Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

First, we present two mechanisms towards equivariance-aware architectural optimization.

The review comment to be evaluated is:

**Strength**
1. The paper is well motivated. Most G-CNN literature assume that the symmetry group of the learning task is explicitly known and perfect. The paper instead uses a equivariance-aware NAS procedure to search for the optimal architecture for (potentially) unknown and imperfect group transformations.
2. The paper is mostly well-written.
3. The experiments are detailed and thorough.

**Weakness**
1. The equivariance relaxation morphism, which relies on reparameterizing the group convolution on a subgroup, seems to be limited to discrete group. Can this be extended to continuous groups?
2. Moreover, the relaxation morphism is also restrictive in the sense that the number of the unstructured channel (c_out) cannot be adjusted after changing to a subgroup. This inevitably increases the model size as group size decreases, which is indeed the case for EquiNAS_D.
3. The clarity of the paper in section 5.1 can benefit from more explanation on the technical terms such as population, pareto selection, pareto front, etc.
4. In MNIST_rot, the review would like to see what happens when $C_8$ and $D_8$ are included in the picture.
5. If I understand correctly, the reported EquiNAS_E result on Galaxy10 in Table 1 might be misleading. Figure 4(a) shows that only one NAS output in the end achieves significantly smaller test error compared to all the remaining results.
6. Are the models involved in EquiNAS_D significantly larger compared to the baseline? After all, a linear combination of layers equivariant to different groups is taken. Does that mean the model is slow to train and (especially) test?

The evidence to be evaluated is:In more diverse approaches, Zhou et al. (2020) and Yeh et al. (2022) represent symmetry-inducing weight sharing via learnable matrices. Romero & Lohit (2022) and van der Ouderaa et al. (2022) learn partial or soft equivariances for each layer.

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer. The proposed morphism, depicted in Figure 1(A) and described in Section 4.1, changes the equivariance constraint from one group to another subgroup while preserving the learned weights of the initial group convolutional operator. The \([G]\)-mixed equivariant layer, shown in Figure 1(B) and presented in Section 4.2, allows for a single layer to represent equivariance to multiple subgroups through a weighted sum. The _equivariance relaxation morphism_ reparametrizes a \(G\)-equivariant group (or lifting) convolutional layer to operate over a subgroup of \(G\), partially removing weight-sharing constraints from the parameter space while maintaining the functionality of the layer.

Given a \(G\)-equivariant group convolutional layer with feature map \(f\) and filter \(\psi\), we define the relaxed feature map \(\tilde{f}\colon G^{\prime}\to\mathbb{R}^{(C_{l-1}\times|R|)}\) and relaxed filter \(\tilde{\psi}\colon G^{\prime}\to\mathbb{R}^{(C_{l}\times|R|)\times(C_{l-1} \times|R|)}\) as follows. For \(c\in[C_{l-1}]\), \(s,t\in R\), \(d\in[C_{l}]\):

We define the _equivariance relaxation morphism_ from \(G\) to \(G^{\prime}\) as the reparametrization of \(\psi\) as \(\tilde{\psi}\) (Eq. 4) and reshaping of \(f\) as \(\tilde{f}\) (Eq. 3). We will show that the new layer, \([\tilde{f}\star_{G^{\prime}}\tilde{\psi}]_{(d,t)}(g^{\prime})\), is equivalent to \([f\star_{G}\psi]_{d}(g^{\prime}t)\) down to reshaping.

\[[f\star_{G}\psi]_{d}(g^{\prime}t) =[f\star_{G}\psi]_{d}(g) \tag{5}\] \[=\sum_{h\in G}\sum_{c=1}^{C_{l-1}}f_{c}(h)\psi_{d,c}(g^{-1}h),\] (6) \[=\sum_{h^{\prime}\in G^{\prime}}\sum_{s\in R}\sum_{c=1}^{C_{l-1} }f_{c}(h^{\prime}s)\psi_{d,c}(t^{-1}g^{\prime-1}h^{\prime}s),\] (7) \[=\sum_{h^{\prime}\in G^{\prime}}\sum_{c=1}^{C_{l-1}}\sum_{s\in R }\tilde{f}_{(c,s)}(h^{\prime})\tilde{\psi}_{(d,t),(c,s)}(g^{\prime-1}h^{\prime }),\] (8) \[=\left[\tilde{f}\star_{G^{\prime}}\tilde{\psi}\right]_{(d,t)}(g^{ \prime}), \tag{9}\]

Thus, the convolution of \(\tilde{f}\) with \(\tilde{\psi}\) is equivariant to \(G\) but parametrized as a \(G^{\prime}\)-equivariant group convolutional layer, where the representatives are expanded into independent channels. This morphism can be viewed as initializing a \(G^{\prime}\)-equivariant layer with a pre-trained prior of equivariance to \(G\), maintaining any previous training.

Towards learning equivariance, we additionally propose partial equivariance via a mixture of layers, each constrained to equivariance to different groups, applied in parallel to the same input then combined via a weighted sum.

For a set of groups \([G]\), such as a subgroup lattice of some group \(G\), we define a \([G]\)_-mixed equivariant layer_ as: \[\left[f^{\hat{\pi}_{[G]}}[\psi]\right]_{(d,t)}(g) =\sum_{G\in[G]}z_{G}\left[f\star_{G^{\prime}}\widetilde{\psi^{G}} \right]_{(d,t)}(g) \tag{10}\] \[=\left[f\star_{G^{\prime}}\sum_{G\in[G]}z_{G}\widetilde{\psi^{G} }\right]_{(d,t)}(g), \tag{11}\] where each element \(z_{G}\) of \([z]\coloneqq\{z_{G}|G\in[G]\}\) is an architectural weighting parameter such that \(\sum_{G\in[G]}z_{G}=1\), \(G^{\prime}\) is a subgroup of all groups in \([G]\), each element \(\psi^{G}\) of \([\psi]\) is a filter with a domain of \(G\), and \(\widetilde{\psi^{G}}\) is the transformation of \(\psi^{G}\) from a domain of \(G\) to \(G^{\prime}\) as defined in Equation 4. Thus, the layer is parametrized by \([\psi]\) and \([z]\), computing a weighted sum of operations that are equivariant to different groups of \([G]\). The layer may be equivalently computed by convolution of the input with the weighted sum of transformed filters, shown in Equation 11.

We present two NAS methods that utilize the presented mechanisms for discovering appropriate equivariance during training: Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) and Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)). Both methods optimize an architecture while learning weights, yielding a final trained network adapted to equivariances present in the training data.

EquiNAS\({}_{E}\), in Section 5.1, searches for networks composed of layers each fully equivariant to possibly different groups, while EquiNAS\({}_{D}\), in Section 5.2, searches for smooth mixtures of equivariant layers.

Offspring are generated from each parent separately by mutation using the relaxation morphism. This preserves the weights of the parametrized equivariance during mutation, allowing for the continuous training of networks over evolution by inheritance from parent individuals. Specifically, mutation reduces a single layer's parametrized equivariance to a subgroup within the constraint that each layer has parametrized equivariance to a subgroup of all preceding layers. This constraint yields local equivariance properties for the network, as shown in Weiler and Cesa (2019) and Elsayed et al. (2020) to be empirically favorable in image classification tasks. The resulting individuals are each trained independently for a given training time, and then this process repeats.

In a contrasting paradigm, the \([G]\)-mixed equivariant layer presented in Section 4.2 allows for smoothly searching across a spectrum of equivariance for each layer via a differentiable NAS algorithm. Our Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)) algorithm, defined in Algorithm 2, is inspired by DARTS (Liu et al., 2018) with significant changes detailed in the following paragraphs.

These two NAS approaches present adaptations of two standard types of NAS, evolutionary and differentiable, to the search for optimal partial equivariance.

We focus on the regular representation of groups and show experiments with reflectional and up to 4-fold rotational symmetry groups applied to image classification tasks. Examples of symmetry groups acting on pixel space, which corresponds to \(\mathbb{Z}^{2}\), include \(T(2)\), which consists of discrete translations in both dimensions; the cyclical groups \(C_{n}\), which consist of \(n\)-fold rotations; and the dihedral groups \(D_{n}\), which consist of reflections with \(n\)-fold rotations, where \(n\in\{1,2,4\}\) for exact symmetry without interpolation.

As standard convolutional layers are already equivariant to \(T(2)\), we refer to layers also equivariant to \(n\)-fold rotations with or without reflections as \(D_{n}\) or \(C_{n}\)-equivariant, respectively. So, a \(C_{1}\) equivariant convolutional layer is a standard translation-equivariant convolutional layer. We use \((C_{1},D_{1},C_{2},D_{2},C_{4},D_{4})\) as the set of potential groups for mutation in EquiNAS\({}_{E}\) and as \([G]\) in EquiNAS\({}_{D}\).

