Venue: NeurIPS 2022 Conference

Reviews:

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 manuscript introduces shape matching framework based on multi-resolution functional maps. the idea is that the Laplace-Beltrami operators and their spectra are computed for shapes at various resolution, and for each resolution the corresponding functional map is computed. In order to make eigenfunctions and functional maps at various resolution comparable, a spectral upscaling scheme is adopted.
With he functional maps to hand, an attention mechanism based on mapped-point residual is adopted. The output of the attention network is a set of weight used to linearly combine the the maps.
strengths_and_weaknesses: The addition of the multi-resolution aspect o functional maps is sound and interesting, and the attention mechanism provides a smart data-driven way to select the resolutions. However, what it lacks is locality: different scales can behave differently in different regions of the shape, especially in conjunction to non-uniform sampling.

I quite like the fact that the approach can work both in a supervised and unsupervised setting, but I believe that a supervised setting where the map is known is a bit unrealistic. Perhaps a setting where known corresponding functions over the two shapes are given would form a better use-case (incidentally, that was the original use-case for functional maps).
questions: Does the approach assume uniform (possibly unequal) sampling of the shapes? I see no indication of the use of the area elements in the spectral decomposition and the multi-resolution approach would probably need to adapt to non-uniform sampling.


limitations: I believe that the negative societal impacts have been assessed correctly. As for Limitations, the performance issue with spectral decomposition is common to all spectral approaches, but they also incur other, arguably more severe limitations regarding their robustness to noise, extreme partiality, and topological deformation. That being said, the present proposal is affected by those limitations more than other spectral approaches because it needs to compute multiple spectral decomposition (one per scale) and then perform the upscaling since in the end it needs to work at the highest resolution.
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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 paper proposes a neural network architecture that exploits multi-resolution functional maps to predict a pointwise correspondence between two shapes. The architecture is composed of two blocks. The first block follows the idea of Deep Functional Maps (REF) and consists of multiple networks each predicting a functional map at a different resolution. The second block takes as input these functional maps and the residual (i.e. the eigenbases alignment error per each point of the two shapes) and computes an attention score with which to derive the final functional map as a weighted sum of all the functional maps. To make it possible to sum the maps at different resolutions, the authors propose to adopt a differentiable upsampling scheme inspired by ZoomOut. The method is validated on common 3D non-rigid datasets composed of humanoid and animal shapes.
strengths_and_weaknesses: The paper addresses a common problem in the design and application of Functional Map-based methods, which consists of the choice of suitable size of the truncated eigenbasis for the problem at hand. The proposed data-driven approach is for sure an interesting solution. 

Unfortunately, an analysis of the obtained attention scores in different scenarios (e.g. near-isometric vs non-isometric) is missing and would have been a strong contribution to the paper. Indeed, as a motivation for this work, the authors suggest that few basis functions are prefered for highly non-isometric shapes while near-isometric shapes can benefit from the use of a wider eigenbasis that includes also higher frequencies. Even if this sounds like a reasonable premise, there isn’t any analysis supporting this claim.

As a last note, I found the paper generally well written, but some details on the method implementation are missing, or at least not perfectly clear.

questions: I’m a bit confused about the use of this attention scheme with the fast version of the network, where lower resolutions are derived as submatrices. In this case, it seems to me that the only difference between the different resolution FMs is given solely by the upsampling strategy. Are the upsampled matrices improving the final result or is most of the attention put on the larger FM? If for ZoomOut it is reasonable that, with few corresponding functions, solving for a smaller optimization problem and then upsampling can lead to better results, in this case, the method solves the least-square problem for the wider basis-function and then takes a submatrix. It is not obvious to me that this leads to the same result.

Also, following the previous point, is  L_{inter} for the fast version computed only for the wider FM? If so, why the huge difference in tab 3?

