Many data symmetries can be described in terms of group equivariance and the most common way of encoding group equivariances in neural networks is by building linear layers that are group equivariant. In this work we investigate whether equivariance of a network implies that all layers are equivariant. On the theoretical side we find cases where equivariance implies layerwise equivariance, but also demonstrate that this is not the case generally. Nevertheless, we conjecture that CNNs that are trained to be equivariant will exhibit layerwise equivariance and explain how this conjecture is a weaker version of the recent permutation conjecture by Entezari et al. [2022]. We perform quantitative experiments with VGG-nets on CIFAR10 and qualitative experiments with ResNets on ImageNet to illustrate and support our theoretical findings. These experiments are not only of interest for understanding how group equivariance is encoded in ReLU-networks, but they also give a new perspective on Entezari et al.'s permutation conjecture as we find that it is typically easier to merge a network with a group-transformed version of itself than merging two different networks.
Paper
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Summary
The paper considers networks with ReLU activations and investigates in which specific way their internal activations learn to be equivariant given an invariant data distribution. Section 2 presents a theoretical result, applying specifically to two-layer networks with non-singular weight matrices. It is shown that the network's equivariance requires 1) that its input and output activations are acted on by scaled permutation representations, and 2) that the network is layerwise equivariant with a related scaled permutation representation acting on the hidden features. Example 2.1 clarifies that this result does not necessarily hold in the case of non-singular weights. The third section proposes the conjecture that ReLU-CNNs that are trained on a reflection-invariant data distribution are close to regular GCNNs, i.e. GCNNs whose group representations are permutation representations. This conjecture is based on Entezari et al.'s conjecture that the activations of neural networks can always be permuted such that the networks are closely connected in the loss landscape. The authors argue that there always exists a CNN with exactly flipped kernels at initialization, which is due to the assumed invariant distribution preserved throughout training. Since these nets can be Entezari's conjecture be aligned via channel permutations, it follows that the CNN has either invariant kernels or kernels which are related by reflections - they are hence layerwise equivariant regular GCNNs. The experimental section investigates whether these two results hold in practice, using an activation matching technique to find reflected kernels. The results suggest that layerwise equivariance holds indeed.
Strengths
The paper is well written and gives insight into the nature of networks which are trained to be equivariant. This is an original idea since most publications on equivariant networks are rather investigating models which are by design constrained to be equivariant. The actual necessity of regular group convolutions is clarified by the submission. Where possible, the networks layerwise equivariance is undoubtably proven with an analytical approach. While this approach does not generalize to networks with more than two layers or singular weight matrices, the authors use a different approach to tackle these cases. All results and conjectures are supported empirically. The extent of the paper is comprehensive, containing multiple additions in the supplementary material.
Weaknesses
The strong assumptions in the analytical result seem initially like a weakness, but are addressed by the alternative approach based on Entezari's conjecture. The argument that this conjecture implies layerwise equivariance seems reasonable, but could be explained better. I am also not sure about the conjecture applies in the first place, however, this is not the scope of the current paper. Technically, I do not see the necessity for the definition of the barrier in Eq. (4) exact at the parameter's average, instead of at the maximizing parameter value. Another weakness is that the authors are only considering reflection-equivariant GCNNs, but not more general symmetry groups.
Questions
Why is section 2 considering linear instead of affine layers? Do the results not hold when biases are summed? Lemma 2.2 claims that the diagonal scaling matrix would be required to have (strictly) positive entries, however, non-negative entries, including zeros, seem sufficient. The resulting set of matrices does then no longer form a group since invertibility is lost, but the original group is contained as subset. The "intertwiner group" should furthermore be renamed to "equivariance group" since intertwiners are by definition linear, which ReLU is not. The activation matching technique should be briefly explained in the current submission instead of only pointing to related work. The paper is currently only mentioning trivial and regular representations, however, everything should apply to more general quotient representations as well, which should be briefly clarified.
Rating
7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
Confidence
3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Soundness
3 good
Presentation
3 good
Contribution
3 good
Limitations
Limitations are clearly addressed in section 1.2. and societal impacts are not to be expected.
