We thank the reviewer for appreciating our work and providing valuable feedback.

Q18a. On the difference from the benchmark of [1]

We thank the reviewer for the opportunity to clarify the difference between the rotated MNIST experiment in [1] and our paper.

[1] Weiler et al., General E(2) - Equivariant Steerable CNNs, NeurIPS 2019

Our experiment aims to measure the models' equivariance guarantee at a finer granularity and evaluate model performance in a limited data setting. Hence, we consider a different experiment setup and use different evaluation metrics. Due to the significant differences in experimental setup and evaluation metrics, a direct comparison with the Tables of [1] is not meaningful.

However, we would like to explain why the experiment is still a meaningful comparison to the method in [1].

First, we want to clarify that, in Table 2, the model without any subgroup subsampling is equivalent to the architecture proposed in [1].

In [1], the training set is composed of $12K$ randomly rotated (+reflected) images of MNIST. The test set is also composed of random rotations of the original MNIST test set. This means that the model in [1] is trained on rotated samples and evaluated on similarly rotated data.

In practice, equivariant CNNs are more applicable for scenarios with limited data, where regular CNN fails and equivariance properties of $G$-CNN models enable better generalization. To reflect this, our experiment uses a more conservative setup designed to simulate a scenario with data scarcity.

In our experiment, the training set is composed of $5K$ unrotated MNIST images.

We test on

1. Unrotated MNIST test set($\tt ACC_{\tt no aug}$)
2. All possible rotation (reflection) of MNIST test set ($\tt ACC_{orbit}$)
3. Rotated test set with angle less than $60^\circ$ ($\tt ACC_{loc}$). We follow the same setup for the CIFAR experiment.

The reasons for such choices are: a. Unlike [1], the model does not observe any rotated samples during training, which prevents the model from learning the symmetry constraints from the training data and reveals the model's actual equivariance property by architectural design (i.e., equivariance constraint imposed by design).

b. Our evaluation metrics make it easy for us to evaluate the model's performance specifically under all possible group actions ($\tt ACC_{orbit}$), group action within some limit ($\tt ACC_{loc}$), and without any group action ($\tt ACC_{\tt no aug}$). On the other hand, if we test on a randomly rotated test set (as done in [1]), we can not explicitly measure the actual consistency of the model under all group action.

In the updated manuscript, we explain the rationale behind our experimental setup (Appendix Sec A7 Lines 1190-1197).

In order to show the limitations of the experiment setup of [1], we now report the results following their setup. We train our models on $12K$ randomly rotated (+refleted for $O(2)$ symmetry) training data.

We report the accuracy on a similary augmented test set. We observe that the performance now matches the performance reported in [1] (see Figure 2 [1]). However, as the training set is large and also contains augmented data, all models achieve high accuracy and the gap between them is minimal. Despite the similar in performance, we still observe that anti-aliasing always improved the performance by a small amount.

SO(2) Symmetry

O(2) Symmetry

We thank the reviewer for appreciating our work and providing valuable feedback.

Q7. On the connection of our approach to existing equivariant architectures

Thanks for the opportunity to clarify our connection with equivariant neural networks. For simplicity, we explain using a $2D$ image as an input to the $G$-CNN models.

In the group equivariant convolution neural network (g-CNN) the input image $f: \mathbb{Z}^2 \rightarrow \mathbb{R}^k$ (k=1 or 3 depending on whether the image is grayscale or colored) are first lifted to the space of roto-translation or dihedral-translation group ($\mathbb{Z}^2 \rtimes C_N$ or $\mathbb{Z}^2 \rtimes D_N$) by the following lifting operation [1]

I appreciate the author's rebuttal. I greatly appreciate the explanation regarding the connection with existing architectures, and I think Figure A.10 is a great visual. (You could also incorporate $\psi$ somehow to make it a bit more complete.) I have one small follow-up on that discussion: under this presentation, 5.1 should still have the form of $f: \mathbb{Z} \to \mathbb{R}$ with the lifting to $\mathbb{Z}\rtimes C_n$, which would still distinguish between the signal entries and the group, but Figure 3 still suggests they are intertwined.

The introduction of the subsampling before the interpolation now makes sense to me, and I agree is reasonable and necessary for the section.

Regardless of a response, I am updating my score slightly as I believe the main disagreement (lifting the group) could be reasonably explained and the subsampling comment was completely addressed.

We thank the reviewer for appreciating our work and providing valuable feedback.

Q12. Providing an illustration to improve clarity

We thank the reviewer for the great suggestion. We understand that some concepts related to group-equivariant neural networks (g-CNNs) can be difficult to grasp for readers unfamiliar with this area. To address this concern, in the updated manuscript, we added a section clarifying "what does it mean to have a signal index by the group elements in g-CNN architectures" and "how the anti-aliasing performed on these functions inside the equivariant CNN architectures." Specifically, we describe the details regarding how images are converted to signals on the group in equivariant CNNs and how the group element can index a function. An illustration to explain the concept better is also provided. Please see Appendix Sec A6 and Fig A2 in the updated manuscript.

We refer the reader to this section from lines 390-393 of the main text.

We hope these changes address your concerns and improve the paper's readability. We also highly appreciate and are open to any further suggestions on improving the paper's clarity.

Q13. Clarification on the order of generators and minimal generating set

We have added a clarification on the existence of the order of generators along with a reference to standard group theory literature (lines 192-194 ).