Minor:
- Maybe I missed it in the paper, but how are the point-wise correspondences computed?
- Description of deep functional map refers to a course. Isn’t there any reference to the original paper proposing this method?
- I’m also not sure of understanding the ablation of \tau. In tab 3 results with 5 and 20 are reported, while the full model should use 10, justifying this choice as a trade-off between speed and performance. Looking at the results I do not see this trend, in fact, both 5 and 20 are way worse than the Full model (with tau=10). Am I missing something?
- In the comparison with other methods, it is worth mentioning which kind of date each method works. Some of the methods take as input point clouds, while the proposed method requires triangular meshes (at least in the shown experiments). Also, even if not critical, it would be worth adding recent SOTA methods for shape registration with transformers (Trappolini et al., Shape registration in the time of transformers, NeurIPS 2021).

limitations: The paper clearly states the main limitations of the proposed approach, including the need for triangled meshes and the lack of experiments on partial ones. (eg. SHREC’16: Partial matching of deformable shapes, Matching deformable objects in clutter)
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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: The paper proposes a learning-based framework for non-rigid shape matching by building upon the functional map representation, and especially its learning-based variant GeomFmaps. The major contribution of the paper is learning the soft thresholding weights to combine functional maps obtained from matching at different spectral resolutions, which allows the network to optimize the spectral resolution. The paper illustrates the effectiveness of the combination on various benchmarks, on which it achieves state-of-the-art or comparable performance. 
strengths_and_weaknesses: Strengths:
Extensive experiments and state-of-the-art performance - the authors perform evaluations on widely adopted human shape matching datasets such as FAUST and SCAPE and a non-isometric dataset SHREC’19. Table 1 shows that on these datasets the paper outperforms the GeoFmaps (the method extended in this paper) and also outperforms all the other methods (presented), with a few exceptions. 
The authors further evaluate on an animal shape matching dataset SMAL and show (Table 2) considerable improvement on GeomFmaps (e.g., in supervised setting: 5.3 vs. 8.4 mean geodesic error X 100) and DeepShells (unsupervised setting 5.3 vs. 29.3 ). Additional evaluation is provided on the challenging humanoid shape dataset from DeformingThings4D  for both near-isometric and
highly non-isometric correspondence. The authors show improved performance intra-class in supervised setting and intra and inter- class in unsupervised setting (Table 2). The authors also provide an ablation study Table 3, that shows the importance of learned spectral attention (linear combination weights of the functional maps).


Weaknesses:
1) Limited novelty of the idea:  lines (54 -  57) "Finally, to train our network, we propose to impose penalties on the intermediate multi-resolution functional maps as well as the final map. This is different from existing approaches [7, 9, 5, 10], to name a few, that penalize a single hand-picked spectral resolution" --Essentially the only novelty of this paper is learning of the soft-thresholding of the / best weighted combination of functional maps obtained from matching at different spectral resolutions. While the benefit to the performance is substantial, the idea is not essentially novel (it makes sense that a learned combination of weights would work better).
2) Extension of an existing work: The authors claim that their method is generic enough to directly benefit from other advances in deep functional map training, such as improved architectures or regularization. Note that this claim seems reasonable based on the architecture introduced in Figure 2 (the blue block can be probably replaced with a different architecture), however it is not supported by experiments. This makes their work seem like an extensions of one specific approach. 
3) There is little deep dive into the results, for example the authors do not elaborate on why the more efficient technique named "Ours-FAST" sometimes outperforms "Ours", which is not intuitive.  The authors only say  lines 293 - 295: "Besides, the fast version of our method gives almost the same results as the complete approach, and  even better results in some cases. This underlines the fact that the approximation made in this version of our model works well in practice.". There is no insight as to why this happens.
questions: 1) line 47 -  "our framework consists **in** two novel.."  -- Did you mean "consists in" or "consists of"? 
2) "Shape 40 in SHREC’19 is a partial shape and removed from the dataset, since it is outside the scope of this work" - it is not clear why partial shape is out of scope of this work, please explain (line 123 actually mentions that attention has been previously applied to partial shape matching) 
3) Please explain why the fast method yields improved results in many cases (Tables 1 and 2).
limitations: To the best of my understanding there is no negative societal impact to the paper. The authors mention human surveillance, which does not  seem to be a direct application of the shape matching.
Otherwise the authors discuss general limitations -  computationally non-efficient computation of spectral decomposition for each input shape. (lines 360 - 363 "While the pre-computation is efficient for moderately sized shapes, down-sampling or remeshing may be needed for large-size input").
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Meta-review: metareview: All reviewers voted for acceptance of the paper. Reviewers acknowledge that the paper addresses an important problem: choosing the size of the truncated Eigenbasis for matching using functional maps. Also strong empirical performance on a number of datasets was noted. The rebuttal also addressed many points raised by reviewers and generally improved our impression of the paper. Overall this paper is a nice mix of theoretical contribution and practical performance. Therefore the paper is recommended for acceptance.
We ask the authors to incorporate the feedback given in the review and discussion phase.
confidence: Certain
Decision: Accept