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:

Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) utilizes the equivariance relaxation morphism in a greedy evolutionary algorithm, dynamically relaxing constraints throughout the training process.

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

Liu et al. (2018) approaches this difficult bi-level optimization by creating a large super-network containing all possible elements and continuously relaxing the discrete architectural parameters to enable search by gradient descent. Other NAS approaches include evolutionary algorithms (Real et al., 2017; Lu et al., 2019; Elsken et al., 2017) and reinforcement learning (Zoph & Le, 2017), which search over discretely represented architectures.

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.

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. The Evolutionary Equivariance-Aware NAS (EquiNAS\({}_{E}\)) algorithm, given in Algorithm 1, is similar to other evolutionary NAS methods such as Elsken et al. (2017) with pareto selection as in Falanti et al. (2022). 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.

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.

The resulting individuals are each trained independently for a given training time, and then this process repeats.

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. Thus, the final optimized architecture and trained weights are output from the single search process.

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

These tasks exhibit varying levels of rotational and reflectional symmetry, motivating architectural optimization to determine the most effective application of equivariance constraints.

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.

Both search methods, EquiNAS\({}_{E}\) and RPP, outperform all baseline models on Galaxy10, and by a large margin for EquiNAS\({}_{E}\).

the selected population maintains a fully equivariant network in every generation. The final selected population originates from two main lineages, one staying fully equivariant until the last generations and the other diverging from the fully equivariant network midway through, showing that training with dynamically constrained parametrizations can produce performant models.

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.

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. 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.

EquiNAS\({}_{E}\) is innately greedy: at each selection step, the population is evaluated by known current performance rather than unknown final performance, biased to architectures that train quickly. 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.

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\({}_{D}\). We investigate how dynamic equivariance achieved by these algorithms affects the training and performance of models across multiple image classification tasks of varying complexity and assumed symmetry, demonstrating that these techniques can search for performant architectures and weights even on noisy tasks.

Evolutionary algorithms are optimization methods inspired by evolution in biology, where individuals in a population compete with their phenotypic traits in order to pass on their genotypic traits to offspring. The _population_ is the current collection of individuals. Each _individual_ is an instance of the object to be optimized and has a genotype that is decoded into a phenotype. 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. The individual continues training on the task before competing against other individuals to be selected as a parent to mutate to generate the next population. 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.

The implementation of regular group convolutional layers can be viewed as a special case of our proposed equivariance relaxation morphism. With the preliminaries given in Section 4.1 and the case of \(G^{\prime}=T(2),\tilde{f}\) and \(\tilde{\psi}\) are computed such that \(\tilde{f}_{(c,s)}(g^{\prime})\coloneqq f_{c}(g^{\prime}s)\) and \(\tilde{\psi}_{(d,t),(c,s)}(g^{\prime})\coloneqq\psi_{d,c}(t^{-1}g^{\prime}s)\) for each \(g^{\prime}\in T(2)\), \(c\in[C_{l-1}]\), \(s,t\in R\), and \(d\in[C_{l}]\). Let \(S_{G}:=|R|\). The learnable parameters of the \(G_{l}\)-equivariant \(l^{\text{th}}\) layer with \(C_{l}\) output channels, corresponding to \(\psi\), are stored as a tensor of size \(C_{l}\times C_{l-1}\times S_{G_{l}}\times K_{l}\times K_{l}\). 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 \(C_{l}S_{G_{l}}\times C_{l-1}S_{G_{l}}\times K_{l}\times K_{l}\). The input tensor to the \(l^{\text{th}}\) layer, corresponding to \(f\), is in the shape of \(B\times C_{l-1}\times S_{G_{l}}\times H_{l-1}\times W_{l-1}\), which is reshaped to \(B\times C_{l-1}S_{G_{l}}\times H_{l-1}\times W_{l-1}\) and convolved with the expanded filter.

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 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.

For EquiNAS\({}_{E}\), the parent selection size is 5, the training time per generation is 0.5 epochs, and the number of generations is 50 for all tasks.

The first ablates smart architecture search: EquiNAS\({}_{E}\) Random Select works as described in Algorithm 1 but with random parent selection (instead of pareto-front selection) and EquiNAS\({}_{D}\) Random \(Z\) works as described in Algorithm 2 but with random \(Z\) updates (instead of gradient descent) by shuffling \(Z\) gradients.

