A general solution of the kernel space constraint is given for arbitrary representations of the Euclidean groupE(2) and its subgroups.

We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic classification of all existing G-CNNs in terms of their symmetry group, base space, and field type. We also answer a fundamental question: what is the most general kind of equivariant linear map between feature spaces (fields) of given types? We show that such maps correspond one-to-one with generalized convolutions with an equivariant kernel, and characterize the space of such kernels.

We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic classification of all existing G-CNNs in terms of their symmetry group, base space, and field type. We also answer a fundamental question: what is the most general kind of equivariant linear map between feature spaces (fields) of given types? We show that such maps correspond one-to-one with generalized convolutions with an equivariant kernel, and characterize the space of such kernels.

I would give an accept score if I were able to have a look at the new version and be happy with it (as is possible in openreview settings for example). However since improving the presentation usually takes a lot of work and it is not possible for me to verify in which way the improvements have actually been implemented, I will bump it to a 5. I do think readability and clarity is key for impact as written in my review, which is the main reason I gave a much lower score than other reviewers, some of whom have worked on exactly this intersection of algebra and G-CNNs themselves and provided valuable feedback on the content from an expert's perspective. The following comments are based on the reviewer's personal definition of clarity and good quality of presentation: that most of the times when following the paper from start to end it is clear to the reader why each paragraph is written and how it links to the objective of the main results of the paper, here claimed e.g. in the last sentence to be the development of new equivariant network architectures. The paper is one long lead-up of three pages of definitions of mathematical terms and symbols to the theorems in section 6 on equivariant kernels which represent the core results of the paper. In general, I appreciate rigorous frameworks which generalize existing methods, especially if they provide insight and enable the design of an arbitrary new instance that fits in the framework (in this case transformations on arbitrary fields).

We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic classification of all existing G-CNNs in terms of their symmetry group, base space, and field type. We also answer a fundamental question: what is the most general kind of equivariant linear map between feature spaces (fields) of given types? We show that such maps correspond one-to-one with generalized convolutions with an equivariant kernel, and characterize the space of such kernels.

We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivariant maps between spaces of fields. The theory enables a systematic classification of all existing G-CNNs in terms of their symmetry group, base space, and field type. We also answer a fundamental question: what is the most general kind of equivariant linear map between feature spaces (fields) of given types? We show that such maps correspond one-to-one with generalized convolutions with an equivariant kernel, and characterize the space of such kernels.