These tasks exhibit varying levels of rotational and reflectional symmetry, motivating architectural optimization to determine the most effective application of equivariance constraints. Across all experiments, the architectures are designed to have consistent channel dimensions once expanded to a standard translation-equivariant convolutional layer for each layer across models. Thus, constrained equivariance to a larger symmetry group results in fewer learnable parameters. A layer constrained to \(C_{4}\) equivariance has \(|C_{4}\setminus D_{4}|=2\) times as many independent channels and as many parameters as a layer constrained to \(D_{4}\) equivariance. This is a notably different paradigm than other works that equate parameter counts across architectures with different equivariance properties.

In addition to the normally initialized static baselines, we also train and test baselines that are initialized with priors to larger symmetry groups. These are implemented by initializing all layers to be constrained to the prior symmetry group, then using the equivariance relaxation morphism on each layer. EquiNAS\({}_{E}\) searches for relaxation schedules that yield trained priors on equivariance, while these additional baselines yield untrained priors.

To our knowledge, this is the first work which proposes search methods for networks with dynamically constrained equivariance. Many NAS approaches separately search for an architecture and then reinitialize and retrain the weights, while our two proposed approaches find an optimal architecture with trained weights in a single process, notably with dynamically constrained weights.

Our two equivariance-aware NAS approaches have distinct approaches: EquiNAS\({}_{E}\) searches for architectures composed of discretely equivariant layers, while EquiNAS\({}_{D}\) searches for continuous mixtures of equivariance within each layer.

We present two mechanisms towards equivariance-aware architectural optimization, the equivariance relaxation morphism and the \([G]\)-mixed equivariant layer, and two NAS algorithms that respectively implement these mechanisms evolutionarily as EquiNAS\({}_{E}\) and differentiably as EquiNAS\({}_{D}\).

* Yeh et al. (2022) Raymond A Yeh, Yuan-Ting Hu, Mark Hasegawa-Johnson, and Alexander Schwing. Equivariance discovery by learned parameter-sharing.

* Zhou et al. (2020) Allan Zhou, Tom Knowles, and Chelsea Finn. Meta-learning symmetries by reparameterization. In _International Conference on Learning Representations_, 2020.

Group convolutional layers The implementation of regular group convolutional layers can be viewed as a special case of our proposed equivariance relaxation morphism.

Equivariance relaxation morphism To implement the equivariance relaxation morphism, the new filter tensor is initialized by applying Equation 4 such that result of applying the preceding filter transformation is equivalent. Our implementation of group actions relies on group channel indexing to represent the order of group elements: to ensure this is consistent before and after the morphism, the appropriate reordering of the output and input channels of the expanded filter are applied upon expansion.

The \([G]\)-mixed equivariant layer is built on top of this implementation, also using proper input and output channel reordering between layers to ensure correct mixing of group channels.

For both EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) experiments, we use the same backbone architecture, such that the static baselines have the same architecture across experiments. The architectures have a lifting layer followed by 7 group convolutional layers, for a total of 8 convolutional layers.

This is enabled by the \([G]\)-mixed equivariant layer, which is a contribution of this work.

Our methods enable searching for both architecture and parameters concurrently in a single training process.

Input: Figure 1: Visualizing the equivariance relaxation morphism and the [G]-mixed equivariant layer, using the C4 group. In (A), the learnable parameters of a C4-equivariant convolutional layer are expanded using each group element action so the expanded filter can be used in a standard convolutional layer. The equivariance relaxation morphism reparametrizes the layer to be architecturally constrained toC2 equivariance, initialized to functionalC4 equivariance. In (B), convolutional operations equivariant to subgroups ofC4 are summed with learnable architectural weighting parameters.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 3: Architecture weighting parameters by layer for one selected trial from RotMNIST, Galaxy10, and ISIC. For other trials, see Figures 10-12 in Appendix D.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Looking at the claim "First, we present two mechanisms towards equivariance-aware architectural optimization," I find this to be well-stated and accurately reflects the evidence without exaggeration.

The evidence clearly demonstrates:
1. The equivariance relaxation morphism is thoroughly explained with mathematical formulations and visualized in Figure 1(A)
2. The [G]-mixed equivariant layer is detailed mathematically and shown in Figure 1(B)

The evidence provides comprehensive technical details, mathematical foundations, and visual representations supporting both mechanisms. The authors do not overstate capabilities and acknowledge limitations (as noted in reviewer comments about discrete groups). The claim simply states what is actually presented without embellishment.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

First, we present two mechanisms towards equivariance-aware architectural optimization.

The review comment to be evaluated is:

## Strengths
- Equivariant networks have more hyperparameters to tune due to the choice of symmetry group.  For example, in E(2)-CNN (https://arxiv.org/abs/1911.08251), the authors experiment with symmetry groups $C_n$ and $D_n$ for many values of n before finding n=12 to work best.  This paper proposes a method for automatically determining the correct level of symmetry.
- Moreover, on many problems end-to-end strict symmetry is not desirable if the symmetry in the domain is only approximate or if symmetry constraints interfere with optimization.  The proposed method adds to the relatively few methods which learn relaxed symmetry constraints and adapt to the level of symmetry in the data.
- I found the equivariance relaxation morphism to be a simple and effective idea for relaxing symmetry constraints in networks on a per-layer basis.
- The [G]-mixed equivariant layer also seems like a reasonable idea for selecting the degree of equivariance by differentiably choosing weights over kernels with different levels of equivariance.  In theory this allows the model to learn a specific equivariance or an approximate equivariance which is in-between.  It also allows the network to reduce the symmetry constraint in later layers, which is what is done effectively by CNNs in practice by downsampling with stride.
- Experimental results on the learned subgroup weights (Fig. 2) reveal interesting trends showing that the networks do select for higher equivariance in early layers and lower equivariance in later layers which matches what has found to be effective in practice for equivariant networks.

## Weaknesses / Questions
- I am not sure of the framing of the method.  I am not an expert on NAS and am open to correction, but it seems to me both proposed methods find a trained network, not an architecture, since they are training the network weights and selecting an architecture at the same time.  The final architecture may not have great performance if trained from scratch.  In this sense, it seems the method is more a special training procedure than NAS method.   EquiNAS_D, in particular, is simply an architecture in which some weights are optimized alternatingly.
- Given the evolutionary algorithm starts with a single network, how do we know to what extent its evolution is guided by architecture versus the specific weights in the mutated networks?  In experiments, I think both population level variance and initialization variance should be accounted for.
- One of the biggest issues is the fact that the relaxation morphism increases the number of parameters making it difficult to distinguish performance increases are from relaxing constrains or increasing parameter count.  The authors do discuss this issue and suggest their optimization strategy over both parameter count and accuracy helps.  It would be better, however, if the number of parameters could be preserved by the relaxation.
- The experimental results are not completely convincing of when this method would add significant value over simply doing hyperparameter search over equivariant networks.  In Table 1, an equivariant baseline is best on rotMNIST and ISIC, and a non-rotationally equivariant network with equivariant initialization is best for Galaxy10.  In table 2, EquiNAS_D does do best for rotMNIST, but the score is much lower than the best values in E(2)-CNN after full tuning of the best equivariant method.  In Galaxy10, EquiNAS_D does outperform, but with very high variance.
- I have some issues with the clarity of the description of 4.2.  The condition that all groups in $[G]$ be subgroups or supergroups of all other groups in $[G]$ does not appear to be met by the expermental example since $D_2$ is not a subgroup or supergroup of $C_4$.  It's also not clear to me why this condition is necessary.  It is a fairly strict condition, although I think it would include useful cases.  I am also a bit unclear on the form of the input signal $f$.  In order to be convolved with $\tilde{\Psi}$ and have $G$-equivariance for each $G$ in $[G]$, $f$ should be defined over a supergroup of all $G \in [G]$.  Is this the case?   I am correct that if $z_G$ is a one-hot vector on the group $G$, then the operation is $G$-equivariant?

## Questions
- If the evolution algorithm can only mutate in the direction of symmetry constrain relaxation, doesn't this bias the algorithm towards relaxing all the symmetry constraints?
- Does this work have a relation to diffstride (https://arxiv.org/abs/2202.01653) in that stride is typically a hyperparameter that corresonds to a choice of the subgroup of the translation group and can be shown to differentiably optimized?
- After equivariance is broken in a given layer, there isn't much point in principle to persevering it downstream.  Are mutations automatically applied to all following layers or did your method discover this?  If so, I think that is interesting enough to be considered a strength.

## Minor
- In Eqn 1, if you reverse the order of $f$ and $\Psi$ and drop the sum over c and indices $c,d$, the equation can be simplified using matrix multiplication.
- 4.1, para 2, "neutral element" -> "identity element"
- 4.1 may benefit from writing as a proposition
- 4.1, Para 3, Line 5,  $h \in G'$ -> $h' \in G'$
- Sec 5., Line 4, typo "optimizffe"

The evidence to be evaluated is:In more diverse approaches, Zhou et al. (2020) and Yeh et al. (2022) represent symmetry-inducing weight sharing via learnable matrices. Romero & Lohit (2022) and van der Ouderaa et al. (2022) learn partial or soft equivariances for each layer.