Since Random Static trains static architectures while EquiNAS\({}_{E}\) and EquiNAS\({}_{D}\) dynamically search for both architectures and parameters, we take the best 5 architectures for each and retrain their param eters from scratch as in the standard NAS paradigm, labeled as EquiNAS\({}_{E}\) Retrain and EquiNAS\({}_{D}\) Retrain, respectively, for fair comparison to Random Static.

Shown in Figure 5, EquiNAS\({}_{E}\) outperforms EquiNAS\({}_{E}\) Random Select, showing the benefit of using informed selection to guide the relaxation of equivariance constraints over training. Additionally, EquiNAS\({}_{E}\) Retrain outperforms the Random Static baseline, showing that using compute in an informed search is more beneficial than just randomly searching the space of static architecture constraints.

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. This approach is more efficient than NAS approaches that only search for architectures, requiring a retraining process within the resulting architecture for evaluation.

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 5: Test errors for experiments of Section 6.1.

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, the claim about EquiNAS_E appears to be partially overstated. While the evidence shows that EquiNAS_E does use an equivariance relaxation morphism in an evolutionary algorithm, the characterization of "dynamically relaxing constraints throughout the training process" is not fully accurate. The evidence indicates it's more of a discrete, stepwise process where mutation reduces a single layer's parametrized equivariance to a subgroup at specific points, rather than a continuous dynamic relaxation. The process follows a structured evolutionary approach with fixed training time per generation and specific selection criteria.

<score>3</score>

Justification:
1. The evidence confirms the use of equivariance relaxation morphism and evolutionary algorithm, as stated in: "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."

2. However, the "dynamically relaxing" aspect is overstated:
- Training occurs in discrete steps: "The training time per generation is 0.5 epochs, and the number of generations is 50 for all tasks."
- Mutations are discrete: "mutation reduces a single layer's parametrized equivariance to a subgroup"
- The review comments did not specifically address this aspect of the claim.

3. The process is more structured and constrained than the term "dynamically relaxing" suggests, with specific selection criteria and mutation rules.


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:

Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)) implements \([G]\)-mixed equivariant layers throughout a network to learn the appropriate approximate equivariance of each layer, in addition to their optimized weights, during training.

The evidence to be evaluated is:Although no other works on approximate equivariance explicitly study architectural optimization, some approaches are architectural in nature.

The main contributions of Wang et al. (2022) and Finzi et al. (2021) are similar to our proposed \([G]\)-mixed equivariant layer. Wang et al. (2022) also uses a weighted sum of filters, but uses the same group for each filter and defines the weights over the domain of group elements. Finzi et al. (2021) uses an equivariant layer in parallel to a linear layer with weighted regularization, thus only using two layers in parallel and weighting them by regularization rather than parametrization.

Romero & Lohit (2022) and van der Ouderaa et al. (2022) learn partial or soft equivariances for each layer. Neural architecture search Neural architecture search (NAS) aims to optimize both the architecture and its parameters for a given task. Liu et al. (2018) approaches this difficult bi-level optimization by creating a large super-network containing all possible elements and continuously relaxing the discrete architectural parameters to enable search by gradient descent. Other NAS approaches include evolutionary algorithms (Real et al., 2017; Lu et al., 2019; Elsken et al., 2017) and reinforcement learning (Zoph & Le, 2017), which search over discretely represented architectures.

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 \(\psi\colon G\to\mathbb{R}^{C_{l}\times C_{l-1}}\). For each output channel \(d\in[C_{l}]\), where \([C]:=\{1,\ldots,C\}\), and group element \(g\in G\), the layer's output is defined as: \[[f\star_{G}\psi]_{d}(g)=\sum_{h\in G}\sum_{c=1}^{C_{l-1}}f_{c}(h)\psi_{d,c}(g^{-1 }h). \tag{1}\]

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer.

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.

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.

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.

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.

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. Thus, the final optimized architecture and trained weights are output from the single search process.

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.

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.

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.

EquiNAS\({}_{D}\) achieves better test accuracy than the other comparable methods on RotMNIST and Galaxy10.

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. 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. The EquiNAS\({}_{D}\) algorithm avoids many known problems in differentiable NAS such as the discretization gap that occurs when searching over a continuous relaxation of a discrete architectural search space (Xie et al., 2021), such as that of EquiNAS\({}_{E}\).

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.

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\({}_{D}\). We investigate how dynamic equivariance achieved by these algorithms affects the training and performance of models across multiple image classification tasks of varying complexity and assumed symmetry, demonstrating that these techniques can search for performant architectures and weights even on noisy tasks.

