Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper proposed an unsupervised shape matching method, where the core idea is to first generate noisy maps from existing unsupervised shape matching methods, and refine them by learning pointwise features from the noisy maps, and build a new map by matching those features. The new map is empirically more accurate than the input noisy maps. Experiments are conducted to support the claim.

strengths_and_weaknesses: The reviewer finds the work novel and interesting. The main concerns are as follows:

1. How is the loss in line 260 differentiable with respect to the network parameters? Since it appears that Pi_MN lies in the set of permutations, which is a discrete set.
2. In section 3.2 it says early stopping is needed for training the random-initialized network to learn features. However, it appears that early stopping criterion is only proposed in 4.2 but not 4.1. Is it true that 4.2 is not an optional step of denoising, but actually an important step for early stopping?


Minor issues on presentation: 
3. The notations of Theorem 1 (lines 201-204) look a bit confusing. Does N(M)_x mean the mapping N restricted on M applied on x \in M? 
If so, I wonder if N(M)_x a standard notation, or if N|_M(x) is more standard? Since in the former, N(M) could mean the image of M under N.
If not, what does this notation mean? 
4. The fonts in the figures could be improved, in particular, they are way too small compared to the body text, and it seems that there is space.

questions: 1. Is it possible if one repeats the process of learning features from a map and matching the features to produce a map, i.e., steps 2-5, would one get further improvement? 
2. What is the main difference between 4.1 NCP and 4.2 test time denoising? In 4.2 do you train an extra network beyond that already trained in 4.1 NCP? Is the purpose of 4.2 to propose an early stopping criterion for NCP?

limitations: N/A
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: This paper applies the idea of deep prior to predict 3D correspondences. It does so using two ideas, which are demonstrated experimentally in the first part of the paper: first, it shows that randomly intialized diffusionNet features actually provide better results than spectral and SHOT   descriptors, and combined with fonctional maps even improve over learned metods; second, it shows that when training a network to predict features to fit noisy correspondences, the quality of correspondences/features obtained at the beginning of training are actually better than the noisy target. This naturally leads to a method to predict correspondences, obtaining features from a randomly initialized network, then training a second one to predict them and stoping early (using a cycle consistency loss to decide when)
strengths_and_weaknesses: I liked this paper, the idea is simple, well explained and well demonstrated. To the best of my knowledge using neural prior for shape correspondence is novel, if this is indeed the case I think the paper should clearly be accepted.

As for weaknesses:
- Why aren't correspondences evaluated on FAUST? I would really want to see this evaluation, which would also ease comparison with other methods.
- smaller weaknesses:
* I liked the separation of section 3 and 4, but then section 4 points several times to section 5 for important details about the method, I think this makes the reading more complicated than needed, please move all details about the method to section 4.
* I am not a fan of theorem 1. It is actually not related directly to the fact that the mapping are neural nets, it would need a demonstration that this is the general case (i.e. with random weights) for the neural net architecture considered, and both formulation and demonstration in the supmat are very handwavy (for example the theorem is clearly false for d<3, which doesn't appear anywhere). I would thus suggest to completely remove it, or at least not call it a theorem for which I would like a much more rigorous demonstration
* too many unnecessary abbreviations make the text harder to read (FR, SR, NCP, NCP-UN, UM, RN not counting the ones for previous methods ), I would remove at least 50% of them and be sure to regularly remind the reader of the meaning of the ones kept (none, or maybe only NCP would be great)
* fig 2 is not readable in the printed version
questions: See weaknesses (in particular FAUST evaluation)
limitations: yes
review_title: null

final_score: rating: 7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
review_content: summary: The authors analyze the power of random features from 3D encoders and use it to provide methods for point cloud matching, non-isometric shape matching on meshes, part segmentation transfer and keypoint transfer. It is shown that the random features of several architectures provide a strong prior that automatically resolves matching errors in the early iterations of training. Together with a novel stopping criteria, this leads to a method for refining dense correspondences between shapes.
strengths_and_weaknesses: Strengths:
- The paper presents a lot of insight into features extracted from 3D operators. The strength of random features was known before (in CNNs) but this is the first time, it was analyzed for 3D architectures on point clouds/meshes
- The described neural correspondence prior in combination with the cycle loss as early-stopping criteria is an interesting and strong mechanism
- There are a lot of state-of-the-art results beaten here (or on par): point cloud matching, shape matching on non-isometric shapes, part segementation transfer, 3D keypoint detection on point clouds
- The paper is well written

Weaknesses:
- The paper raises a lot of questions (see below), I am not sure if I fully understood the effect of the second stage of the network. It would be good if the authors made an additional effort describing the main intuitions.
- Theorem 1 seems to be not that important to me. The crucial parts are the assumption of smooth encoders and injectivity, which are often not given for existing methods (especially local encoders like PointNet). The consequence of smooth encoders and injectivity is pretty straight-forward. It would be more interesting if the authors would additionally show that the used encoders are indeed injective and smooth.
- It seems to me that the term "noise" is used in a hand-wavy fashion in this work. I would appreciate a more clear definition of "noise" and comparisons between different types of noise if they are applied artificially. I am not convinced that the output of stage 1 of the approach should be called "noisy". The errors seem to be more systematic.
- This paper seems to be divided into to main messages, (1) the analysis of random features/neural correspondence prior, and (2) Theorem 1 and its application using the respective networks (DiffusionNet and PointMLP + fmaps). This makes the paper hard to follow in some parts.