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer. The proposed morphism, depicted in Figure 1(A) and described in Section 4.1, changes the equivariance constraint from one group to another subgroup while preserving the learned weights of the initial group convolutional operator. The \([G]\)-mixed equivariant layer, shown in Figure 1(B) and presented in Section 4.2, allows for a single layer to represent equivariance to multiple subgroups through a weighted sum. The _equivariance relaxation morphism_ reparametrizes a \(G\)-equivariant group (or lifting) convolutional layer to operate over a subgroup of \(G\), partially removing weight-sharing constraints from the parameter space while maintaining the functionality of the layer.

Given a \(G\)-equivariant group convolutional layer with feature map \(f\) and filter \(\psi\), we define the relaxed feature map \(\tilde{f}\colon G^{\prime}\to\mathbb{R}^{(C_{l-1}\times|R|)}\) and relaxed filter \(\tilde{\psi}\colon G^{\prime}\to\mathbb{R}^{(C_{l}\times|R|)\times(C_{l-1} \times|R|)}\) as follows. For \(c\in[C_{l-1}]\), \(s,t\in R\), \(d\in[C_{l}]\):

We define the _equivariance relaxation morphism_ from \(G\) to \(G^{\prime}\) as the reparametrization of \(\psi\) as \(\tilde{\psi}\) (Eq. 4) and reshaping of \(f\) as \(\tilde{f}\) (Eq. 3). We will show that the new layer, \([\tilde{f}\star_{G^{\prime}}\tilde{\psi}]_{(d,t)}(g^{\prime})\), is equivalent to \([f\star_{G}\psi]_{d}(g^{\prime}t)\) down to reshaping.

\[[f\star_{G}\psi]_{d}(g^{\prime}t) =[f\star_{G}\psi]_{d}(g) \tag{5}\] \[=\sum_{h\in G}\sum_{c=1}^{C_{l-1}}f_{c}(h)\psi_{d,c}(g^{-1}h),\] (6) \[=\sum_{h^{\prime}\in G^{\prime}}\sum_{s\in R}\sum_{c=1}^{C_{l-1} }f_{c}(h^{\prime}s)\psi_{d,c}(t^{-1}g^{\prime-1}h^{\prime}s),\] (7) \[=\sum_{h^{\prime}\in G^{\prime}}\sum_{c=1}^{C_{l-1}}\sum_{s\in R }\tilde{f}_{(c,s)}(h^{\prime})\tilde{\psi}_{(d,t),(c,s)}(g^{\prime-1}h^{\prime }),\] (8) \[=\left[\tilde{f}\star_{G^{\prime}}\tilde{\psi}\right]_{(d,t)}(g^{ \prime}), \tag{9}\]

Thus, the convolution of \(\tilde{f}\) with \(\tilde{\psi}\) is equivariant to \(G\) but parametrized as a \(G^{\prime}\)-equivariant group convolutional layer, where the representatives are expanded into independent channels. This morphism can be viewed as initializing a \(G^{\prime}\)-equivariant layer with a pre-trained prior of equivariance to \(G\), maintaining any previous training.

Towards learning equivariance, we additionally propose partial equivariance via a mixture of layers, each constrained to equivariance to different groups, applied in parallel to the same input then combined via a weighted sum.

For a set of groups \([G]\), such as a subgroup lattice of some group \(G\), we define a \([G]\)_-mixed equivariant layer_ as: \[\left[f^{\hat{\pi}_{[G]}}[\psi]\right]_{(d,t)}(g) =\sum_{G\in[G]}z_{G}\left[f\star_{G^{\prime}}\widetilde{\psi^{G}} \right]_{(d,t)}(g) \tag{10}\] \[=\left[f\star_{G^{\prime}}\sum_{G\in[G]}z_{G}\widetilde{\psi^{G} }\right]_{(d,t)}(g), \tag{11}\] where each element \(z_{G}\) of \([z]\coloneqq\{z_{G}|G\in[G]\}\) is an architectural weighting parameter such that \(\sum_{G\in[G]}z_{G}=1\), \(G^{\prime}\) is a subgroup of all groups in \([G]\), each element \(\psi^{G}\) of \([\psi]\) is a filter with a domain of \(G\), and \(\widetilde{\psi^{G}}\) is the transformation of \(\psi^{G}\) from a domain of \(G\) to \(G^{\prime}\) as defined in Equation 4. Thus, the layer is parametrized by \([\psi]\) and \([z]\), computing a weighted sum of operations that are equivariant to different groups of \([G]\). The layer may be equivalently computed by convolution of the input with the weighted sum of transformed filters, shown in Equation 11.

We present two NAS methods that utilize the presented mechanisms for discovering appropriate equivariance during training: Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) and Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)). Both methods optimize an architecture while learning weights, yielding a final trained network adapted to equivariances present in the training data.

EquiNAS\({}_{E}\), in Section 5.1, searches for networks composed of layers each fully equivariant to possibly different groups, while EquiNAS\({}_{D}\), in Section 5.2, searches for smooth mixtures of equivariant layers.

Offspring are generated from each parent separately by mutation using the relaxation morphism. This preserves the weights of the parametrized equivariance during mutation, allowing for the continuous training of networks over evolution by inheritance from parent individuals. Specifically, mutation reduces a single layer's parametrized equivariance to a subgroup within the constraint that each layer has parametrized equivariance to a subgroup of all preceding layers. This constraint yields local equivariance properties for the network, as shown in Weiler and Cesa (2019) and Elsayed et al. (2020) to be empirically favorable in image classification tasks. The resulting individuals are each trained independently for a given training time, and then this process repeats.

In a contrasting paradigm, the \([G]\)-mixed equivariant layer presented in Section 4.2 allows for smoothly searching across a spectrum of equivariance for each layer via a differentiable NAS algorithm. Our Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)) algorithm, defined in Algorithm 2, is inspired by DARTS (Liu et al., 2018) with significant changes detailed in the following paragraphs.

These two NAS approaches present adaptations of two standard types of NAS, evolutionary and differentiable, to the search for optimal partial equivariance.

We focus on the regular representation of groups and show experiments with reflectional and up to 4-fold rotational symmetry groups applied to image classification tasks. Examples of symmetry groups acting on pixel space, which corresponds to \(\mathbb{Z}^{2}\), include \(T(2)\), which consists of discrete translations in both dimensions; the cyclical groups \(C_{n}\), which consist of \(n\)-fold rotations; and the dihedral groups \(D_{n}\), which consist of reflections with \(n\)-fold rotations, where \(n\in\{1,2,4\}\) for exact symmetry without interpolation.

As standard convolutional layers are already equivariant to \(T(2)\), we refer to layers also equivariant to \(n\)-fold rotations with or without reflections as \(D_{n}\) or \(C_{n}\)-equivariant, respectively. So, a \(C_{1}\) equivariant convolutional layer is a standard translation-equivariant convolutional layer. We use \((C_{1},D_{1},C_{2},D_{2},C_{4},D_{4})\) as the set of potential groups for mutation in EquiNAS\({}_{E}\) and as \([G]\) in EquiNAS\({}_{D}\).

These tasks exhibit varying levels of rotational and reflectional symmetry, motivating architectural optimization to determine the most effective application of equivariance constraints. Across all experiments, the architectures are designed to have consistent channel dimensions once expanded to a standard translation-equivariant convolutional layer for each layer across models. Thus, constrained equivariance to a larger symmetry group results in fewer learnable parameters. A layer constrained to \(C_{4}\) equivariance has \(|C_{4}\setminus D_{4}|=2\) times as many independent channels and as many parameters as a layer constrained to \(D_{4}\) equivariance. This is a notably different paradigm than other works that equate parameter counts across architectures with different equivariance properties.

In addition to the normally initialized static baselines, we also train and test baselines that are initialized with priors to larger symmetry groups. These are implemented by initializing all layers to be constrained to the prior symmetry group, then using the equivariance relaxation morphism on each layer. EquiNAS\({}_{E}\) searches for relaxation schedules that yield trained priors on equivariance, while these additional baselines yield untrained priors.

To our knowledge, this is the first work which proposes search methods for networks with dynamically constrained equivariance. Many NAS approaches separately search for an architecture and then reinitialize and retrain the weights, while our two proposed approaches find an optimal architecture with trained weights in a single process, notably with dynamically constrained weights.

Our two equivariance-aware NAS approaches have distinct approaches: EquiNAS\({}_{E}\) searches for architectures composed of discretely equivariant layers, while EquiNAS\({}_{D}\) searches for continuous mixtures of equivariance within each layer.

We present two mechanisms towards equivariance-aware architectural optimization, the equivariance relaxation morphism and the \([G]\)-mixed equivariant layer, and two NAS algorithms that respectively implement these mechanisms evolutionarily as EquiNAS\({}_{E}\) and differentiably as EquiNAS\({}_{D}\).

