CONVOLUTIONAL NEURAL NETWORKS ON NON- UNIFORM GEOMETRICAL SIGNALS USING EUCLIDEAN SPECTRAL TRANSFORMATION
Convolutional Neural Networks (CNN) have been successful in processing data signals that are uniformly sampled in the spatial domain (e.g., images). However, most data signals do not natively exist on a grid, and in the process of being sampled onto a uniform physical grid suffer significant aliasing error and information loss. Moreover, signals can exist in different topological structures as, for example, points, lines, surfaces and volumes. It has been challenging to analyze signals with mixed topologies (for example, point cloud with surface mesh). To this end, we develop mathematical formulations for Non-Uniform Fourier Transforms (NUFT) to directly, and optimally, sample nonuniform data signals of different topologies defined on a simplex mesh into the spectral domain with no spatial sampling error. The spectral transform is performed in the Euclidean space, which removes the translation ambiguity from works on the graph spectrum. Our representation has four distinct advantages: (1) the process causes no spatial sampling error during the initial sampling, (2) the generality of this approach provides a unified framework for using CNNs to analyze signals of mixed topologies, (3) it allows us to leverage state-of-the-art backbone CNN architectures for effective learning without having to design a particular architecture for a particular data structure in an ad-hoc fashion, and (4) the representation allows weighted meshes where each element has a different weight (i.e., texture) indicating local properties. We achieve results on par with the state-of-the-art for the 3D shape retrieval task, and a new state-of-the-art for the point cloud to surface reconstruction task.
Overall Thoughts:
I think this paper addresses an interesting topic and that the community as a whole are interested on the application of learning algorithms to non-Euclidean domains. It is nice to see the application of Fourier sampling to geometric primitives in a sensible manner and I am positive about that part of the paper. However, in its current form, I have quite a few questions about the approach and the empirical studies - I would need to here more information from the authors on the points below.
Specific Comments/Questions:
Sec1: The authors make a number of assertions about the representational errors that occur in other approaches - I feel that these claims should be supported by specific references.
Contributions: It is not clear to me that the experiments show that the method “preserves maximal information content” - in my understanding, information content has a formal definition and I don’t see where this is presented in the results?
Sec2: Before CNNs there has been a substantial analysis of Fourier methods applied to shape models, e.g. elliptical Fourier series for shape contours (and signed distance representations) by Prisacariu et al.
Sec2: It is also worth noting that there is substantial literature on non-uniform Fourier methods including the non-uniform Fourier transform and a number of accelerations (e.g. NUFFT) as well as consideration of the implications of band-limiting the sampled spectra.
Sec3: The mathematical derivation all makes sense to me and makes use of results for piecewise uniform signals. Please could the authors provide more details on how the spectra are represented? These discontinuous signals (esp. delta functions) will have infinite bandwidth in the spectral domain so how they are stored would seem very important to me. Are the signals band limited at some point? If so, how does this affect the approximation and should filtering/windowing be used? Otherwise is there not a difficult storage problem? The final paragraph suggests all the analytic signals might be stored but this has a big impact on how efficient the algorithm is and really is far too important a point to just have a single sentence - please can the authors expand on how this is actually implemented and what the computational considerations are (and resulting impacts on performance)?
Sec4: Please can the authors add error bars (at the least) to all the tables/plots in the results. It is entirely unreasonable to make any statements about how significant the results may be without even the most basic of analysis. Ideally we should see histograms of the results for the retrieval and shape reconstruction results. 
Sec4: How is the downsampling performed in the MNIST experiment? It would seem very important to take care with this for the purpose of comparison. A significant disparity between the signed distance function and the NUFT would seem slightly surprising to me? Again, without error bars we really cannot say very much about the results on the right of Fig3(b).
Sec4: Could the Fig4 results not be provided as histograms? We also need many more details about how the results were obtained and procedures to ensure that the results are meaningful and robust (e.g. repeated tests and partitions of the data, etc..)
Arch and Training Details: How are we to know that these choices provide fair comparisons to previous approaches?
Sec3/4: All the results seem to require the input to be reconstructed back into a dense sampling domain (via inverse FT) - is this the case? Would it not be more efficient to perform the convolutions in the spectral domain where the signal is sparse?
Sec4: It seems pretty unfair to train and test on a single category of shapes in the shape test since the data is known not to be very diverse? Particularly when, unless I’ve misunderstood, the baselines on the shape recovery test to not involve learning and so (while helpful to have) they are not really fair baselines compared to other learning approaches? Also, please can the authors provide much more information about the extra processing applied (e.g. the part starting with the “extra mesh thickness”) since there seem to be some extra steps that are nothing to do with the rest of the method and may impact the results significantly. 
Sec4: It is interesting that Table 3 (again difficult to say without error bars) indicates that there are times when the method performs better with the additional of noise - this seems counter-intuitive - please could the authors comment on this?
Other Points:
I’m afraid that there are quite a few grammatical errors in the text (too many to list here) so I would recommend another round of proof-reading.