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 generalizes implicit neural representation (INR) to arbitrary topological spaces beyond the regular Euclidean space. Technically, the authors utilize a graph as an approximation of topological space, and compute its spectral embedding as the point-wise encoding to regress an INR. In their experiments, they show the learned INR can be transferable to other graph approximations of the topological space and their techniques are also applicable to conditional INRs.
strengths_and_weaknesses: Strengths:

+ The paper is clearly written and easy to follow.

+ The authors put forward an interesting question, i.e., how to learn an INR over the domain of arbitrary topology. They further propose a simple yet effective and well motivated method. It also has lots of practical usage in geometric, molecular, and climate data representation.

+ The experimental discussion is comprehensive. It demonstrates different applications and verifies the transferability of the learned INR across graphs sampled from the same topological space.

Weaknesses:

- The reviewer cannot agree that the represented signals are still continuous. Approximating a topological space via a graph discretizes the signal space, which makes the input points restricted to the vertex set of the graph. To show the interpolation ability, the authors may first construct an over-dense graph, and then only train the INR on a subset of vertices.

- In regards to the experiment, there is no comparison with other methods. To demonstrate the benefit of involving topological information, authors may compare their model with Euclidean domain INR. For example, also show the texture map regression results on SIREN.

questions: - Solving spectral embedding suffers from the problem of scalability, stability, multiplicity and ambiguity of sign. How did authors solve these practical challenges in this paper?

- Typical INR maps spatial coordinates to the corresponding value. How to bind generalized INR with positional information?

- Can generalized INR solve differential/integral equations like SIREN?

limitations: Generalizing INR to arbitrary topology seems a promising direction. However, the strategy of  approximation via graphs seems to have some limitations. In vanilla INR, arbitrary coordinates in Euclidean space can be directly fed into the INR and query for the value. However, graph discretizes the signal and thus only vertices in the graph can be fed into the INR. That being said, the represented signals are no more continuous. Authors may discuss this limitation and provide preliminary solutions to it.
review_title: null

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 to extend implicit neural representation (INR) to non-euclidean settings, in particular graphs. This is achieved by using the top K eigenvectors of the graph laplacian to replace euclidean coordinates in INR. Experiments on several datasets, including protein and social networks, demonstrate the ability of the generalized INR to represent the signal and generalize to similar graphs faithfully.

strengths_and_weaknesses: **Strengths**

- The paper has a polished and easy-to-read presentation, with many figures that help present the main idea and experimental results. 
- The paper does a good job of reviewing related works both on INR and related developments in geometric deep learning. 
- A wide range of experiments considering different tasks and families of graphs are used for evaluation.

**Weaknesses**

- Some experiments lack proper baselines and quantitative evaluation. Also, although quite pleasing to the eye, figures do not convey the quality of results since comparison to ground truth is missing.   
- Discussions and experiments put more emphasis on reconstruction rather than generalization in using INR, especially generalization to new graphs where this approach can be most interesting could receive more attention. 

questions: - The paper claims that the proposed model is “equivariant” to domain symmetries. I understand that the generation process is “invariant” to the symmetry transformations that leave the graph invariant. What am I missing? Also, these transformations are not necessarily the symmetries of the original manifold but those of the graph.

- For experiments on conditional generation with the Stanford bunny, the figure (figure 9) does not convey the error in the generation process. The same is true for weather modelling (figure 10). Since you have the ground truth, please report the error in your approach. You can also compare to a simple baseline that, for example, uses averaging for interpolation? 

limitations: Limitations are not currently discussed.
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 propose to parameterize implicit neural representations (INRs) as neural networks that take as input the Laplacian eigenvectors of a non-Euclidean domain. This allows for learning INRs over spaces without an external coordinate system. These INRs are tested over various application areas, each of which defines a suitable discrete graph for computing the Laplacian eigenvectors from.

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After reading the authors replies, I have increased my score to a 7. The comparison with other choices / baselines, and the improved exposition / comparison to literature have greatly improved the paper in my eyes.
strengths_and_weaknesses: Strengths:
1. The paper is simple and straightforward, in a good way.
2. This is a natural generalization of the Euclidean case, in which signals are usually defined on a grid. INRs have found much success in the Euclidean case, and they can presumably be very useful for non-Euclidean domains.

Weaknesses:
1. Please be more precise with the claim that the INRs are ``equivariant under the symmetry group of the domain.'' In particular, it would be great to include a reference, or to include a precise statement and proof. I think what you mean is that the eigenvectors are equivariant under the symmetry group, and / or the entire INR algorithm (i.e. the mapping from input to final trained model) is equivariant.
2. Would be nice to compare different choices of Laplacian normalizations, or to compare against using the ground truth eigenfunctions of a continuous surface like the sphere (see Questions section for more). E.g. various normalized graph Laplacians are often used for many applications.
3. The ad-hoc manual choice of sign for the eigenvectors seems a bit hacky for the super-resolution task. Past work has given ways to get rid of sign ambiguity (there is also a basis ambiguity for higher dimensional eigenspaces), see [Lim et al. 2022], which also tests on a generalized INR task on non-Euclidean data.

Minor notes:
1. For graph positional encodings, in addition to random walk PEs there are various other PEs that are sign invariant, see [Lim et al. 2022] Proposition 3 for examples. These include the well known Heat Kernel Signature [Sun et al. 2009].
2. May be worth pointing out that the sinusoidal positional encodings / Fourier features used in Euclidean INRs are eigenfunctions of the Laplace operator on Euclidean space. Thus, your method is a direct generalization.

References:  
[Lim et al. 2022] Sign and Basis Invariant Networks for Spectral Graph Representation Learning. https://arxiv.org/abs/2202.13013  
[Sun et al. 2009] A Concise and Provably Informative Multi-Scale Signature Based on Heat Diffusion. Eurographics Symposium on Geometry Processing.

questions: 1. For the Bunny mesh, do you use the cotangent Laplacian scheme, or do you still use the combinatorial graph Laplacian of the mesh? I'm curious as the cotangent Laplacian scheme is widely used for meshes, but there appears to be no mention of specific Laplacian normalization for this mesh.
2. You treat the Earth as a sphere, correct? If so, it would be possible to evaluate the exact Laplacian eigenfunctions, how does this compare?
limitations: Are the authors aware of [Koestler et al. 2022]? These two papers do the same thing. Given that the idea is so simple, this would be an issue for novelty. I see that it is from about 2 months before the NeurIPS submission deadline, so we may ignore the effect on novelty it if the policy requires.

References  
[Koestler et al. 2022] Intrinsic Neural Fields: Learning Functions on Manifolds. https://arxiv.org/abs/2203.07967
review_title: null

Meta-review: metareview: There is a clear consensus amongst the reviewers that the results are interesting, the manuscript is easy to read, and would be interesting to the NeurIPS community.  One of the main criticisms was the lack of baselines to compare the authors results against, and this was resolved by the authors in revision which encouraged one reviewer to advance their recommendation from weak accept to accept; this can be expected to also have been appreciated by other reviewers.
confidence: Less certain
Decision: Accept