Venue: ICLR 2020 Conference

Reviews:

final_score: rating: 6: Weak Accept
review_content: This paper proposes a neural network architecture which that enables the implementation of group convolutional neural networks for arbitrary Lie groups. This lifts a significant limitation of such models which were previously confined to discrete or continuous compact groups due to tractability issues. 
I'm afraid that this paper is over my head. It relies heavily on field-specific terminology and as such is likely to be accessible to a relatively small subset of researchers. This looks to me like a solid contribution, however I'm really not qualified to judge.
review_title: Official Blind Review #3

final_score: rating: 3: Weak Reject
review_content: The paper proposes an (approximately) equivariant neural network architecture for data lying on homogeneous spaces of Lie groups. In contrast to the Gauge equivariant and Fourier approaches that have recently appeared, here the authors simply put a B-spline basis on local patches of the homogeneous space and move the basis elements around explicitly by applying the group action. 

The approach is appealing in its simplicity and generality. No need to worry about irreducible representations and Fourier transforms, the formalism works for virtually any Lie group, no problem with non-compact groups. However, there is a constant need for interpolation. What is more more significant is that both the homogeneous space and the group need to be discretized and in general that cannot be done in a regular manner (no notion of a uniform grid on SO(3) for example). The authors assure us that "we find that it is possible to find approximately uniform B-splines... e.g. by using a repulsion model". I am not sure that it is so simple. This is one of those things where the idea is straightforward but the devil is in the details.

Theorem 1 seems important but it is a bit cryptic. What is the statement "a kernel satisfying such and such properties gives rise to an equivariant CNN"? Or "A CNN is equivariant if and only the kernel satisfies such and such properties"?

Concerningly, the paper is closely related to a few other papers using the spline CNN idea or at least the idea of taking a fixed set of functions and moving it around on the homogeneous space by acting on it with select group elements, most notably "Roto-translational convolutional neural networks for medical image analysis" by Bekkers et al.. The main difference of the present paper relative to that one is that the idea is fleshed out in a little more detail and is generalized from SE(2) to arbitrary Lie groups. However, conceptually there is little that is new.

In such a situation it would be important to present convincing experiments. Unfortunately in the present paper, results are only presented on 2 datasets, and the algorithm is basically only compared to different versions of itself, rather than state of the art competitors.

The paper is clearly written but the intuitive nature of the core ideas could be better conveyed e.g. by fancy diagrams.
review_title: Official Blind Review #2

final_score: rating: 8: Accept
review_content: In this paper, a framework for building group CNN with an arbitrary Lie group G is proposed. Generally, such a group CNN consists of 3 types of layers: a lifting layer which lifts a 2D image to a 3D data (G-image) whose domain is G; a group correlation layer which computes a 3D G-image from a 3D G-image; and a projection layer from a 3D G-image to a 2D image. To implement the convolutions in the lifting layer and group correlation layer which are defined in the continuous setting, the B-Spline basis functions are applied to expand the convolution kernels. Experimental results on tumor clarification and landmark localization show the superiority over CNN.

Advantages:
1. A flexible framework for group convolutional neural network is proposed with strong theoretical support in Theorem 1.
2. Familiar properties of convolutions from classical CNN design (like localized, atrous, and deformable convolutions) can also be implemented in G-CNN using specified B-Spline basis functions.
3. In comparison with standard CNN, the effectiveness of the B-Spline-based G-CNN is validated through experiments on two typical data sets. 

Weakness:
1. [Readability] For readers who are not familiar with Lie groups, this paper is very hard to follow. 
(1)	For Theorem 1, the authors are suggested to give some illustrative explanation. Besides, what is “Stab_G”? 
(2)	The architecture of G-CNN, i.e., the 3 types of layers, are directly given in Eqs. (5)-(7) without examples, illustrative examinations, or visual illustrations.
(3)	Fig. 1 can be modified for better readability. 
 
2. [Experiments] The proposed G-CNN has some similarities with data augmentation (like rotation, scaling) based CNN. Then, how better can the G-CNN perform than CNN with data augmentation? More experiments on this point are suggested, and relevant theoretical explanations will be appreciated.  

3. [Implementation] Considering the complicated mathematics in this paper, I am afraid that implementation of the proposed G-CNN is also very hard. It would be better for the authors to discuss the implementation. In my mind, if the implementation is not so hard, then the formulation of G-CNN can also be simplified for better readability. 

review_title: Official Blind Review #1

Meta-review: The paper describes principles for endowing a neural architecture with invariance with respect to a Lie group. The contribution is that these principles can accommodate discrete and continuous groups, through approximation via a base family (B-splines). 

The main criticisms were related to the intelligibility of the paper and the practicality of the approach, implementation-wise. Significant improvements have been done and the paper has been partially rewritten during the rebuttal period.

Other criticisms were related to the efficiency of the approach, regarding how the property of invariance holds under the approximations done. These comments were addressed in the rebuttal and the empirical comparison with data augmentation also supports the merits of the approach.

This leads me to recommend acceptance. I urge the authors to extend the description and discussion about the experimental validation. 

Decision: Accept (Poster)