Summary
The paper is theoretical (no experiments) and describes two things. Firstly, it shows that if two signals are equal (up to some epsilon error) through an affine transformation, then representations obtained by group convolutional neural networks are also similar up to some (scaled) epsilon. This follows from the equivariance principle (theorems 1-3). Secondly, the paper shows how group convolutions for affine groups can be efficiently computed through 1) seperable factorization of the kernel $k(x,A)$ into $k_1(x)k_2(A)$ and 2) making use of the fact that integration over the GL2 part of the affine transformation (over A) can be further split via a QR decomposition. This result is summarized in Theorem 4.
Weaknesses
1. Often times, the paper is not very clear. Some definitions are not very precise and often the notation is sloppy (def 2 what is a symmetry of a Riemannian manifold, in def 3 alpha is introduced but not used, example 3 contains a type, equation above (2) uses symbol h, but the expression should not depend on h and moreover h is not defined yet, etc.)
2. The overall structure of the paper is hard to follow. What precisely is the goal of the paper? The paper loosely talks about the problem of matching of functions f1 and f2 without making concrete how this happens. It only defines the notion epsilon-affine invariance but not how we determine when to functions are epsilon invariant. Moreover, the paper on several occasions says that it it wants to avoid solving complex optimization problems (please be more precise on what kind of problems) but does not explain how the paper does it instead. I suppose the motivation is that one can learn representations of functions through affine equivariant mappings, and then measure similarity between those learned representations, but I can only guess.
3. Theorems 1 to 3 all seem to be intimately related and could probably be merged, in particular 1 and 2 seem almost identical. Could these not be merged by stating that affine invariance is maintained for any equivariant mapping $\Phi$ (with the property $\Phi[\rho(g) f] = \rho'(g) \Phi[f]$, with $\rho$ and $\rho'$ representations for the in and output resp.). An intuitive explanation up front would also help; why are these theorems relevant?
4. Section 2.2 then seems to be the most important result but feels a bit rushed and the essence of it is hard to grasp (I hope my summary above is correct).
5. The objective of the paper seems to be efficiency, but there is no proof of principle on the actual usefulness of the approach. I.e., I imagine that splitting the convolution over 4 integrals, including forward and backward Fourier transforms and change of variables might not be more efficient than applying out of the box group convolutions.
6. Example 4 is incorrect and is the result from an earlier typo.
Smaller comments:
1. The abstract uses $G_2$ as a symbol, but it is unclear what it represents. (The affine Lie group $R^2 \rtimes GL_2(\mathbb{R}^2)$, I imagine). I don't think it is ever defined.
2. There are 3 papers which might be relevant based on the notion of separability of group convolutions. [Knigge et al. 2022] Uses separability convolutions to achieve equivariance to scale-rotation-translation transforms. [Chen et al. 2021] uses separability for efficient implementations for SE(3) equivariance. [Mironenco and Forre 2023] have almost exactly the same results as this paper (decomposition of the GL group), and also builds on the works from the group of Taylor, so it seems. This last reference however appeared after the ICLR submission deadline so I of course do not expect you to have known about this paper, it could however a helpful reference in improving the narrative of the paper.
[knigge et al] Knigge, D. M., Romero, D. W., & Bekkers, E. J. (2022, June). Exploiting redundancy: Separable group convolutional networks on lie groups. In International Conference on Machine Learning (pp. 11359-11386). PMLR.
[chen et al] Chen, H., Liu, S., Chen, W., Li, H., Hill, R.: Equivariant point network
for 3d point cloud analysis, 14514–14523 (2021)
[Mironenco and Forre] Mironenco, M., & Forré, P. (2023). Lie Group Decompositions for Equivariant Neural Networks. arXiv preprint arXiv:2310.11366.
3. The contributions again say rather then focussing on complex optimization problems, we study invariance through convolution integrals. I do not follow this logic. Simply studying invariance obviously does not require an optimization problem. What precisely is the alternative to the complex optimization problem? This is unclear to me. I do understand the implications of the theorems, but no where in the paper do I learn how these results are to be used in practice.
4. What is the reason of writing presenting the kernel transformation (above definition 1)? I think part of the story is missing that tells that under and equivariance constraint this operator reduces to a group convolution? I am guessing this is the story because the paper so far closely follows the paper Bekkers 2019 as well as his lecture series on Youtube (might be a useful reference to add? https://uvagedl.github.io)
5. Example 3 contains a typo. It should be $f(R^{-1}_\theta (y - x)$, the parentheses were missing. This mistake is later on used in Example 4. The resulting kernel operator in example 4 is therefore also false.
6. In definition 2 it is unclear what is meant with symmetries of a Riemannian manifold, it seems like an odd way to define a Lie group.
7. In definition 4, alpha is defined but not used. Instead $\odot$ is used but not defined.
8. In section 1.2 the isotropic kernel convolution is defined, but this definition should not have the 1/|det h| in it, because the left-hand side does not depend on $h$, and moreover, $|\det h| = 1$ if $H=SO(d)$ (which is also not defined).
9. Example 4 is incorrect as $g^{-1} \tilde{x} = R^{-1}(\tilde{x} - x)$ and the resulting $\mathcal{K}$ should be $(\mathcal{K} f)(x,h)=f(x)/|\det h|$.
10. In section 2.1 when $G_2$ finally gets defined, you should also define the group product. Without a definition of a group product the definition of the group is not complete. Currently only the action on $\mathbb{R}^d$ is given. I know this is a bit nit-picky because the full group product merely involves the product of the GL2 part (A.B).
11. The short discussion on the Clebsch-Gordan theory is a bit odd. It is not necessarily used for "categorizing kernels", rather I would say it is a framework for working with irreducible representations and tensor products relative to basis for vector spaces that transform via the irreps. And then, I do not see the connection to the subsequent sentence "In contrast to the traditional approach used to establish...". Some logic here is missing.
12. In theorem 1, there are two types of $g$... One is used as input coordinates to the feature maps, the other is used to parametrize the transformation.