* Yeh et al. (2022) Raymond A Yeh, Yuan-Ting Hu, Mark Hasegawa-Johnson, and Alexander Schwing. Equivariance discovery by learned parameter-sharing.

* Zhou et al. (2020) Allan Zhou, Tom Knowles, and Chelsea Finn. Meta-learning symmetries by reparameterization. In _International Conference on Learning Representations_, 2020.

Group convolutional layers The implementation of regular group convolutional layers can be viewed as a special case of our proposed equivariance relaxation morphism.

Equivariance relaxation morphism To implement the equivariance relaxation morphism, the new filter tensor is initialized by applying Equation 4 such that result of applying the preceding filter transformation is equivalent. Our implementation of group actions relies on group channel indexing to represent the order of group elements: to ensure this is consistent before and after the morphism, the appropriate reordering of the output and input channels of the expanded filter are applied upon expansion.

The \([G]\)-mixed equivariant layer is built on top of this implementation, also using proper input and output channel reordering between layers to ensure correct mixing of group channels.

For both EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) experiments, we use the same backbone architecture, such that the static baselines have the same architecture across experiments. The architectures have a lifting layer followed by 7 group convolutional layers, for a total of 8 convolutional layers.

This is enabled by the \([G]\)-mixed equivariant layer, which is a contribution of this work.

Our methods enable searching for both architecture and parameters concurrently in a single training process.

Input: Figure 1: Visualizing the equivariance relaxation morphism and the [G]-mixed equivariant layer, using the C4 group. In (A), the learnable parameters of a C4-equivariant convolutional layer are expanded using each group element action so the expanded filter can be used in a standard convolutional layer. The equivariance relaxation morphism reparametrizes the layer to be architecturally constrained toC2 equivariance, initialized to functionalC4 equivariance. In (B), convolutional operations equivariant to subgroups ofC4 are summed with learnable architectural weighting parameters.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 3: Architecture weighting parameters by layer for one selected trial from RotMNIST, Galaxy10, and ISIC. For other trials, see Figures 10-12 in Appendix D.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
Looking at the evidence, the claim "we present two mechanisms towards equivariance-aware architectural optimization" appears well-stated and accurately reflects what is demonstrated in the paper. The evidence clearly shows:

1. The equivariance relaxation morphism, which is thoroughly described mathematically and visually illustrated in Figure 1(A)
2. The [G]-mixed equivariant layer, which is detailed mathematically and shown in Figure 1(B)

The reviewer feedback also confirms these as novel and effective mechanisms, noting them as "simple and effective" and "reasonable" ideas. The extensive experimental results shown in Figures 2,3,10-12 demonstrate these mechanisms being successfully applied. The claim makes no exaggerated statements about capabilities or performance, simply stating what is presented.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

First, we present two mechanisms towards equivariance-aware architectural optimization.

The review comment to be evaluated is:

### Strengths:
1. The problem of finding the correct equivariance that can exploit symmetries in the data is important for the development of general-purpose NAS methods.
2. The proposed approaches are simple and natural procedures.
3. The writing is fairly clear and the experimental results are interesting.
4. Code will be released upon acceptance.

### Weaknesses:
1. The experimental evaluation is somewhat limited. Given the fairly direct extension of prior work (morphisms and weight-sharing) to actually developing the search algorithms, I would expect more emphasis to be placed on experimental depth. This could either come in the form of evaluation on interesting benchmarks where we might expect finding symmetries to outperform regular CNNs (e.g. Tu et al. (2022)) and/or more detailed analysis on more involved search spaces of whether the proposed methods are recovering the “right” symmetry for the dataset, even if synthetic.
2. The authors do not consider the random search baseline for their method. More generally, it would be useful to follow the NAS reproducibility checklist here (Lindauer & Hutter, 2020).

### References:
- Lindauer, Hutter. Best practices for scientific research on neural architecture search. JMLR 2020.
- Tu, Roberts, Khodak, Shen, Sala, Talwalkar. NAS-Bench-360: Benchmarking neural architecture search on diverse tasks. NeurIPS 2022.

The evidence to be evaluated is:In more diverse approaches, Zhou et al. (2020) and Yeh et al. (2022) represent symmetry-inducing weight sharing via learnable matrices. Romero & Lohit (2022) and van der Ouderaa et al. (2022) learn partial or soft equivariances for each layer.

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer. The proposed morphism, depicted in Figure 1(A) and described in Section 4.1, changes the equivariance constraint from one group to another subgroup while preserving the learned weights of the initial group convolutional operator. The \([G]\)-mixed equivariant layer, shown in Figure 1(B) and presented in Section 4.2, allows for a single layer to represent equivariance to multiple subgroups through a weighted sum. The _equivariance relaxation morphism_ reparametrizes a \(G\)-equivariant group (or lifting) convolutional layer to operate over a subgroup of \(G\), partially removing weight-sharing constraints from the parameter space while maintaining the functionality of the layer.

Given a \(G\)-equivariant group convolutional layer with feature map \(f\) and filter \(\psi\), we define the relaxed feature map \(\tilde{f}\colon G^{\prime}\to\mathbb{R}^{(C_{l-1}\times|R|)}\) and relaxed filter \(\tilde{\psi}\colon G^{\prime}\to\mathbb{R}^{(C_{l}\times|R|)\times(C_{l-1} \times|R|)}\) as follows. For \(c\in[C_{l-1}]\), \(s,t\in R\), \(d\in[C_{l}]\):

We define the _equivariance relaxation morphism_ from \(G\) to \(G^{\prime}\) as the reparametrization of \(\psi\) as \(\tilde{\psi}\) (Eq. 4) and reshaping of \(f\) as \(\tilde{f}\) (Eq. 3). We will show that the new layer, \([\tilde{f}\star_{G^{\prime}}\tilde{\psi}]_{(d,t)}(g^{\prime})\), is equivalent to \([f\star_{G}\psi]_{d}(g^{\prime}t)\) down to reshaping.

\[[f\star_{G}\psi]_{d}(g^{\prime}t) =[f\star_{G}\psi]_{d}(g) \tag{5}\] \[=\sum_{h\in G}\sum_{c=1}^{C_{l-1}}f_{c}(h)\psi_{d,c}(g^{-1}h),\] (6) \[=\sum_{h^{\prime}\in G^{\prime}}\sum_{s\in R}\sum_{c=1}^{C_{l-1} }f_{c}(h^{\prime}s)\psi_{d,c}(t^{-1}g^{\prime-1}h^{\prime}s),\] (7) \[=\sum_{h^{\prime}\in G^{\prime}}\sum_{c=1}^{C_{l-1}}\sum_{s\in R }\tilde{f}_{(c,s)}(h^{\prime})\tilde{\psi}_{(d,t),(c,s)}(g^{\prime-1}h^{\prime }),\] (8) \[=\left[\tilde{f}\star_{G^{\prime}}\tilde{\psi}\right]_{(d,t)}(g^{ \prime}), \tag{9}\]

Thus, the convolution of \(\tilde{f}\) with \(\tilde{\psi}\) is equivariant to \(G\) but parametrized as a \(G^{\prime}\)-equivariant group convolutional layer, where the representatives are expanded into independent channels. This morphism can be viewed as initializing a \(G^{\prime}\)-equivariant layer with a pre-trained prior of equivariance to \(G\), maintaining any previous training.

Towards learning equivariance, we additionally propose partial equivariance via a mixture of layers, each constrained to equivariance to different groups, applied in parallel to the same input then combined via a weighted sum.

For a set of groups \([G]\), such as a subgroup lattice of some group \(G\), we define a \([G]\)_-mixed equivariant layer_ as: \[\left[f^{\hat{\pi}_{[G]}}[\psi]\right]_{(d,t)}(g) =\sum_{G\in[G]}z_{G}\left[f\star_{G^{\prime}}\widetilde{\psi^{G}} \right]_{(d,t)}(g) \tag{10}\] \[=\left[f\star_{G^{\prime}}\sum_{G\in[G]}z_{G}\widetilde{\psi^{G} }\right]_{(d,t)}(g), \tag{11}\] where each element \(z_{G}\) of \([z]\coloneqq\{z_{G}|G\in[G]\}\) is an architectural weighting parameter such that \(\sum_{G\in[G]}z_{G}=1\), \(G^{\prime}\) is a subgroup of all groups in \([G]\), each element \(\psi^{G}\) of \([\psi]\) is a filter with a domain of \(G\), and \(\widetilde{\psi^{G}}\) is the transformation of \(\psi^{G}\) from a domain of \(G\) to \(G^{\prime}\) as defined in Equation 4. Thus, the layer is parametrized by \([\psi]\) and \([z]\), computing a weighted sum of operations that are equivariant to different groups of \([G]\). The layer may be equivalently computed by convolution of the input with the weighted sum of transformed filters, shown in Equation 11.