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).

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. The individual continues training on the task before competing against other individuals to be selected as a parent to mutate to generate the next population. 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.

The implementation of regular group convolutional layers can be viewed as a special case of our proposed equivariance relaxation morphism. With the preliminaries given in Section 4.1 and the case of \(G^{\prime}=T(2),\tilde{f}\) and \(\tilde{\psi}\) are computed such that \(\tilde{f}_{(c,s)}(g^{\prime})\coloneqq f_{c}(g^{\prime}s)\) and \(\tilde{\psi}_{(d,t),(c,s)}(g^{\prime})\coloneqq\psi_{d,c}(t^{-1}g^{\prime}s)\) for each \(g^{\prime}\in T(2)\), \(c\in[C_{l-1}]\), \(s,t\in R\), and \(d\in[C_{l}]\). Let \(S_{G}:=|R|\). The learnable parameters of the \(G_{l}\)-equivariant \(l^{\text{th}}\) layer with \(C_{l}\) output channels, corresponding to \(\psi\), are stored as a tensor of size \(C_{l}\times C_{l-1}\times S_{G_{l}}\times K_{l}\times K_{l}\). 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 \(C_{l}S_{G_{l}}\times C_{l-1}S_{G_{l}}\times K_{l}\times K_{l}\). The input tensor to the \(l^{\text{th}}\) layer, corresponding to \(f\), is in the shape of \(B\times C_{l-1}\times S_{G_{l}}\times H_{l-1}\times W_{l-1}\), which is reshaped to \(B\times C_{l-1}S_{G_{l}}\times H_{l-1}\times W_{l-1}\) and convolved with the expanded filter.

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.

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.

After the final group convolutional layer is a group-dimension average pooling followed by two linear layers to the output dimension.

To explore the symmetry discovery of EquiNAS\({}_{D}\), we apply it to six augmentations of the MNIST dataset (LeCun et al., 1998), where each augmentation applies the group actions of each group in \(\{C_{1},C_{2},C_{4},D_{1},D_{2},D_{4}\}\) respectively. 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.

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

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

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 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:
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 the claim about EquiNAS_D implementing [G]-mixed equivariant layers throughout a network is well-supported by the extensive technical evidence provided. The paper clearly demonstrates the mathematical foundations, implementation details, and experimental results showing how EquiNAS_D uses [G]-mixed equivariant layers to learn appropriate approximate equivariance during training. The visualizations in Figures 3, 4, 10-12 provide concrete evidence of how the architectural weighting parameters evolve across different layers during training. The claim accurately reflects the methodology 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:

Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)) implements \([G]\)-mixed equivariant layers throughout a network to learn the appropriate approximate equivariance of each layer, in addition to their optimized weights, during training.

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:Although no other works on approximate equivariance explicitly study architectural optimization, some approaches are architectural in nature.

The main contributions of Wang et al. (2022) and Finzi et al. (2021) are similar to our proposed \([G]\)-mixed equivariant layer. Wang et al. (2022) also uses a weighted sum of filters, but uses the same group for each filter and defines the weights over the domain of group elements. Finzi et al. (2021) uses an equivariant layer in parallel to a linear layer with weighted regularization, thus only using two layers in parallel and weighting them by regularization rather than parametrization.

Romero & Lohit (2022) and van der Ouderaa et al. (2022) learn partial or soft equivariances for each layer. Neural architecture search Neural architecture search (NAS) aims to optimize both the architecture and its parameters for a given task. Liu et al. (2018) approaches this difficult bi-level optimization by creating a large super-network containing all possible elements and continuously relaxing the discrete architectural parameters to enable search by gradient descent. Other NAS approaches include evolutionary algorithms (Real et al., 2017; Lu et al., 2019; Elsken et al., 2017) and reinforcement learning (Zoph & Le, 2017), which search over discretely represented architectures.

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 \(\psi\colon G\to\mathbb{R}^{C_{l}\times C_{l-1}}\). For each output channel \(d\in[C_{l}]\), where \([C]:=\{1,\ldots,C\}\), and group element \(g\in G\), the layer's output is defined as: \[[f\star_{G}\psi]_{d}(g)=\sum_{h\in G}\sum_{c=1}^{C_{l-1}}f_{c}(h)\psi_{d,c}(g^{-1 }h). \tag{1}\]

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer.