We have also clarified the concept of the "minimal generating set": In general, there can be more than one minimal generating set. However, in most practical cases, they are equivalent. For example, for any cyclic group $C_n$ generated by the minimal generating set $S=\{r\}$, $S'=\{r^b\}$ is also a minimal generating set when $b$ and $n$ are relatively prime. (line 127-129 and line 740-742)

Q14. Suggestions on Notations

Thanks for pointing this out. we now use different notation ($U$) for vector space. And we now represent all downsampling rates by $R$.

Q15. Checking for minimal generating set

Algorithm to check the minimality of a generating set $S$:

Let $S$ be a generating set of group $G$ and $b \in S$. If we remove all the edges corresponding to the element $b$ from the Cayley graph of $G$, and even after removing these edges, the Cayley graph remains connected, then $b$ is a redundant element, and the generating set $S$ is not minimal.

To prove $S$ is a minimal generating set, we need to do this test $\forall b \in S$.

The algorithm can be expensive if the Cayley graph is very dense, the group size is large, and the number of elements in the generating set is also large. However, in practice, we already know the minimal generating sets of common groups.

Q16. Clarification on Example 1

Yes. For the Example 1, $S=\{ \Delta t\}$ and $s_d = \Delta t$. As only one generator exists, we have to fix $s_d = \Delta t$.

Q17. Clarification on Perfect Reconstruction

We defined "perfect reconstruction" in lines 291-294. It means when the original signal $x$ and reconstructed signal $\mathcal{I} \mathcal{S}x$ matches exaclty, i.e., $| x - \mathcal{I} \mathcal{S}x|_2^2=0$. In other words, we can get back the original signal on the larger group from the subsampled signal on the subgroup.

We thank the reviewer for appreciating our work and providing valuable feedback.

Q1. How this work differs from related works on Subgroup subsampling

We appreciate the opportunity to clarify how our work differs from prior literature.

We first clarify how our work differs from existing work [1,2] on subgroup subsampling:

[1] Xu et al., Group equivariant subsampling, NeurIPS 2021

[2] Cohen & Welling, Group equivariant convolutional networks, ICML 2016

In the paper, we highlighted four main differences from prior work [1,2] on subgroup subsampling:

a) [1,2] assume that the subgroup of the subsampling operation is given. (See line 44). We do not make such an assumption.

b) As the subgroup is given, they do not consider any notion of selecting a subgroup or sampling rate. In contrast, we propose an approach to determine a subgroup given a desired subsampling rate. (See Line 061, the first bullet.)

c) [1,2] did not analyze the effect of subsampling from a signal processing perspective. This work introduces the notion of band-limitedness and the conditions for which perfect reconstruction can be achieved. (See Lin 063, the second bullet.)

d) [1,2] did not consider any form of anti-aliasing for their subsampling layer. While [1] did mention anti-aliasing in their related work, their approach actually does not address it. In fact, without a concrete notion of bandlimited-ness, it is not possible to discuss anti-aliasing. Following the bandlimited-ness introduced in our work, we further propose an anti-aliasing operation. (See Line 066, the third bullet point.)

In hindsight, anti-aliasing for subsampling on a group seems quite intuitive. However, to the best of our knowledge, we are not aware of any existing work discussing anti-aliasing for group equivariant models.

Next, we now discuss how our work differs from prior work [4,5,6] relating to the sampling theorem:

[4] Chen et al., Discrete Signal Processing on Graphs: Sampling Theory, IEEE TSP 2015

[5] Dodson et al., Groups and the sampling theorem, Sampling Theory in Signal and Image Processing, 2007

[6] McEwen et al., A novel sampling theorem on the rotation group, IEEE Signal Processing Letters, 2015

We believe that the closest works are the ones that extend the classic sampling theorem. This includes the studying of bandlimited-ness on graphs [4], or the study of sampling theorem on cyclic and abelian groups [5,6]. (See Line, 083 in related works). In this work, we propose a subgroup sampling theorem for general finite groups that guarantees the perfect reconstruction of the signal on the whole group from the subsampled signal on the subgroup. In Example 3, we also demonstrated that the existing Sampling theorem for the cyclic group is a special case of our proposed theorem.

To summarize, our contribution bridges this gap by extending classical signal processing concepts—such as Shannon's sampling theorem—to group-equivariant models. Specifically, we propose a novel sampling theorem for functions over groups and introduce an anti-aliasing operator that improves both accuracy and equivariance consistency without adding any new learning parameters or significant computational overhead. Also, we provide a subgroup subsampling algorithm, which returns the subgroups at a fixed rate. We also show that the existing concept of bandlimited-ness of the cyclic group is a special case of our proposed theorem (see Example 3).

The key message: We extend the techniques from traditional signal processing to group equivariant architecture, establishing a connection between sampling rate and subgroup subsampling. We propose a generalization of the sampling theorem for functions over a group. This holds great theoretical significance. We show that, by considering proper signal processing analysis, we can design a downsampling layer to improve accuracy and consistency without introducing any additional parameters.

Regarding baseline comparisons: In Tab 2, all columns where $\mathcal{P}_\mathcal{M*}$ is marked X is the baseline with the naive subsampling procedure [2] (i.e., without proper anti-aliasing).

Also, in Appendix Sec A2.1, we discuss how the primary goals of [1] differ from ours and involve equivariant index selection, not considering the problems mentioned above, i.e., our work and [1] targets two very different problems. In Table A1, we show that when our theory is applied to their approach, we achieve better results. Further, proof of the varsity and adaptability of our proposed technique.

We believe this addresses the concern regarding comparison with previous works, baselines, and key ideas of our work. We appreciate and are open to any suggestions to further improve the work.