We present two NAS methods that utilize the presented mechanisms for discovering appropriate equivariance during training: Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) and Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)). Both methods optimize an architecture while learning weights, yielding a final trained network adapted to equivariances present in the training data.

EquiNAS\({}_{E}\), in Section 5.1, searches for networks composed of layers each fully equivariant to possibly different groups, while EquiNAS\({}_{D}\), in Section 5.2, searches for smooth mixtures of equivariant layers.

Offspring are generated from each parent separately by mutation using the relaxation morphism. This preserves the weights of the parametrized equivariance during mutation, allowing for the continuous training of networks over evolution by inheritance from parent individuals. Specifically, mutation reduces a single layer's parametrized equivariance to a subgroup within the constraint that each layer has parametrized equivariance to a subgroup of all preceding layers. This constraint yields local equivariance properties for the network, as shown in Weiler and Cesa (2019) and Elsayed et al. (2020) to be empirically favorable in image classification tasks. The resulting individuals are each trained independently for a given training time, and then this process repeats.

In a contrasting paradigm, the \([G]\)-mixed equivariant layer presented in Section 4.2 allows for smoothly searching across a spectrum of equivariance for each layer via a differentiable NAS algorithm. Our Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)) algorithm, defined in Algorithm 2, is inspired by DARTS (Liu et al., 2018) with significant changes detailed in the following paragraphs.

These two NAS approaches present adaptations of two standard types of NAS, evolutionary and differentiable, to the search for optimal partial equivariance.

We focus on the regular representation of groups and show experiments with reflectional and up to 4-fold rotational symmetry groups applied to image classification tasks. Examples of symmetry groups acting on pixel space, which corresponds to \(\mathbb{Z}^{2}\), include \(T(2)\), which consists of discrete translations in both dimensions; the cyclical groups \(C_{n}\), which consist of \(n\)-fold rotations; and the dihedral groups \(D_{n}\), which consist of reflections with \(n\)-fold rotations, where \(n\in\{1,2,4\}\) for exact symmetry without interpolation.

As standard convolutional layers are already equivariant to \(T(2)\), we refer to layers also equivariant to \(n\)-fold rotations with or without reflections as \(D_{n}\) or \(C_{n}\)-equivariant, respectively. So, a \(C_{1}\) equivariant convolutional layer is a standard translation-equivariant convolutional layer. We use \((C_{1},D_{1},C_{2},D_{2},C_{4},D_{4})\) as the set of potential groups for mutation in EquiNAS\({}_{E}\) and as \([G]\) in EquiNAS\({}_{D}\).

These tasks exhibit varying levels of rotational and reflectional symmetry, motivating architectural optimization to determine the most effective application of equivariance constraints. Across all experiments, the architectures are designed to have consistent channel dimensions once expanded to a standard translation-equivariant convolutional layer for each layer across models. Thus, constrained equivariance to a larger symmetry group results in fewer learnable parameters. A layer constrained to \(C_{4}\) equivariance has \(|C_{4}\setminus D_{4}|=2\) times as many independent channels and as many parameters as a layer constrained to \(D_{4}\) equivariance. This is a notably different paradigm than other works that equate parameter counts across architectures with different equivariance properties.

In addition to the normally initialized static baselines, we also train and test baselines that are initialized with priors to larger symmetry groups. These are implemented by initializing all layers to be constrained to the prior symmetry group, then using the equivariance relaxation morphism on each layer. EquiNAS\({}_{E}\) searches for relaxation schedules that yield trained priors on equivariance, while these additional baselines yield untrained priors.

To our knowledge, this is the first work which proposes search methods for networks with dynamically constrained equivariance. Many NAS approaches separately search for an architecture and then reinitialize and retrain the weights, while our two proposed approaches find an optimal architecture with trained weights in a single process, notably with dynamically constrained weights.

Our two equivariance-aware NAS approaches have distinct approaches: EquiNAS\({}_{E}\) searches for architectures composed of discretely equivariant layers, while EquiNAS\({}_{D}\) searches for continuous mixtures of equivariance within each layer.

We present two mechanisms towards equivariance-aware architectural optimization, the equivariance relaxation morphism and the \([G]\)-mixed equivariant layer, and two NAS algorithms that respectively implement these mechanisms evolutionarily as EquiNAS\({}_{E}\) and differentiably as EquiNAS\({}_{D}\).

* Yeh et al. (2022) Raymond A Yeh, Yuan-Ting Hu, Mark Hasegawa-Johnson, and Alexander Schwing. Equivariance discovery by learned parameter-sharing.

* Zhou et al. (2020) Allan Zhou, Tom Knowles, and Chelsea Finn. Meta-learning symmetries by reparameterization. In _International Conference on Learning Representations_, 2020.

Group convolutional layers The implementation of regular group convolutional layers can be viewed as a special case of our proposed equivariance relaxation morphism.

Equivariance relaxation morphism To implement the equivariance relaxation morphism, the new filter tensor is initialized by applying Equation 4 such that result of applying the preceding filter transformation is equivalent. Our implementation of group actions relies on group channel indexing to represent the order of group elements: to ensure this is consistent before and after the morphism, the appropriate reordering of the output and input channels of the expanded filter are applied upon expansion.

The \([G]\)-mixed equivariant layer is built on top of this implementation, also using proper input and output channel reordering between layers to ensure correct mixing of group channels.

For both EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) experiments, we use the same backbone architecture, such that the static baselines have the same architecture across experiments. The architectures have a lifting layer followed by 7 group convolutional layers, for a total of 8 convolutional layers.

This is enabled by the \([G]\)-mixed equivariant layer, which is a contribution of this work.

Our methods enable searching for both architecture and parameters concurrently in a single training process.

Input: Figure 1: Visualizing the equivariance relaxation morphism and the [G]-mixed equivariant layer, using the C4 group. In (A), the learnable parameters of a C4-equivariant convolutional layer are expanded using each group element action so the expanded filter can be used in a standard convolutional layer. The equivariance relaxation morphism reparametrizes the layer to be architecturally constrained toC2 equivariance, initialized to functionalC4 equivariance. In (B), convolutional operations equivariant to subgroups ofC4 are summed with learnable architectural weighting parameters.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 3: Architecture weighting parameters by layer for one selected trial from RotMNIST, Galaxy10, and ISIC. For other trials, see Figures 10-12 in Appendix D.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the claim "First, we present two mechanisms towards equivariance-aware architectural optimization" and the evidence provided:

The evidence strongly supports this claim as being well-stated. The paper clearly demonstrates and details two distinct mechanisms:
1. The equivariance relaxation morphism (thoroughly explained with equations 3-9 and visualized in Figure 1A)
2. The [G]-mixed equivariant layer (detailed in equations 10-11 and illustrated in Figure 1B)

The mechanisms are not only presented but also thoroughly mathematically formalized and implemented in two different NAS approaches (EquiNAS_E and EquiNAS_D). The claim is precise and accurately reflects what is delivered in the evidence without exaggeration.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

First, we present two mechanisms towards equivariance-aware architectural optimization.

The review comment to be evaluated is:

**Strengths:**
1. The formulation of partial/ approximate equivariances/ invariances $-$ as a mixture over subgroups is interesting and an important problem to tackle.
2. Strong experimental results for both their learning paradigms.
3. For a person with sufficient group theory background - the paper is easy to understand.

**Weaknesses and correspnding questions/ suggestions**:
1. Lack of comparison of the evolutionary paradigm with a similar approach [1] which searches over the subspace lattice  to identify te groups to be equivariant/ invariant to.
2. It is hard to see how this extends to non-finite groups which are not compact. Would suggest that the authors comment on this.
3. While the paper is understandable for someone who has the necessary group theory background - it is almost impossible for general audience - would recommend the authors add necessary background and preliminaries to the appendix.
4. Studies in out of distribution settings are missing (for e.g. train only contains images rotated by 0-15 degrees, but test has those between 60-180 degrees) - this is will give us more insight into the weights learned.

**Minor:**
1. Please make sure the title on openreview matches the title on the paper

**References**
1. Mouli, S. Chandra, and Bruno Ribeiro. "Neural Networks for Learning Counterfactual G-Invariances from Single Environments." ICLR 2021

The evidence to be evaluated is:In more diverse approaches, Zhou et al. (2020) and Yeh et al. (2022) represent symmetry-inducing weight sharing via learnable matrices. Romero & Lohit (2022) and van der Ouderaa et al. (2022) learn partial or soft equivariances for each layer.

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer. The proposed morphism, depicted in Figure 1(A) and described in Section 4.1, changes the equivariance constraint from one group to another subgroup while preserving the learned weights of the initial group convolutional operator. The \([G]\)-mixed equivariant layer, shown in Figure 1(B) and presented in Section 4.2, allows for a single layer to represent equivariance to multiple subgroups through a weighted sum. The _equivariance relaxation morphism_ reparametrizes a \(G\)-equivariant group (or lifting) convolutional layer to operate over a subgroup of \(G\), partially removing weight-sharing constraints from the parameter space while maintaining the functionality of the layer.