Summary
This work investigates the relationship between end-to-end equivariance of a network and layerwise equivariance. It theoretically investigates when we can guarantee that an equivariant network is layerwise equivariant, and also cases where layerwise equivariant is not guaranteed or is harmful. In the case of CNNs, the authors draw a connection between horizontal flip invariance and the permutation conjecture for linear mode connectivity.
Strengths
1. Good exposition in Sections 1 and 2 2. Appendices C and D have nice observations and are illustrative. I quite like the examples in C and C.1., which concretely show ways in which layerwise equivariance is not good enough when network capacity is low. 3. Neat simpler proof of the lemma from Godfrey et al. 2022 4. Nice connection drawn to the permutation conjecture that I would not have expected. 5. The trained VGGs are remarkably similar to GCNNs.
Weaknesses
1. The point about layerwise equivariance is covered in some prior works that are not cited. Much of appendix D.1. in particular is discussed in depth in [1]. Limitations of linear layerwise equivariance is discussed in [Finzi et al. 2021]. 2. As noted by the authors, the restriction to horizontal flips is restrictive for empirical evaluation, though to be fair horizontal flips consistently improve many vision systems. [Shakerinava et al. 2022] Structuring Representations Using Group Invariants. NeurIPS 2022. [Finzi et al. 2021] A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups. ICML 2021
Questions
What do you mean on Page 5 when you say that the results for Godfrey et al. 2022 for other nonlinearities "also straightforwardly translate to the group equivariance case"? Minor: * Page 4: "and then $\tilde f$ is equivariant" should be "... layerwise equivariant" * Can you give a citation for the existence of dead neurons on page 4? * Page 6, several times you refer to the numerator of (4) when you mean the denominator.
Rating
7: Accept: Technically solid paper, with high impact on at least one sub-area, or moderate-to-high impact on more than one areas, with good-to-excellent evaluation, resources, reproducibility, and no unaddressed ethical considerations.
Confidence
3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Soundness
4 excellent
Presentation
3 good
Contribution
3 good
Limitations
Good discussion of limitations in Sectino 1.2.
Summary
This paper provides an investigation on whether equivariance of a trained deep neural network with ReLU activations implies that each of its learned layers are equivariant. The authors show that this should be true in some sense, i.e., some kind of group action must be present in the intermediate feature spaces (Line 141-143 and Appendix D.1), as the network should be able to always encode the group transformation of the input in some way to achieve overall equivariance. However, this applies to intermediate feature spaces where the clamping behavior of (ReLU) activations are involved; the authors argue that, when considering the learned linear layers, this is not always true (Example 2.1). Regarding ReLU activation, the authors show that if ReLU is G equivariant, the representations on the input and output must be scaled permutation representations (Lemma 2.2). From that, for two-layer networks with a strong assumption of invertible weight matrices, the authors show that overall equivariance implies layerwise equivariance where representations on intermediate features are scaled permutation representations (Proposition 2.3). Based on that, the authors investigate how the (requirement for) scaled permutation representations provides an implication of how horizontal flipping symmetry (related to the representation of S2 group) can be encoded in CNNs. The key intuition here is that permutation representation exactly appears in the characterization of GCNNs (Cohen et al., 2016), where the kernel constraint gives that an element of S2 group acts on the group convolution kernel through the joint action of spatial permutation representation (horizontal flipping here) and channel permutation representation (allocation). The authors further make a connection to the invariance of neural networks under permutation of neurons, or channels in case of CNNs due to spatial structure of convolutions, and the permutation conjecture (Entezari et al., 2022) that networks of the same type trained on the same data would lie in the same loss basin up to some permutation of neurons (channels). Then, the authors logically combine the requirement for permutation representations on ReLU for G equivariance, the spatial and channel permutation representation on group convolution filters, and the permutation conjecture, leading to the following conjecture: CNNs trained on a data distribution invariant to horizontal flips would be close to being GCNNs. The authors empirically test their conjecture using VGG and ResNet architectures on CIFAR10 and ImageNet, which supports the proposed conjecture. Cohen et al., Group Equivariant Convolutional Networks (2016) Entezari et al., The role of permutation invariance in linear mode connectivity of neural networks (2022)
Strengths
S1. I think this is a solid work that establishes a creative combination of ideas and findings from multiple sub-areas of deep learning theory, and from that proposes some original theoretical findings that combines into a very interesting conjecture that is potentially impactful in the field of equivariant deep learning. S2. In addition to the above, the proposed conjecture is equipped with a proper empirical support involving multiple deep architectures and datasets, which is an important strength of this work.
