__ Summary and Contributions__: This work proposes a novel template-free fully convolutional auto encoder for arbitrary registered meshes. A spatially-varying convolution kernel is used to deal with for irregular sampling and connectivity. It performs experiments on D-FAUST to show the advantage over existing methods.

__ Strengths__: The major novelty is a mesh AE for arbitrary topology which captures effectively captures local geometry property.
Decomposing weights to vertex-specific weighted basis functions is the key to success.
Experiments are thorough and convincing.

__ Weaknesses__: Is the local neighborhood N defined based on Euclidean distance or geodesic distance? It is not clear.

__ Correctness__: Yes.

__ Clarity__: Yes.

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The paper proposes a fully convolutional mesh autoencoder. The key ideas include a spatially varying kernel formed by a linear combination of basis kernels, as well as a more adaptive pooling operation. The method shows improved performance for autoencoder based mesh reconstruction.

__ Strengths__: The ideas are generally plausible.
The method outperforms state-of-the-art Neural 3DMM.

__ Weaknesses__: While using spatially varying kernels may be new for meshes, the idea has been well studied in image-based CNN, and the extension seems fairly straightforward. The paper should refer to the image-based work, e.g.
Adaptive Convolutional Kernels, ICCV Workshop 2019.
and clarify the contribution of the paper .
Comparison with MeshCNN in this paper is a bit unfair as the method is not designed for reconstruction. Other than this, the paper only compares with one method, and could benefit from comparison with more state-of-the-art methods to be convincing.

__ Correctness__: The method seems generally correct.
The paper claims that the capability of handling non-manifold etc. is a strength of the method. However, existing methods based on graph CNN can also be applied to non-manifold cases.

__ Clarity__: The writing is fine.

__ Relation to Prior Work__: As discussed above, the key idea of using spatially varying kernels has been considered in 2D CNN, so this should be discussed:
Adaptive Convolutional Kernels, ICCV Workshop 2019.

__ Reproducibility__: Yes

__ Additional Feedback__: Post-rebuttal comments: The rebuttal has promised to make the requested changes, so I will keep the positive score with the paper.

__ Summary and Contributions__: The paper proposes a new convolution technique for graphs that share the same connectivity. It is based on learning a global "imaginary" linear basis for the convolutional filters. Every neighborhood can define its own kernel by learning coefficients for this basis.
This permits to have an entirely convolutional network, giving also interpretability of the latent representation.

__ Strengths__: Contribution
==========
I think the proposed technique is novel and it proposes a different point of view. I appreciated the effort of defining all the needed layers (Pooling, UnPooling, UpRes, DownRes). I find it really interesting the idea to have an "imaginary basis"; it is somehow related to learning the Gaussian parameters of [25], but with more degrees of freedom.
This idea can be easily extended to several representations as shown in the paper (tetrahedrons, non-manifold mesh). I was wondering: would it be possible to extend it on point clouds, e.g. by using a Gaussian distance to define the neighborhood?
Localized Latent feature Interpolation
========
I think this is a nice point: the capability to have an interpretable latent representation is a hot topic in this research field. While in the video of the hands there are some unnatural artifacts, in Figure 2 the latent representation looks like a skeleton and it seems there are potentially nice connections with standard skinning systems. I would love to see some interpolation examples between humans, in both pose (i.e. same subject in different poses) and identity (i.e. same pose with different subjects) interpolations.

