Capsule networks are constrained by the parameter-expensive nature of their layers, and the general lack of provable equivariance guarantees. We present a variation of capsule networks that aims to remedy this. We identify that learning all pairwise part-whole relationships between capsules of successive layers is inefficient. Further, we also realise that the choice of prediction networks and the routing mechanism are both key to equivariance. Based on these, we propose an alternative framework for capsule networks that learns to projectively encode the manifold of pose-variations, termed the space-of-variation (SOV), for every capsule-type of each layer. This is done using a trainable, equivariant function defined over a grid of group-transformations. Thus, the prediction-phase of routing involves projection into the SOV of a deeper capsule using the corresponding function. As a specific instantiation of this idea, and also in order to reap the benefits of increased parameter-sharing, we use type-homogeneous group-equivariant convolutions of shallower capsules in this phase. We also introduce an equivariant routing mechanism based on degree-centrality. We show that this particular instance of our general model is equivariant, and hence preserves the compositional representation of an input under transformations. We conduct several experiments on standard object-classification datasets that showcase the increased transformation-robustness, as well as general performance, of our model to several capsule baselines.
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)?