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.

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.

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.

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.

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. Thus, the final optimized architecture and trained weights are output from the single search process.

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.

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.

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.

EquiNAS\({}_{D}\) achieves better test accuracy than the other comparable methods on RotMNIST and Galaxy10.

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. 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. The EquiNAS\({}_{D}\) algorithm avoids many known problems in differentiable NAS such as the discretization gap that occurs when searching over a continuous relaxation of a discrete architectural search space (Xie et al., 2021), such as that of EquiNAS\({}_{E}\).

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.

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\({}_{D}\). We investigate how dynamic equivariance achieved by these algorithms affects the training and performance of models across multiple image classification tasks of varying complexity and assumed symmetry, demonstrating that these techniques can search for performant architectures and weights even on noisy tasks.

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).

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. The individual continues training on the task before competing against other individuals to be selected as a parent to mutate to generate the next population. 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.

The implementation of regular group convolutional layers can be viewed as a special case of our proposed equivariance relaxation morphism. With the preliminaries given in Section 4.1 and the case of \(G^{\prime}=T(2),\tilde{f}\) and \(\tilde{\psi}\) are computed such that \(\tilde{f}_{(c,s)}(g^{\prime})\coloneqq f_{c}(g^{\prime}s)\) and \(\tilde{\psi}_{(d,t),(c,s)}(g^{\prime})\coloneqq\psi_{d,c}(t^{-1}g^{\prime}s)\) for each \(g^{\prime}\in T(2)\), \(c\in[C_{l-1}]\), \(s,t\in R\), and \(d\in[C_{l}]\). Let \(S_{G}:=|R|\). The learnable parameters of the \(G_{l}\)-equivariant \(l^{\text{th}}\) layer with \(C_{l}\) output channels, corresponding to \(\psi\), are stored as a tensor of size \(C_{l}\times C_{l-1}\times S_{G_{l}}\times K_{l}\times K_{l}\). 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 \(C_{l}S_{G_{l}}\times C_{l-1}S_{G_{l}}\times K_{l}\times K_{l}\). The input tensor to the \(l^{\text{th}}\) layer, corresponding to \(f\), is in the shape of \(B\times C_{l-1}\times S_{G_{l}}\times H_{l-1}\times W_{l-1}\), which is reshaped to \(B\times C_{l-1}S_{G_{l}}\times H_{l-1}\times W_{l-1}\) and convolved with the expanded filter.

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.

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.

After the final group convolutional layer is a group-dimension average pooling followed by two linear layers to the output dimension.

To explore the symmetry discovery of EquiNAS\({}_{D}\), we apply it to six augmentations of the MNIST dataset (LeCun et al., 1998), where each augmentation applies the group actions of each group in \(\{C_{1},C_{2},C_{4},D_{1},D_{2},D_{4}\}\) respectively. 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.

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

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

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 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:
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, the claim that "EquiNAS_D implements [G]-mixed equivariant layers throughout a network to learn appropriate approximate equivariance of each layer during training" appears to be well-stated. The evidence thoroughly demonstrates this through:

1. Detailed mathematical formulation of [G]-mixed equivariant layers
2. Experimental results showing dynamic adaptation of equivariance constraints across different layers during training (visible in Figures 3,4,10-12)
3. Implementation details confirming the concurrent optimization of architecture and weights
4. Empirical validation across multiple datasets showing the method successfully learns appropriate equivariance patterns

The claim accurately reflects the technical capabilities without overstating the results or ignoring limitations.

<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:

Differentiable Equivariance-Aware NAS (EquiNAS\({}_{D}\)) implements \([G]\)-mixed equivariant layers throughout a network to learn the appropriate approximate equivariance of each layer, in addition to their optimized weights, during training.

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:Although no other works on approximate equivariance explicitly study architectural optimization, some approaches are architectural in nature.

The main contributions of Wang et al. (2022) and Finzi et al. (2021) are similar to our proposed \([G]\)-mixed equivariant layer. Wang et al. (2022) also uses a weighted sum of filters, but uses the same group for each filter and defines the weights over the domain of group elements. Finzi et al. (2021) uses an equivariant layer in parallel to a linear layer with weighted regularization, thus only using two layers in parallel and weighting them by regularization rather than parametrization.