Given a \(G\)-equivariant group convolutional layer with feature map \(f\) and filter \(\psi\), we define the relaxed feature map \(\tilde{f}\colon G^{\prime}\to\mathbb{R}^{(C_{l-1}\times|R|)}\) and relaxed filter \(\tilde{\psi}\colon G^{\prime}\to\mathbb{R}^{(C_{l}\times|R|)\times(C_{l-1} \times|R|)}\) as follows. For \(c\in[C_{l-1}]\), \(s,t\in R\), \(d\in[C_{l}]\):

We define the _equivariance relaxation morphism_ from \(G\) to \(G^{\prime}\) as the reparametrization of \(\psi\) as \(\tilde{\psi}\) (Eq. 4) and reshaping of \(f\) as \(\tilde{f}\) (Eq. 3). We will show that the new layer, \([\tilde{f}\star_{G^{\prime}}\tilde{\psi}]_{(d,t)}(g^{\prime})\), is equivalent to \([f\star_{G}\psi]_{d}(g^{\prime}t)\) down to reshaping.

\[[f\star_{G}\psi]_{d}(g^{\prime}t) =[f\star_{G}\psi]_{d}(g) \tag{5}\] \[=\sum_{h\in G}\sum_{c=1}^{C_{l-1}}f_{c}(h)\psi_{d,c}(g^{-1}h),\] (6) \[=\sum_{h^{\prime}\in G^{\prime}}\sum_{s\in R}\sum_{c=1}^{C_{l-1} }f_{c}(h^{\prime}s)\psi_{d,c}(t^{-1}g^{\prime-1}h^{\prime}s),\] (7) \[=\sum_{h^{\prime}\in G^{\prime}}\sum_{c=1}^{C_{l-1}}\sum_{s\in R }\tilde{f}_{(c,s)}(h^{\prime})\tilde{\psi}_{(d,t),(c,s)}(g^{\prime-1}h^{\prime }),\] (8) \[=\left[\tilde{f}\star_{G^{\prime}}\tilde{\psi}\right]_{(d,t)}(g^{ \prime}), \tag{9}\]

Thus, the convolution of \(\tilde{f}\) with \(\tilde{\psi}\) is equivariant to \(G\) but parametrized as a \(G^{\prime}\)-equivariant group convolutional layer, where the representatives are expanded into independent channels. This morphism can be viewed as initializing a \(G^{\prime}\)-equivariant layer with a pre-trained prior of equivariance to \(G\), maintaining any previous training.

Towards learning equivariance, we additionally propose partial equivariance via a mixture of layers, each constrained to equivariance to different groups, applied in parallel to the same input then combined via a weighted sum.

For a set of groups \([G]\), such as a subgroup lattice of some group \(G\), we define a \([G]\)_-mixed equivariant layer_ as: \[\left[f^{\hat{\pi}_{[G]}}[\psi]\right]_{(d,t)}(g) =\sum_{G\in[G]}z_{G}\left[f\star_{G^{\prime}}\widetilde{\psi^{G}} \right]_{(d,t)}(g) \tag{10}\] \[=\left[f\star_{G^{\prime}}\sum_{G\in[G]}z_{G}\widetilde{\psi^{G} }\right]_{(d,t)}(g), \tag{11}\] where each element \(z_{G}\) of \([z]\coloneqq\{z_{G}|G\in[G]\}\) is an architectural weighting parameter such that \(\sum_{G\in[G]}z_{G}=1\), \(G^{\prime}\) is a subgroup of all groups in \([G]\), each element \(\psi^{G}\) of \([\psi]\) is a filter with a domain of \(G\), and \(\widetilde{\psi^{G}}\) is the transformation of \(\psi^{G}\) from a domain of \(G\) to \(G^{\prime}\) as defined in Equation 4. Thus, the layer is parametrized by \([\psi]\) and \([z]\), computing a weighted sum of operations that are equivariant to different groups of \([G]\). The layer may be equivalently computed by convolution of the input with the weighted sum of transformed filters, shown in Equation 11.

We present two NAS methods that utilize the presented mechanisms for discovering appropriate equivariance during training: Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) and Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)). Both methods optimize an architecture while learning weights, yielding a final trained network adapted to equivariances present in the training data.

EquiNAS\({}_{E}\), in Section 5.1, searches for networks composed of layers each fully equivariant to possibly different groups, while EquiNAS\({}_{D}\), in Section 5.2, searches for smooth mixtures of equivariant layers.

Offspring are generated from each parent separately by mutation using the relaxation morphism. This preserves the weights of the parametrized equivariance during mutation, allowing for the continuous training of networks over evolution by inheritance from parent individuals. Specifically, mutation reduces a single layer's parametrized equivariance to a subgroup within the constraint that each layer has parametrized equivariance to a subgroup of all preceding layers. This constraint yields local equivariance properties for the network, as shown in Weiler and Cesa (2019) and Elsayed et al. (2020) to be empirically favorable in image classification tasks. The resulting individuals are each trained independently for a given training time, and then this process repeats.

In a contrasting paradigm, the \([G]\)-mixed equivariant layer presented in Section 4.2 allows for smoothly searching across a spectrum of equivariance for each layer via a differentiable NAS algorithm. Our Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)) algorithm, defined in Algorithm 2, is inspired by DARTS (Liu et al., 2018) with significant changes detailed in the following paragraphs.

These two NAS approaches present adaptations of two standard types of NAS, evolutionary and differentiable, to the search for optimal partial equivariance.

We focus on the regular representation of groups and show experiments with reflectional and up to 4-fold rotational symmetry groups applied to image classification tasks. Examples of symmetry groups acting on pixel space, which corresponds to \(\mathbb{Z}^{2}\), include \(T(2)\), which consists of discrete translations in both dimensions; the cyclical groups \(C_{n}\), which consist of \(n\)-fold rotations; and the dihedral groups \(D_{n}\), which consist of reflections with \(n\)-fold rotations, where \(n\in\{1,2,4\}\) for exact symmetry without interpolation.

As standard convolutional layers are already equivariant to \(T(2)\), we refer to layers also equivariant to \(n\)-fold rotations with or without reflections as \(D_{n}\) or \(C_{n}\)-equivariant, respectively. So, a \(C_{1}\) equivariant convolutional layer is a standard translation-equivariant convolutional layer. We use \((C_{1},D_{1},C_{2},D_{2},C_{4},D_{4})\) as the set of potential groups for mutation in EquiNAS\({}_{E}\) and as \([G]\) in EquiNAS\({}_{D}\).

These tasks exhibit varying levels of rotational and reflectional symmetry, motivating architectural optimization to determine the most effective application of equivariance constraints. Across all experiments, the architectures are designed to have consistent channel dimensions once expanded to a standard translation-equivariant convolutional layer for each layer across models. Thus, constrained equivariance to a larger symmetry group results in fewer learnable parameters. A layer constrained to \(C_{4}\) equivariance has \(|C_{4}\setminus D_{4}|=2\) times as many independent channels and as many parameters as a layer constrained to \(D_{4}\) equivariance. This is a notably different paradigm than other works that equate parameter counts across architectures with different equivariance properties.

In addition to the normally initialized static baselines, we also train and test baselines that are initialized with priors to larger symmetry groups. These are implemented by initializing all layers to be constrained to the prior symmetry group, then using the equivariance relaxation morphism on each layer. EquiNAS\({}_{E}\) searches for relaxation schedules that yield trained priors on equivariance, while these additional baselines yield untrained priors.

To our knowledge, this is the first work which proposes search methods for networks with dynamically constrained equivariance. Many NAS approaches separately search for an architecture and then reinitialize and retrain the weights, while our two proposed approaches find an optimal architecture with trained weights in a single process, notably with dynamically constrained weights.

Our two equivariance-aware NAS approaches have distinct approaches: EquiNAS\({}_{E}\) searches for architectures composed of discretely equivariant layers, while EquiNAS\({}_{D}\) searches for continuous mixtures of equivariance within each layer.

We present two mechanisms towards equivariance-aware architectural optimization, the equivariance relaxation morphism and the \([G]\)-mixed equivariant layer, and two NAS algorithms that respectively implement these mechanisms evolutionarily as EquiNAS\({}_{E}\) and differentiably as EquiNAS\({}_{D}\).

* Yeh et al. (2022) Raymond A Yeh, Yuan-Ting Hu, Mark Hasegawa-Johnson, and Alexander Schwing. Equivariance discovery by learned parameter-sharing.

* Zhou et al. (2020) Allan Zhou, Tom Knowles, and Chelsea Finn. Meta-learning symmetries by reparameterization. In _International Conference on Learning Representations_, 2020.

Group convolutional layers The implementation of regular group convolutional layers can be viewed as a special case of our proposed equivariance relaxation morphism.