Weaknesses
W1. While reading the paper, I was quite confused about the implication of formulation in Appendix D.1. It seems the results, in particular Proposition D.4, explicitly proves that equivariance of a network implies layerwise equivariance, even under non-invertibility of individual layers. Because of this, I was confused when reading the discussion on Example 2.1 as well as Line 158 that overall equivariance might not lead to layerwise equivariance, as these seem like conflicting arguments. Am I missing something? W2. The discussion on identifying an equivalent equivariant network with layerwise equivariance (Line 132-136), while understandably leading to discussion in Appendix C, seems not critical in describing the main conjecture of the paper (please correct me if I am wrong). It might be better in terms of readability to contain the relevant discussion in some separated section. W3. In describing Eq. (3), it might be beneficial to explain how the transformation of filter relates to transformation of input and output (in context of equivariance) for readers not familiar with GCNN. W4. The description of why and how the REPAIR algorithm is used in Line 289-296 was hard to understand, I think it has room for improvement.
Questions
Q1. Reading Line 260-268, a question I had was to which extent the proposed conjecture depends on the particular nature of S2, i.e., how specifically it may generalize to other discrete groups such as 90-degree rotations. May I ask for an explanation on this?
Rating
8: Strong Accept: Technically strong paper, with novel ideas, excellent impact on at least one area, or high-to-excellent impact on multiple areas, with excellent evaluation, resources, and reproducibility, and no unaddressed ethical considerations.
Confidence
3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Soundness
4 excellent
Presentation
2 fair
Contribution
4 excellent
Limitations
The authors have clarified the limitation of the work in Section 1.2.
Summary
Exploring the symmetries of representations and parameters in neural networks is crucial. This paper provides several valuable contributions. First, the authors theoretically proved that for Relu-Networks equivariance implies layer-wise equivariance, but not vice versa. Second, inspired from the conjecture by [22], the authors discussed a weaker version that connects CNNs with GCNNs. Finally, quantitative experiments were performed and showed that a ReLU-network that has been trained to be equivariant will be layer-wise equivariant.
Strengths
1. This paper is well written. I enjoy the writing. While the concepts particularly those related to group representation are complicated and non-trivial, the authors did a great job in introducing the relevant definitions, examples and propositions in an elegant and compact way. The authors also provided necessary intuitive understanding, which greatly help readers to digest the motivation of how the investigation is performed. 2. The conclusion by Proposition 2.3 is interesting and valuable, even the proof is a direct result of Lemma 2.2 by [14]. I have checked [14] and find no similar conclusion. The authors also provided abundant theoretical understandings and derivations in the supplementary materials including the discussion of the invariant case. 3. The discussion of how GCNNs are related with the notion of Conjecture 3.1 [14] is insightful and inspiring. 4. The experimental evaluations are scrupulously carried out and are able to support the claims by the authors to a certain extent.
Weaknesses
I have no major concern. There are still some questions below: 1. Proposition 2.3 shows that layer-wise equivariance is NOT a necessary condition of equivariance. But the experiments did show that CNN with equivariant training boils down to be layer-wise equivariant. So, why we have this observation and does it mean that the examples derived from Proposition 2.3 are just corner cases that rarely happen in practice? Moreover, how does Proposition 2.3 influence the design of GCNNs? Or particularly, what is the relationship between the equivariant constraint in Proposition 2.3 and the derived kernel constraint (Eq. 3) by [31,30]? I would expect the authors to explain more on these points? 2. Would the authors explain why they have the claims in Lines 151-155 that equivariance can hurt the performance? Why in Table 2, the unconstrained models obtained better performance than those constrained or equivariant ones? 3. Table 1 only defined the models trained with invariant losses, which indicates that the performance in Table 2 can only answer Q1? How about the experiments of models trained with horizontal flipping data for Q2? Is Table 3 mainly for Q2?