Even if the weaknesses list is longer, I think this is a good paper. I liked reading it and it gave me several insights and a few ideas, which is a good sign. In order to make it really great I would encourage the authors to work a bit more on the structure, discussions of main implications and several details that need to be answered, see below.
questions: - Are many-to-one mappings theoretically allowed / is it just assumed that there is an injective mapping from shape to feature space, as in the requirement of Theorem 1?
- Related to the last question: DiffusionNet is a method that incorporates global information through the global diffusion mechanism. However, other networks, such as PointNet and variants are inherently local, meaning that many-to-one feature mappings will be much more likely, even for isometric shapes. Does the method require a global backend, such as as DiffusionNet (or seemingly PointMLP, not so sure on this one) to work, do the insights in Section 3 only hold for those backends, and in case of a local operator, would an additional consensus method (like [1]) help?
- Would the following statement be correct: The first part of the method finds dense correspondences based on a global feature extractor and fmaps and relies on the near-isometry assumption. Refining the random features during test time is able to resolve errors made due to non-isometric correspondences (in this case, "noise" would not be a reasonable description of such errors). Is this due to the local nature of the random features? 
- Many correspondence methods first do local feature matching and then resolve mistakes by introducing global information in a second stage. Here, it seems to be the other way around. Could the authors discuss the implications of this and which advantages/downsides this variant has?
- I would like to know where the proposed method breaks - how mismatching do the shapes have to be? Since the method seem to rely on a global feature extractor backend, how well does it work on partial meshes? Could a similar technique as in the keypoint extractor be applied?
- I think the paper would benefit from an analysis of different initialization schemes, when using random features. In the end, this seems to be the main part of the paper.

[1] Fey et al.: Deep Graph Matching Consensus, ICLR 2020

Minor:
- l. 61: Correspondnece
- l 281: does not relies
- l 285: "point cloud dataset that provide"
- Figure 2 labels are way too small
- Content of the supplementary materials should be explicitly referenced in the main paper (most important: keypoint application).
limitations: The authors shortly mention some limitations of the method, as it depends on the results of the applied step 1. Discussion of limitations could be improved.
Potential negative societal impact is not discussed but I also see no immediate reason to do so.
review_title: null

final_score: rating: 3: Reject: For instance, a paper with technical flaws, weak evaluation, inadequate reproducibility and incompletely addressed ethical considerations.
review_content: summary: An unsupervised shape correspondence method is proposed :

1. Inputs : training set of shapes (without ground truth)
2. Train an off-the-shelf unsupervised matching "teacher" network on the training set
3. Train a "student" network from scratch on the training set in a supervised manner using the "teacher" predictions as ground truth
4. Use the "student" network to establish correspondences between unseen shape pairs

This method is motived by two empirical observations :

1. "Neural bias" : It experimentally shows that features extracted with a randomly initialized DiffusionNet produce good matching performances on FAUST remeshed and SCAPE-remeshed datasets .
2. "Neural Correspondence Prior" : It experimentally shows that the "Deep Image Prior" holds for a shape correspondence problem.

The proposed method either outperforms or is on par with the sota methods on the KeyPointNet dataset, the SHREC20 dataset,  the SMAL dataset and the PartNet dataset.
strengths_and_weaknesses: ### Strengths

1. Extending the "Deep Image Prior" to shape matching is an interesting idea.
2. The paper is well written.
3. The overall approach is simple.

### Weaknesses

#### 3.1 Neural bias for pointwise features

1. This part of the paper experimentally shows that features extracted with a randomly initialized diffusionNet produce good matching performances on FAUST remeshed and SCAPE-remeshed datasets. I believe the results of a randomly initialized pointwise MLP should be added, otherwise the section title should be changed to "diffusionNet bias for pointwise features".
2. The link between this neural bias (i.e the fact that an untrained network produces good matches) and the proposed method is unclear to me, since both the teacher and the student networks are trained.

#### 4.1 NCP within a shape collection

1. Why not using the standard terminology teacher network / student network ? I think it would make the whole paper easier to read.
2. I was expecting the teacher network to be an untrained diffusionNet, especially after reading section 3.1. It seems to me that section 3.1 is not tightly connected to the rest of the paper.

#### 5.1.1 Correspondences on Point Clouds

1. The proposed method outperforms the sota methods but I could not find the performances of the teacher network (i.e unsupervised functional map method + PointMLP backbone).

#### 5.1.2 Correspondences between non-rigid meshes

1. Table 2 shows that the proposed method only slightly outperforms the DiffusionNet-based methods, for instance "WSupFMNeT + DiffusionNet" is an unsupervised method that is only slighly outperformed by the proposed method.
2. Figure 4 shows the results of NeuroMorph which is the worst unsupervised method in Table 2. Why not showing the results of "WSupFMNeT + DiffusionNet" instead ?

Overall the proposed method is probably interesting for the shape matching community but the novelty is limited. Moreover, it seems the proposed method only slightly outperforms Diffusion-based methods.
questions: Please see "Weaknesses" above.
limitations: -
review_title: null

Meta-review: metareview: This paper received mixed scores, with three reviewer recommending acceptance and one rejection. The reviewers appreciated the simplicity and effectiveness of the method, but nonetheless raised many questions about the method, requesting the authors to clarify several points. The authors' feedback addressed most of these questions. During the discussion, 2gYm, the most negative reviewer, mentioned that they found the contributions interesting for the shape matching community but not significant enough to be published in NeurIPS. Considering that three reviewers found this work sufficiently interesting to recommend acceptance, the AC deems this to be a secondary concern. The AC nonetheless strongly encourages the authors to revise their paper based on their feedback for the camera-ready version.
confidence: Certain
Decision: Accept