Venue: ICLR 2020 Conference

Reviews:

final_score: rating: 8: Accept
review_content: This paper combines CapsuleNetworks and GCNNs with a novel formulation. First they modify the CapsNet formulation by replacing the linear transformation between two capsule layers with a group convolution. Second they share the group equivarient convolution filters per all capsules of the lower layer.  Third, they change the similarity metric from a lower-upper similarity into a pairwise lower similarity and aggregation which makes it keep the equivarience. Since the cij does not depend on upper capsule anymore they only perform 1 routing iteration (no modification of the routing factors).

One assumption in CapsNets is that each part belongs to one whole. Therefore, the normalization in Alg.2 usually is division by degree^k_i. The proposed normalization formula for c_ij seems to encourage that each upper capsule only receives one part. Is this a typo or is there a justification for this?

The discussion on ideal graph on page 5 is interesting. But the points made are not used later on. I expected the results to have an analysis or at least a show case that indeed if you transform the resultant graphs stay isomorphic. 

One goal for CapsuleNetworks vs GCNNs is the hope for handling different transformations and not only rotations that one can grid with group convolutions. But, the experiments only report on rotation, translation as a transformation. Reporting results by training on MNIST, testing on AFFNIST could shed light on this aspect of SOVNETs.

Conditioned that the last two points will be addressed in the rebuttal I vote for accepting this paper since they suggest a novel formulation that brings some measures of rotation equivarience guarantee into CapsNets. Also their results suggest that there is no need for per Capsule filter bank and several refinements to get rotation robustness (it would be interesting to check the performance of a simple capsnet with shared parameters).  In the appendix there is a comparison with GCNNs on fashion MNIST which shows they have better performance than GCNNs. I would advise reporting GCNNs for all the experiments in the main paper. 


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Thank you for updating and expanding the paper. The extra experiments, isomorphism analysis and their response regarding the attention vs part-whole makes the paper much stronger. Therefore, I am increasing my score.
review_title: Official Blind Review #2

final_score: rating: 6: Weak Accept
review_content: In this work, a method was proposed to train capsule network by projectively encoding the manifold of pose-variations, termed the space-of-variation (SOV), for every capsule-type of each layer. Thereby, the proposed method aims to improve equivariance of capsule nets with respect to translation (rotation and scaling).

The proposed method is interesting and the initial results are promising. However, there are various major and minor problems with the work:

- There are various undefined functions and mathematical notation, such as the following:

- Please give formal and precise definitions of groups and group representations for readers who are not familiar with mathematical groups.

- What are GetWeights and Agreement used in Algorithm 1?

- Please define “routing among capsules” more precisely.

- How do you calculate Pool() more precisely?

- In the paper, the results are given for a general class of groups. However, it is not clear how these results generalize even for some popularly employed groups, such as Z^n, S_n, SO(n), SE(n) etc., with different symmetry properties, base space, and field type.

- Please check the following paper for a detailed discussion on group equivariant CNNs with different group structures, and elaborate the theoretical results for particular groups (e.g. at least for p4m used in experiments) :

T.S. Cohen, M. Geiger, M. Weiler, A General Theory of Equivariant CNNs on Homogeneous Spaces, NeurIPS 2019

- Please define the norms used in Algorithm 2.

- How do you calculate accuracy of models? Are these numbers calculated for a single run, or for an average of multiple runs? If it is the former, please repeat the results for multiple runs, and provide average accuracy with variance/standard deviation. If it is the latter, please provide the variance/standard deviation as well.

- Have you performed analyses using larger datasets such as Cifar 100 or Imagenet? It would be great to provide some results for larger datasets to explore their scalability. 

- Please define accuracy given in tables more precisely, use dot "." at the end of sentences in captions.

- There are several typo/grammatical errors, such as the following:

-- Homegenous -> homogeneous

--  for with prediction networks

-- Please proof-read the paper in detail and fix the typo etc.

After the discussions:

Most of my questions were addressed and the paper was improved in the discussion period. Therefore, I increase my rating.

However, some parts of the paper still need to be clarified. For instance;

- GetWeights: To ensure that the predictions are combined in a meaningful manner, different methods can be used to obtain the weights. The role of GetWeights is to represent any such mechanism. 

-> Please define these methods in detail and more precisely, at least in the Supp. mat.

- Agreement : The Agreement function represents any means of evaluating such consensus.

-> This is also a very general concept, which should be more precisely defined. 

- Our theoretical results (Theorem 2.1 and Theorem 2.2) hold for all general groups, and the particular group representation  defined in the main paper. 

-> Could you please give a concrete discussion on generalization of these results and the proposed algorithms for discrete and continues groups? For instance, how do these algorithms and results generalize with Z^n and SO(n)?

review_title: Official Blind Review #1

Meta-review: This paper combine recent ideas from capsule networks and group-equivariant neural networks to form equivariant capsules, which is a great idea. The exposition is clear and the experiments provide a very interesting analysis and results. I believe this work will be very well received by the ICLR community.
Decision: Accept (Talk)