Equivariance relaxation morphism To implement the equivariance relaxation morphism, the new filter tensor is initialized by applying Equation 4 such that result of applying the preceding filter transformation is equivalent. Our implementation of group actions relies on group channel indexing to represent the order of group elements: to ensure this is consistent before and after the morphism, the appropriate reordering of the output and input channels of the expanded filter are applied upon expansion.

The \([G]\)-mixed equivariant layer is built on top of this implementation, also using proper input and output channel reordering between layers to ensure correct mixing of group channels.

For both EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) experiments, we use the same backbone architecture, such that the static baselines have the same architecture across experiments. The architectures have a lifting layer followed by 7 group convolutional layers, for a total of 8 convolutional layers.

This is enabled by the \([G]\)-mixed equivariant layer, which is a contribution of this work.

Our methods enable searching for both architecture and parameters concurrently in a single training process.

Input: Figure 1: Visualizing the equivariance relaxation morphism and the [G]-mixed equivariant layer, using the C4 group. In (A), the learnable parameters of a C4-equivariant convolutional layer are expanded using each group element action so the expanded filter can be used in a standard convolutional layer. The equivariance relaxation morphism reparametrizes the layer to be architecturally constrained toC2 equivariance, initialized to functionalC4 equivariance. In (B), convolutional operations equivariant to subgroups ofC4 are summed with learnable architectural weighting parameters.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 3: Architecture weighting parameters by layer for one selected trial from RotMNIST, Galaxy10, and ISIC. For other trials, see Figures 10-12 in Appendix D.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the claim and evidence, I find that "we present two mechanisms towards equivariance-aware architectural optimization" is well-stated and supported by the evidence. The paper clearly demonstrates and thoroughly explains both mechanisms:

1. The equivariance relaxation morphism (shown in Figure 1A and detailed mathematically)
2. The [G]-mixed equivariant layer (shown in Figure 1B and formalized with equations)

Both mechanisms are thoroughly developed with mathematical foundations, visual explanations, and implementation details. The review comments also acknowledge the strength of both learning paradigms. The claim is precise, accurately reflects the scope of the work, and avoids exaggeration.

<score>1</score>


Input:
Your role is to assess whether a claim is well-stated or overstated based on the available evidence.

The Input Information will include:
1. Original Claim: The claim under evaluation.
2. Evidence: Research findings, including figures, tables, or other relevant data supporting the claim.

Optional. review comment: Reviewer feedback relevant to the claim's validity.

The claim should be assesed with the following:
Score: Evaluate the claim on a scale of 1 to 5, where:
- 1: Well-stated (accurately reflects the evidence without exaggeration)
- 5: Overstated (exaggerates findings' significance, magnitude, or certainty)

Justification: Provide a detailed explanation that includes:
- Instances of exaggerated wording, insufficient experiments, lack of experimental details, gaps in knowledge, weak grounding in evidence, or lack of acknowledgment of limitations.
- Direct citations from relevant evidence that substantiate your reasoning.
- If review comment is included only take into account the relevant comments towards the claim.

The claim to be assessed is:

The _equivariance relaxation morphism_ for group convolutional layers partially expands the representation and parameters of the layer to enable less constrained learning with a prior on symmetry.

The evidence to be evaluated is:Basu et al. (2021) also utilizes subgroup decomposition but instead algorithmically builds up equivariances from smaller groups, while our work focuses on relaxing existing constraints.

Mouli & Ribeiro (2021) also progressively relaxes equivariance constraints, but with regularized rather than parametrized constraints.

Romero & Lohit (2022) and van der Ouderaa et al. (2022) learn partial or soft equivariances for each layer.

For discrete group \(G\), the \(l^{\text{th}}\)\(G\)-equivariant group convolutional layer (Cohen & Welling, 2016) of a group convolutional neural network (G-CNN) convolves1 the feature map \(f\colon G\to\mathbb{R}^{C_{l-1}}\) from the previous layer with a filter with kernel size \(k\) represented as learnable parameters \(\psi\colon G\to\mathbb{R}^{C_{l}\times C_{l-1}}\).

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer. The proposed morphism, depicted in Figure 1(A) and described in Section 4.1, changes the equivariance constraint from one group to another subgroup while preserving the learned weights of the initial group convolutional operator.

The _equivariance relaxation morphism_ reparametrizes a \(G\)-equivariant group (or lifting) convolutional layer to operate over a subgroup of \(G\), partially removing weight-sharing constraints from the parameter space while maintaining the functionality of the layer.

We define the _equivariance relaxation morphism_ from \(G\) to \(G^{\prime}\) as the reparametrization of \(\psi\) as \(\tilde{\psi}\) (Eq. 4) and reshaping of \(f\) as \(\tilde{f}\) (Eq. 3). We will show that the new layer, \([\tilde{f}\star_{G^{\prime}}\tilde{\psi}]_{(d,t)}(g^{\prime})\), is equivalent to \([f\star_{G}\psi]_{d}(g^{\prime}t)\) down to reshaping.

Thus, the convolution of \(\tilde{f}\) with \(\tilde{\psi}\) is equivariant to \(G\) but parametrized as a \(G^{\prime}\)-equivariant group convolutional layer, where the representatives are expanded into independent channels. This morphism can be viewed as initializing a \(G^{\prime}\)-equivariant layer with a pre-trained prior of equivariance to \(G\), maintaining any previous training.

The equivariance relaxation morphism generalizes this concept to any subgroup.

Towards learning equivariance, we additionally propose partial equivariance via a mixture of layers, each constrained to equivariance to different groups, applied in parallel to the same input then combined via a weighted sum. The equivariance relaxation morphism provides a mapping of group elements between a group and a subgroup.

where each element \(z_{G}\) of \([z]\coloneqq\{z_{G}|G\in[G]\}\) is an architectural weighting parameter such that \(\sum_{G\in[G]}z_{G}=1\), \(G^{\prime}\) is a subgroup of all groups in \([G]\), each element \(\psi^{G}\) of \([\psi]\) is a filter with a domain of \(G\), and \(\widetilde{\psi^{G}}\) is the transformation of \(\psi^{G}\) from a domain of \(G\) to \(G^{\prime}\) as defined in Equation 4. Thus, the layer is parametrized by \([\psi]\) and \([z]\), computing a weighted sum of operations that are equivariant to different groups of \([G]\).

Towards finding the optimal full equivariance per layer, the equivariance relaxation morphism presented in Section 4.1 is applied as the genetic operator in an evolutionary hill-climbing algorithm.

A population of networks, which starts with an individual with all layers equivariant to the largest possible group, undergoes mutation via equivariance relaxation and selection based on accuracy and parameter count to optimize neural architecture while learning network parameters.

Specifically, mutation reduces a single layer's parametrized equivariance to a subgroup within the constraint that each layer has parametrized equivariance to a subgroup of all preceding layers. This constraint yields local equivariance properties for the network, as shown in Weiler and Cesa (2019) and Elsayed et al. (2020) to be empirically favorable in image classification tasks.

Accuracy-based selection alone would necessarily prefer larger networks as mutation via the equivariance relaxation morphism results in two networks with identical performance but different size, the relaxed network having more parameters, until training; potentially short-term increases in validation accuracy after training would then result in the selection of individuals with more parameters.

In our formulation, this is not necessary as any mixed operation can be equivalently expressed as a single layer equivariant to any group \(G^{\prime}\) that is a common subgroup to all groups of the mixed operation (Eq. 11): in our experimental case, this is a standard translation-equivariant convolutional layer, so the final model can be equivalently expressed as a standard convolutional model with encoded partial equivariance.

These two NAS approaches present adaptations of two standard types of NAS, evolutionary and differentiable, to the search for optimal partial equivariance.

We focus on the regular representation of groups and show experiments with reflectional and up to 4-fold rotational symmetry groups applied to image classification tasks. Examples of symmetry groups acting on pixel space, which corresponds to \(\mathbb{Z}^{2}\), include \(T(2)\), which consists of discrete translations in both dimensions; the cyclical groups \(C_{n}\), which consist of \(n\)-fold rotations; and the dihedral groups \(D_{n}\), which consist of reflections with \(n\)-fold rotations, where \(n\in\{1,2,4\}\) for exact symmetry without interpolation. The \(p4\) group consists of discrete translations and multiples of \(90^{\circ}\) rotations and may be represented as \(T(2)\rtimes C_{4}\). The \(p4m\) group consists of discrete translations, reflections, and multiples of \(90^{\circ}\) rotations and may be represented as \(T(2)\rtimes D_{4}\). As standard convolutional layers are already equivariant to \(T(2)\), we refer to layers also equivariant to \(n\)-fold rotations with or without reflections as \(D_{n}\) or \(C_{n}\)-equivariant, respectively.

These tasks exhibit varying levels of rotational and reflectional symmetry, motivating architectural optimization to determine the most effective application of equivariance constraints. Across all experiments, the architectures are designed to have consistent channel dimensions once expanded to a standard translation-equivariant convolutional layer for each layer across models. Thus, constrained equivariance to a larger symmetry group results in fewer learnable parameters. A layer constrained to \(C_{4}\) equivariance has \(|C_{4}\setminus D_{4}|=2\) times as many independent channels and as many parameters as a layer constrained to \(D_{4}\) equivariance.