__ Weaknesses__: Geometry
========
From a geometrical point of view, this method takes into account only the connectivity; the shapes are seen as a graph with features (i.e. the 3D coordinates). Also, the vertices need to be ordered, i.e. it assumes to have fixed connectivity in input. I think these are the two major drawbacks. Working with meshes, several applications (e.g. point-to-point matching) cannot be addressed. For example, [25] does not suffer from this problem. Can you think of any strategy to overcome these limits?
I also think that setting the M parameters relies on a further assumption that the connectivity is almost evenly distributed on all the graphs. How does the method perform in case of high-variance in the connectivity?
Experiments
========
More details in the "Correctness" box

__ Correctness__: The paper seems correct to me, from a theoretical perspective. I have only one point about learning the Weight Basis and locally variant coefficients at the same time; it looks like a min-min problem (e.g. finding the best coefficient for the basis, and find the best basis for the coefficients) and in general it is not directly optimizable. I am wondering if sometimes it rises some undefined solutions during training, e.g. basis collapsing in zero-vectors. Have you experienced any issues with this? A visualization of the learned imaginary basis would be definitely useful.
However, my major concern is on the experimental setup. The ablation study is extensive, but the method has been compared with only two previous works, in only one task. No baseline has been provided, and there is a lack of applications (e.g. Analogies, Extrapolation, Latent space samplings present in [7]). I find a bit disappointing that the capability of localized interpolation has so few space in the paper. I would see more on the semantic capability of the learned representation.

__ Clarity__: I think the paper have some issues to fix.
1) As stated in the "Prior work" box, I find the introduction and related work a bit repetitive. I would suggest merging the two sections
2) Table 1 is a bit too small in the printed version, and I think could be better organized.
3) I would rename section 4.4 as "Ablation study", for clarity
4) It is a bit hard to appreciate the trees in Figure 3. Also, it is not super evident how they are non-manifold. I would suggest having a larger figure just for them, maybe with some zoomed detail
5) I would love to have a visual insight into the learned Weight Basis.
6) I would suggest changing the word "topology" with "connectivity"; the first means several different things and could be misleading.
Minor typos:
- row 41: space missing after "data."
- row 53: "is the use a spatially-varying" -> "is the use of a spatially-varying"
- row 127: "Tasks that requires" -> "Tasks that require"

__ Relation to Prior Work__: I think the prior works are well discussed and I have no comment on this. I find only a bit repetitive the introduction and section 2.2; I would suggest merging the two sections to ease the reading.

__ Reproducibility__: Yes

__ Additional Feedback__: 1) The weight basis B are shared for all the mesh (i.e. all the neighborhoods use the same basis set). Then, the M parameter is a fixed scalar for all the neighborhoods, and I think it is around 8-10 for a genus-0 trimesh. Is it correct? I think this is a useful detail. Also: does highly non-uniform connectivity requires special effort?
2) I have not found any mention of your implementation. Are you planning to release it in the future?
3) Since you learn some imaginary basis, they also could be non-linear (e.g. using an hypernetwork setup). There is some advantage to use a linear combination of the learned basis? Also, have you ever had problems with the linear independence of the learned basis?
4) Shouldn't Equation 1 and 3 multiplications between W_j and x_{i,j} be switched to match the dimensionalities (Since x_{i,j} is \in R^I and W_j \in R^{I \times O})?
5) In the "Conclusion" paragraph is stated that "We plan to extend our work in the future so that it can work on datasets with varying topology". Could you elaborate on this point, especially if there are some specific directions to investigate? This extension does not seem straightforward to me, and actually, it is the major limitation of the architecture.
Rebuttal Comment
=================
I thank the authors for their rebuttal and their effort to reply to my questions.
I still have some concerns:
- Provided interpolation example is a “Latent Algebra” example. While it still interesting and I appreciated it, it does not provide an insight into how the intermediate frames look (i.e., semantic coherence between poses). Also, the rendering makes it difficult to understand the impact on the rest of the body
- In the rebuttal is stated that adding regularizations hurts performance. If authors already performed these experiments, I think would be worthful to include them in the paper. Also, no further study has been performed on the learned basis; it is a pity, and would be interesting to provide some insight on this.
- Rebuttal confirms that the work is placed in contexts where the connectivity (and then the correspondence) among the training set is already solved. This limits the applicability of the method (e.g. shape matching, segmentation).
Concluding, while I think the paper still have some experimental\analysis issues, I appreciate the idea and I think it is interesting for the NIPS audience. For these reasons, I will keep my initial vote.