}" rather than a commutative form ? I cannot see where you give the full term for CEMP other than its reference [17]. These issues are in my view not merely language issues, or poor writing. They reflect the authors may have not spent sufficient time to organize their thoughts and the paper's structure well. The description of the graphical structure used in the paper is also unclear. Was it a complete graph or a singly connected cycle graph ? In the context of SFM (e.g [29]) the graph structure is determined by the visibilities of the multiple input camera views. More experiments are needed on this aspect in order to evaluate the effects of different graph connections. In general, I find the experiment section is thin and fail to valid many of the claims (of advantages) that the paper makes in the main text of the paper.

__ Correctness__: Appears so, though have not checked completely.

__ Clarity__: Not satisfactory... and can be substantially improved. See comments above.

__ Relation to Prior Work__: The current work seems to draw inspiration from multiple prior works. The literature review section (1.1) lacks in depth analysis and comparison. It could have been better written , to place the contribution of this work in context.

__ Reproducibility__: No

__ Additional Feedback__: