Group convolutional neural networks (G-CNNs) can be used to improve classical CNNs by equipping them with the geometric structure of groups. Central in the success of G-CNNs is the lifting of feature maps to higher dimensional disentangled representations in which data characteristics are effectively learned, geometric data-augmentations are made obsolete, and predictable behavior under geometric transformations (equivariance) is guaranteed via group theory. Currently, however, the practical implementations of G-CNNs are limited to either discrete groups (that leave the grid intact) or continuous compact groups such as rotations (that enable the use of Fourier theory). In this paper we lift these limitations and propose a modular framework for the design and implementation of G-CNNs for arbitrary Lie groups. In our approach the differential structure of Lie groups is used to expand convolution kernels in a generic basis of B-splines that is defined on the Lie algebra. This leads to a flexible framework that enables localized, atrous, and deformable convolutions in G-CNNs by means of respectively localized, sparse and non-uniform B-spline expansions. The impact and potential of our approach is studied on two benchmark datasets: cancer detection in histopathology slides in which rotation equivariance plays a key role and facial landmark localization in which scale equivariance is important. In both cases, G-CNN architectures outperform their classical 2D counterparts and the added value of atrous and localized group convolutions is studied in detail.
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.