Questions
See the weakness part above.
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Soundness
3 good
Presentation
4 excellent
Contribution
3 good
Limitations
The authors have adequately discussed the limitations.
Summary
This paper shows that CNNs will be close to G-CNNs if they are trained to be equivariant. In addition, they also provide some theoretical analysis and conjectures regarding the layerwise equivariant of ReLU-networks.
Strengths
1. They show that equivariance implies layerwise equivariance with a scaled permutation representation acting on the feature maps (Proposition 2.3.). 2. They propose a new conjecture 'Most SGD CNN-solutions on image data with a distribution that is invariant to horizontal flips of the images will be close to GCNNs.' (Conjecture 3.2.) They proposed a new measure for closeness to being a GCNN. 3. Experiments on CIFAR-10 and ImageNet support Conjecture 3.2.
Weaknesses
1. In general, the paper may have limited or vague benefits for applications. First, the work focuses on ReLu-networks in the theoretical analysis, ignoring normalization layers. Modern neural network designs heavily rely on different normalization layers. In addition, in the ImageNet experiments, ResNet-50 uses batch normalization, which is not aligned with the theoretical analysis. 2. The paper did not explain what are the insights for network designs if ReLu CNNs are layerwise equivariance. In practice, it even hurts the performance of the model if the invariance loss is applied. Then, the generalization of the model might contradict the actual layerwise equivariance of the model. And the invariance loss adds an extra restriction for learning. 3. The evidence of Q1 is much weaker than Q2 in section 4. The G-CNN Barrier of 'CNN' in table 2 is 20$\times$ larger than 'CNN + inv-loss.' I wonder if it is still valid to say it is close to G-CNN. Maybe a baseline without horizontal data augmentation can better justify this.
Questions
1. What is 'ppt' defined in section 4.2? I did not find explanations regarding it. 2. Could you give more explanations regarding the last sentence of the conclusion 'If a negative GCNN barrier is achievable ...... enable weight space ensembling.'? How 'negative GCNN barrier' is related to 'weight space test time data augmentation', etc?
Rating
5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
Confidence
2: You are willing to defend your assessment, but it is quite likely that you did not understand the central parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Soundness
3 good
Presentation
3 good
Contribution
2 fair
Limitations
Limitations are given.
Thank you for the response! I have no further questions at the time, and will discuss with other reviewers.
Thank your for the reply
I thank the authors for their efforst in addressing my concerns. There are some misunderstanding I have made before. But after checking the feedbacks and other reviewers' comments. I am sure that this is a solid paper and comes valuable for the publication in NeurIPS. I have no any question now.
Thanks for the detailed response. The rebuttal addresses all of my questions and I would like to express again that I vote for accepting the paper. Regarding the "intertwiner group" terminology I think that both of the proposed solutions are fine. I would personally choose "equivariance group" as it should be understood by everyone working on equivariant networks and does not require the additional footnote. Note that not every reader in equivariant DL may be familiar with representation theory, intertwiners or the terminology of Godfrey et al.
I want to thank the authors for their detailed responses. Most of my concerns are addressed and additional explanations are provided. The 'weight space test time data augmentation' may provide an interesting direction for future works, and it seems promising. As a result, I changed my score to positive.
Response to rebuttal
Thank you for the detailed response. I have read through the manuscript again with the response in mind, and the narrative leading to ReLU and permutation representation is more clear now. I recommend the authors to revise Line 141-148 according to the rebuttal to prevent potential misunderstanding (like the one I had) regarding Proposition D.4. Now that my major concern is resolved, I have adjusted my score accordingly.
Post rebuttal
We wish to thank the reviewers for going through our rebuttals and are of course very happy to read that their questions have been answered and that all reviewers recommend acceptance of the paper.
Decision
Accept (poster)