Romero & Lohit (2022) and van der Ouderaa et al. (2022) learn partial or soft equivariances for each layer. Neural architecture search Neural architecture search (NAS) aims to optimize both the architecture and its parameters for a given task. Liu et al. (2018) approaches this difficult bi-level optimization by creating a large super-network containing all possible elements and continuously relaxing the discrete architectural parameters to enable search by gradient descent. Other NAS approaches include evolutionary algorithms (Real et al., 2017; Lu et al., 2019; Elsken et al., 2017) and reinforcement learning (Zoph & Le, 2017), which search over discretely represented architectures.

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 \(\psi\colon G\to\mathbb{R}^{C_{l}\times C_{l-1}}\). For each output channel \(d\in[C_{l}]\), where \([C]:=\{1,\ldots,C\}\), and group element \(g\in G\), the layer's output is defined as: \[[f\star_{G}\psi]_{d}(g)=\sum_{h\in G}\sum_{c=1}^{C_{l-1}}f_{c}(h)\psi_{d,c}(g^{-1 }h). \tag{1}\]

We propose two mechanisms to enable search over subgroups: the equivariance relaxation morphism and a \([G]\)-mixed equivariant layer.

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.

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.

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.

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.

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. Thus, the final optimized architecture and trained weights are output from the single search process.

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.

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.

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.

EquiNAS\({}_{D}\) achieves better test accuracy than the other comparable methods on RotMNIST and Galaxy10.

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. 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. The EquiNAS\({}_{D}\) algorithm avoids many known problems in differentiable NAS such as the discretization gap that occurs when searching over a continuous relaxation of a discrete architectural search space (Xie et al., 2021), such as that of EquiNAS\({}_{E}\).

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.

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\({}_{D}\). We investigate how dynamic equivariance achieved by these algorithms affects the training and performance of models across multiple image classification tasks of varying complexity and assumed symmetry, demonstrating that these techniques can search for performant architectures and weights even on noisy tasks.

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).

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. The individual continues training on the task before competing against other individuals to be selected as a parent to mutate to generate the next population. 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.

The implementation of regular group convolutional layers can be viewed as a special case of our proposed equivariance relaxation morphism. With the preliminaries given in Section 4.1 and the case of \(G^{\prime}=T(2),\tilde{f}\) and \(\tilde{\psi}\) are computed such that \(\tilde{f}_{(c,s)}(g^{\prime})\coloneqq f_{c}(g^{\prime}s)\) and \(\tilde{\psi}_{(d,t),(c,s)}(g^{\prime})\coloneqq\psi_{d,c}(t^{-1}g^{\prime}s)\) for each \(g^{\prime}\in T(2)\), \(c\in[C_{l-1}]\), \(s,t\in R\), and \(d\in[C_{l}]\). Let \(S_{G}:=|R|\). The learnable parameters of the \(G_{l}\)-equivariant \(l^{\text{th}}\) layer with \(C_{l}\) output channels, corresponding to \(\psi\), are stored as a tensor of size \(C_{l}\times C_{l-1}\times S_{G_{l}}\times K_{l}\times K_{l}\). 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 \(C_{l}S_{G_{l}}\times C_{l-1}S_{G_{l}}\times K_{l}\times K_{l}\). The input tensor to the \(l^{\text{th}}\) layer, corresponding to \(f\), is in the shape of \(B\times C_{l-1}\times S_{G_{l}}\times H_{l-1}\times W_{l-1}\), which is reshaped to \(B\times C_{l-1}S_{G_{l}}\times H_{l-1}\times W_{l-1}\) and convolved with the expanded filter.

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.

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.

After the final group convolutional layer is a group-dimension average pooling followed by two linear layers to the output dimension.

To explore the symmetry discovery of EquiNAS\({}_{D}\), we apply it to six augmentations of the MNIST dataset (LeCun et al., 1998), where each augmentation applies the group actions of each group in \(\{C_{1},C_{2},C_{4},D_{1},D_{2},D_{4}\}\) respectively. 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.

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

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

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 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:
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, this claim appears to be well-stated (score of 1). The evidence thoroughly demonstrates that EquiNAS_D does implement [G]-mixed equivariant layers throughout networks to learn appropriate approximate equivariance during training. The figures (particularly Figs 3, 4, 10-12) show the dynamic learning of layer-specific equivariance weights over training steps. The mathematical formulation in Section 4.2 and experimental results validate that the method allows continuous mixing of different equivariance constraints per layer. The reviewer's comments, while noting some limitations, support that the method effectively learns and adapts equivariance constraints during training.

<score>1</score>