However, the \(D_{4}\) baseline fails at the Galaxy10 task, demonstrating that the same equivariant architecture can not always be naively applied. Both search methods, EquiNAS\({}_{E}\) and RPP, outperform all baseline models on Galaxy10, and by a large margin for EquiNAS\({}_{E}\).

These are implemented by initializing all layers to be constrained to the prior symmetry group, then using the equivariance relaxation morphism on each layer. EquiNAS\({}_{E}\) searches for relaxation schedules that yield trained priors on equivariance, while these additional baselines yield untrained priors. The results in Table 1 show that, the \(C_{1}\)-equivariant networks generally improve with either equivariance prior, while the \(C_{4}\) equivariant networks perform better with \(D_{4}\) equivariance initialization only when the \(D_{4}\) constrained baselines also work well.

The general trend of less constrained layers toward the output supports the conjecture of local equivariance being beneficial.

However, our algorithm shows strong preference for a single, more equivariant and thus more expressive layer, notably to \(D_{4}\) or \(C_{4}\) equivariance, without such regularization.

To our knowledge, this is the first work which proposes search methods for networks with dynamically constrained equivariance.

Our two equivariance-aware NAS approaches have distinct approaches: EquiNAS\({}_{E}\) searches for architectures composed of discretely equivariant layers, while EquiNAS\({}_{D}\) searches for continuous mixtures of equivariance within each layer.

Networks with more equivariance constraints tend to learn faster, but equivariance relaxation may yield large gradients for newly unconstrained parameters and thus fast increases in performance.

Beyond NAS, the equivariance relaxation morphism could be used in other applications such as fine-tuning and distillation. Layers of a pre-trained equivariant network could be expanded via equivariance relaxation before fine-tuning on the same or a new task.

We present two mechanisms towards equivariance-aware architectural optimization, the equivariance relaxation morphism and the \([G]\)-mixed equivariant layer, and two NAS algorithms that respectively implement these mechanisms evolutionarily as EquiNAS\({}_{E}\) and differentiably as EquiNAS\({}_{D}\).

Equivariance discovery by learned parameter-sharing.

A _symmetry group_\(G\) is a set of such mappings along with a binary operation \(\cdot\): \(G\times G\to G\), known as the _group product_, that satisfies axioms for closure, associativity, the identity, and the inverse (Herstein, 2006). A group \(G\) acts on a set \(\mathcal{X}\) via the _group action_ \(\cdot:G\times\mathcal{X}\rightarrow\mathcal{X},\ (g,x)\mapsto g.x\) that satisfies axioms for identity and compatibility: \(\mathcal{X}\) is called a _\(G\)-space_. Equivariance is the property of a mapping such that transformation of the input results in equivalent transformation of the output. Formally, a mapping \(h\colon\mathcal{X}\rightarrow\mathcal{Z}\) between two \(G\)-spaces is \(G\)-equivariant if for all \(g\in G\) and \(x\in\mathcal{X}\) we have: \(h(g.x)=g.h(x)\).

In this case, each individual is a neural network, with a genotype that encodes the parametrized equivariance group of each convolutional layer, represented as a vector of integers.

Each parent itself is kept for the next population, as well as each valid child that is generated via the equivariance relaxation morphism, such that they are functionally equivalent to their parent at initialization (although with a different architecture) and thus have the same fitness before training. Each individual in this population is partially trained, such that these children diverge from their siblings and parent, so that the next set of parents may be selected and this process repeats.

Group convolutional layers The implementation of regular group convolutional layers can be viewed as a special case of our proposed equivariance relaxation morphism.

The filter transformation expands this filter tensor by performing the action of each \(r\in R\) on another copy of the tensor to expand its shape along a new dimension, resulting in a tensor of size \(C_{l}\times S_{G_{l}}\times C_{l-1}\times S_{G_{l}}\times K_{l}\times K_{l}\), which is reshaped to \(C_{l}S_{G_{l}}\times C_{l-1}S_{G_{l}}\times K_{l}\times K_{l}\).

To implement the equivariance relaxation morphism, the new filter tensor is initialized by applying Equation 4 such that result of applying the preceding filter transformation is equivalent.

to ensure this is consistent before and after the morphism, the appropriate reordering of the output and input channels of the expanded filter are applied upon expansion. The new filter tensor has a shape of \(C_{l}|R|\times C_{l-1}|R|\times S_{G_{l}}/|R|\times K_{l}\times K_{l}\). The \([G]\)-mixed equivariant layer is built on top of this implementation, also using proper input and output channel reordering between layers to ensure correct mixing of group channels.

After 4 layers, the channel count doubles, from 16 to 32 for a \(D_{4}\) equivariant layer and scaling up for smaller symmetry group equivariance constraints.

The resulting architecture dynamics of this experiment are shown in Figure 4, showing that less augmented versions still have some inherent symmetry, while more augmented versions induce stronger architectural changes towards layers that are equivariant to larger groups. Across all augmentations, earlier layers tended towards more constraints to equivariance.

EquiNAS\({}_{D}\) finds competitive architectures on average and can find architectures which outperform baseline choices like architectures fully equivariant to \(C_{1}\) or \(D_{4}\). From comparing EquiNAS\({}_{D}\) Retrain to EquiNAS\({}_{D}\) results in Figure 8, retraining a resulting architecture is not consistently better or worse than using the final weights from EquiNAS\({}_{D}\), showing that there isn't a disadvantage to training weights during search and avoiding the additional cost of a post-search training step.

On this baseline, EquiNAS\({}_{D}\) greatly outperforms a search of random architectures, demonstrating that EquiNAS\({}_{D}\) can discover the appropriate equivariance for a specific dataset over fixed or randomly selected architectures.

This is enabled by the \([G]\)-mixed equivariant layer, which is a contribution of this work.

This can explain why the EquiNAS\({}_{D}\) results are closer to random baselines than the EquiNAS\({}_{E}\) results, as the search space permits easily finding the appropriate mix of equivariances compared to a discretized search space. Our methods enable searching for both architecture and parameters concurrently in a single training process.

Input: Table 1: Test error (lower is better) in percent of incorrect classifications across tasks and approaches. Statistics are aggregated over the final selected population of 5 individuals for EquiNASE and across 5 random seeds for all other methods. The best and second best average errors per task are highlighted. See Figures 5-6 in Appendix D for individual results and additional experiments.

Input: Figure 4: EquiNASD on classification of six augmentations of MNIST, where each row is a trial on MNIST augmented to exhibit symmetry to the labeled group.

Input: Figure 1: Visualizing the equivariance relaxation morphism and the [G]-mixed equivariant layer, using the C4 group. In (A), the learnable parameters of a C4-equivariant convolutional layer are expanded using each group element action so the expanded filter can be used in a standard convolutional layer. The equivariance relaxation morphism reparametrizes the layer to be architecturally constrained toC2 equivariance, initialized to functionalC4 equivariance. In (B), convolutional operations equivariant to subgroups ofC4 are summed with learnable architectural weighting parameters.

Input: Figure 10: Architecture weighting parameters by layer for all trials on RotMNIST.

Input: Figure 12: Architecture weighting parameters by layer for all trials on ISIC.

Input: Figure 11: Architecture weighting parameters by layer for all trials on Galaxy10.

Input: Figure 8: Test errors for experiments of Section 6.2.

Input: Figure 3: Architecture weighting parameters by layer for one selected trial from RotMNIST, Galaxy10, and ISIC. For other trials, see Figures 10-12 in Appendix D.

Input: Figure 6: Test errors against stored parameter count for experiments of Section 6.1. For EquiNASE , parameter counts of the final models are shown, although training begins with the same parameter count as theD4 baselines as shown in Figures 2 and 7. Although RPP trains with a higher parameter count, the parameters may be stored in a single filter per layer using Equation 11 for testing.

Input:
You should:
1. Carefully review the claim and the text and image evidence. Summarize how the evidence influenced your evaluation of the claim's validity and strength and briefly explain whether the claim is wellstated or overstated on a scale of 1 to 5 (up to 100 words).
2. Provide the final score (choose from 1 to 5), in the format: <score>{score}</score>.

Output:
After reviewing the evidence, I find that the claim about the equivariance relaxation morphism is well-stated and accurately reflects the technical details presented in the paper. The evidence provides extensive mathematical formalization, experimental validation, and clear visualizations (particularly in Figure 1A) demonstrating how the morphism expands representation while maintaining symmetry priors. The paper thoroughly explains how the morphism reparametrizes G-equivariant layers to operate over subgroups while preserving learned weights, which aligns precisely with the claim's description. The results show this enables effective learning with relaxed constraints while maintaining symmetry priors.

<score>1</score>