Thank you for your thoughtful comments. We are glad the reviewer found that our paper presents an innovative method, with impressive performance and improved computational efficiency compared to previous approaches.

Below, we address the reviewer’s comments:

• Thermodynamic and mechanical stability evaluation: This is an interesting point! The issue is that evaluation of thermodynamic or mechanical stability would require very expensive DFT simulations that cannot realistically be run on hundreds of samples for evaluation.
• Efficiency on larger datasets: We agree that our method has potential to scale to larger datasets more efficiently than other methods. To the best of our knowledge, the only larger publicly available dataset of inorganic crystal structures is the MPTS-52 dataset. No other generative modeling method has been able to scale to this dataset. As a direction for future work, we are investigating if our model could be trained on this dataset. This would be a significant breakthrough, but requires more computational resources than what we currently have access to.
• Clarification of binary symmetry encoding: We have clarified this in the updated manuscript to take your feedback into account. We have added statistics to show how data fragmentation is an issue for methods that do not use a shared representation over symmetries. Specifically, for the MP-20 dataset it turns out that 113 space groups out of 169 in the training set have fewer than a hundred samples associated with them, which is a very small amount of data.
• Generation process preserving symmetry and clarification of figure 3: This is a useful comment! We have added a short explanation as to why the diffusion and denoising process symmetry. Essentially, it is because they only operate on the asymmetric unit, which is the minimal representation given a symmetry. In an effort to improve clarity, we have updated figure 3 with additional annotations as you suggested. The goal of the updates is to better highlight some elements specific to the training and sampling from the diffusion model that might have been confusing in the previous version.

We thank you again for your review. Note that all the changes in the revised manuscript have been highlighted in blue for clarity. Please let us know if we have addressed your concerns appropriately or if we can answer any other question.

Thank you for your comments on our paper and your supportive review. We are happy to see your positive assessment of the writing of the paper and the feedback regarding the usefulness of our crystal symmetry and more broadly crystal structure representation.

Below, we address some of your comments more concretely:

• 1-4 (applications on inorganic and organic crystal): This is a useful comment. We have updated our draft to clarify that our main focus relates to inorganic, solid-state materials. Given the crucial importance of symmetry for inorganic materials, we thought this would be a useful start. As you correctly point out, symmetry-informed representations can be useful for a wider range of applications including molecular crystals. While many parts of our method would be directly applicable to those cases, we believe future work is needed to fully develop and validate such a method. The reviewer is correct in pointing out that our method can be applied to multi-component crystals. We have corrected this, we meant to point out polycrystalline materials as an avenue of future work since they cannot be described by a single unit cell. We have also reframed the conclusion to emphasize the advantages of SymmCD related to generation and focus on the limitations and some of the important practicalities beyond generative methods, such as materials synthesis of complex materials.
• Q1 (10 SGs variant): Great suggestion! We initially included the 10SGs version for the sake of being complete. However, the goal with this variant of the model is to show that sampling from the most frequent space groups will result in overall more stable materials, as shown in the SUN metrics in table 3 and structural validity in table 2. Since it is pretty obvious that the effect of conditioning on specific space groups will be to reduce diversity and alignment with the dataset, we have applied the reviewers suggestion of removing this variant from tables 1.
• Q2 (evaluation): We agree that the margins of difference are not large enough to make strong claims about the superiority of SymmCD over other methods when it comes to evaluating stability. Unfortunately it is quite slow to perform structural relaxations. But, in the light of your feedback, we are currently working on performing relaxations on the full 10,000 samples and will update the paper when they are completed.
• Q3 (conventional unit cell model): The comparison is between our proposed SymmCD and DiffCSP with the same architecture and hyperparameters. Note that SymmCD leverages the asymmetric unit and learns the site symmetries while DiffCSP uses the entire conventional unit cell which leads to poor computational efficiency metrics. We previously highlighted this in Section 5.4. However, in the revised version, we have also cited DiffCSP.
• Q4 (anonymized code): Yes! We have uploaded an anonymized version of the repository here: https://anonymous.4open.science/r/SymmCD-596C/. We have also included a link in the paper.

We thank you again for your review. Note that all the changes in the revised manuscript have been highlighted in blue for clarity. Please let us know if we have addressed your concerns appropriately or if we can answer any other question.

Thank you for your comments on our paper. We appreciate your positive feedback on the clarity of our writing, given the complexity of the topic, and on the soundness of the method. Below, we address some of your comments more concretely:

• Introduction: We appreciate your suggestion of tying in our work to a broader context of structure generation. We see our work as a continuation of a more general line of research on generative modeling for science, but applied to the domain of materials design. We focused our approach initially on inorganic materials given the crucial importance of symmetry for that class of materials. You correctly point out that symmetry is relevant for a broader diversity of systems, including molecules (organic materials) and graph generation which often use the discrete diffusion framework. SymmCD’s approach leveraging an asymmetric unit is indeed more applicable to molecules and molecular crystals where such building blocks are commonly available. We have updated the related works section to make this connection clearer. We also agree that there are useful intersections between our method and other design cases from a representation learning perspective and have updated the conclusion to reflect that. We believe that conditioning on symmetry can lead to powerful representations that can be applied to generative methods for a wide range of use cases.
• Reproducibility: This is a valid point! We have uploaded an anonymized version of the repository here to address this: https://anonymous.4open.science/r/SymmCD-596C/. We have also included a link in the paper.
• Experimentation (performance gain): While SymmCD performs on par with other methods on metrics such as stability, validity, and distributions of properties, our main claim is that SymmCD performs significantly better than prior works at generating crystals with realistic, diverse symmetries, as seen in Figure 4 and Table 1. The inability of prior works to address this issue has been noted by others [1] and represents a significant barrier to useful crystal generation, as the vast majority of real-world crystals have complex symmetries. Many properties of crystals (such as piezoelectricity and optical activity) are determined by symmetry, so when searching for practical crystals, a generative model should be able to generate crystals with desired symmetry. The computational speedup is also important in practical settings, as high-throughput materials discovery requires the generation of tens of thousands of crystals.
• Experimentation (validation): We use the same standardized train/validation/test set split as the methods we compare against, for fair comparison. Unfortunately, it would have been too costly to run cross-validation for each baseline method, and this is not standard practice.
• Experimentation (hyperparameters): We performed two hyperparameter sweeps: we first checked all values of $\lambda_k$, and $\lambda_S$, keeping $\lambda_X$ fixed at 1, and selected the loss coefficients that lead to the highest structural validity. Next, we performed a random sweep of other architecture parameters, running 150 different hyperparameter combinations and choosing a model that had high performance on structural validity, compositional validity, and $d_E$. We’ve added details in appendix E.2, and included the hyperparameter sweep configuration file in the repository. For all baselines, we use their reported hyperparameters.

We thank you again for your useful feedback. Note that all the changes in the revised manuscript have been highlighted in blue for clarity. Please let us know if we have addressed your concerns appropriately or if we can answer any other questions.

[1] Anthony K Cheetham and Ram Seshadri. Artificial intelligence driving materials discovery? perspective on the article: Scaling deep learning for materials discovery. Chemistry of Materials, 36 (8):3490–3495, 2024.

Dear authors and fellow reviewers,

I appreciate your rebuttal comments to my and other reviews. Especially, providing the code is an important transparent step.

However, after reading the review of Reviewer Tsh1 and your reply I am now even more concerned about the experimentation, reproducibility and robustness of your presented results.

I have a few follow up questions and concerns and would like to hear your and other reviewer comments.

1. Hyperparameter selection. I am very concerned about the fact that the authors do not carry out any hyperparameter search for the baseline but do a hyperparameter search for their own method. I have compared the baseline performances reported in the DiffCSP++ paper by Jie et al. and the FlowMM paper by Miller et al. to the ones the authors reported in Table 2 of this manuscript and find quite a difference in Property distribution metrics reported. The metrics of the baselines here appear to be mostly lower. I cannot evaluate how these metrics relate to the template statistics reported in Table 1 but since there is no crossvalidation of the presented work and hyperparameter search for the baselines I am concerned that the baselines may be systematically disadvantaged in Table 1 as well.

2. You state in your rebuttal that "SymmCD performs significantly better than prior works at generating crystals with realistic, diverse symmetries,". How do the authors come to the conclusion that their model performs significantly better without any statistical test or even crossvalidation?

3. The authors stated that crossvalidation is too costly. What are the cost exactly? Considering the computational cost that the authors themselves describe in Table 4 it appears quite reasonable to crossvalidate in my opinion, especially in light of the reproducibility questions.

4. The other papers e.g. FLowMM and DiffCSP++ report results on other datasets, e.g. Perov-5 and MPTS-52, Carbon-24 . Is there a reason why other datasets are not experimented with here? It appears to be the standard practice in the field. I am particularly curious since these datasets are part of the uploaded anonymous GitHub repo from the authors. Showing comprehensive results on multiple datasets would also be an option to strengthen the confidence in the empirical results.

5. Model selection. In your rebuttal you state "Next, we performed a random sweep of other architecture parameters, running 150 different hyperparameter combinations and choosing a model that had high performance on structural validity, compositional validity, and d_E" What do the authors mean by "a model"? In my opinion it would be good practice to define a specific model selection rule and choose all models in crossvalidation and hyperparameter search space for all models including baselines.

I would be curious to hear what the authors and other reviewers think about my concerns.