Venue: ICLR 2023 Conference

Reviews:

final_score: rating: 6: marginally above the acceptance threshold
review_content: summary_of_the_paper: This paper studies the discrete symmetries of robotics systems. The authors present a set of results: (1) the identification of the system’s morphological symmetry Group G, (2) the characterization of how the group acts upon the system state variables and any relevant measurement living in the Euclidean space, and (3) the exploitation of data symmetries through the use of G-equivariant/G-invariant Neural Networks. However, the motivations and insights are not clear to me.
strength_and_weaknesses: Pros: 
+ The paper is well-written and easy to read.
+ The literature review is comprehensive. The presented results are well-positioned in the existing works.
+ The presented theoretical foundations for robotics systems are novel and make sense. The authors also present detailed proofs.

Cons:
- The presented results lack a clear motivation. The authors are encouraged to discuss what the presented results can contribute to. The paper says “addressing the memory
and computational complexity drawbacks of using this type of architecture.” However, no theory or experiments are give to support this. Overall, I am concerned about the impact of this paper.

clarity,_quality,_novelty_and_reproducibility: See above.
summary_of_the_review: I find the results are novel and interesting but I am concerned about the motivation and impact of the results.
review_title: null

final_score: rating: 5: marginally below the acceptance threshold
review_content: summary_of_the_paper: this paper proposes to mimic physically unrealizable symmetries such as reflections, by reconfiguration of dynamics such as left/right legs shift in bipedal. 
strength_and_weaknesses: Strength:
1. DMS is an interesting idea
2. fabulous representation of the background

Weaknesses:
As a whole this paper is good. It does contribute to the field and combines different advanced approaches. My main concerns are 
1. that the DBS itself is a tiny part of the paper, which is built on and plugged in to the existing methods.
2. DBS can not be applicable to many dynamics, which is mentioned also in "Imitating reflections of space requires symmetric rigid bodies and/or modularity"
3. DBS requires significant domain knowledge (when and how to mimic the original symmetry). there is no general principle how to calculate it in an arbitrary dynamics. it is not unique, as there can be multiple reconfigurations mimicking the original symmetry. 
4. the experiments are limited to supervised learning, even though the author method in the text uncontrollable and consolable trajectories, which alluded there would be application to control problems. 





small comments:
typos: "Kinematic three"," an angle $\theta$ unfeasible".  
clarity,_quality,_novelty_and_reproducibility: suggestion: to simplify the second sentence in Section 4, which is a 7 lines sentence. It is the main point of the paper, which could be more readable. I will reconsider my score, provided the authors address my concerns in "Weaknesses".


summary_of_the_review: as a whole it is a good paper. the idea is interesting. the novelty is limited. the applicability is limited.
review_title: null

final_score: rating: 5: marginally below the acceptance threshold
review_content: summary_of_the_paper: This study focuses on the morphological symmetries in dynamical systems; for example, a bipedal robot is symmetric to the reflection. When applying data-driven method to such systems, the function is desired to have corresponding symmetries. To this end, this study uses the method proposed in ICML,2021 to build symmetric neural networks.

strength_and_weaknesses: **Strengths**

The problem setting is clear, and the proposal is a nice application of Finzi et al., ICML, 2021.

**Weakness**

Novelty and significance appear to be limited. Several studies have already investigated the symmetries in dynamical systems.
Especially, Finzi et al., ICML, 2021, on which the proposed method is based, already examined dynamical systems symmetric to continuous groups such as $O(2)$. Discrete symmetries have also examined in the references in Finzi et al., ICML, 2021. It is not clear what main difficulties were and how the proposed method resolved.

The morphological discrete symmetries in robots have also examined in Finzi et al., "Residual Pathway Priors for Soft Equivariance Constraints", NeurIPS, 2021, which is not cited in the present manuscript.

Also, the differences from previous studies (especially, Weissenbacher et al., 2022) are still unclear. Please introduce Limitations in the Appendix in more detail.


In Section3, the definition and expression about the velocity contain several abuses of notions. $TQ$ is often used to represent a tangent bundle (the space of the pair of position and velocity). $T_qQ$ is the notion for a tangent space (the space of the velocity) at point $q\in Q$. The map $\dot g$ accepts $q$ as an argument, but if so, its domain should be $Q$, not $TQ$. Nothing can map a tangent vector $\dot q$ independently from the position $q$. The map should be the pushforward by $g$ at the position $q$ and can be denoted by (but not limited to) $g^*_q$.


clarity,_quality,_novelty_and_reproducibility: Quality: The definition and expression can be improved.

Novelty: The problem statement and the proposed method seem straightforward combinations of existing studies. It is better to what is non-trivial.

summary_of_the_review: While the problem statement and the proposed method are clear, they seem combinations of existing studies. It is better to emphasize the novelty.
review_title: null

Meta-review: metareview:_summary,_strengths_and_weaknesses: The final reviewer scores are 5 (tbqU), 5 (Yn9H), 6 (vevv), putting this paper in borderline territory. Unfortunately reviewers Yn9H and vevv have not engaged in discussions. Therefore I have carefully read the paper and all responses, leading me to conclude the following:

Reviewers agree the paper is well written and gives an excellent introduction to the relevant background, much of which will be less familiar to the ML community than to physicists and roboticists. Personally, I found the paper interesting, though somewhat challenging due to (to me) somewhat unfamiliar background material. The analysis of discrete morphological symmetries in robotics systems / dynamical systems is one of the main contributions of the paper. The experimental results show decent performance, though they don't seem to reflect a major improvement.

Reviewer vevv mentions lack of motivation as the main weakness. The updated manuscript has made some improvements in this regard. 

Reviewer Yn9H mentions:
- "DBS can not be applicable to many dynamics": as noted in the paper, many existing robots do satisfy some DBS.
- "DBS requires significant domain knowledge", "there is no general principle how to calculate it in an arbitrary dynamics". The updated paper does try to address this, though it remains true that finding DMSs requires domain knowledge and calculating by hand. This is probably more acceptable to roboticists than ML researchers.
- "the experiments are limited to supervised learning" [and don't include control]. I agree it would be nice to see such experiments.

Reviewer tbqU is mainly concerned about novelty relative to Finzi et al., ICML 2021, where a method for computing linear layers of equivariant networks for arbitrary groups is proposed. Indeed this paper does not introduce a new method for equivariant networks, though some novel tricks are introduced, e.g. for computing the basis B for sparse representations (this could be useful in other applications), though unfortunately these contributions are scattered throughout the paper and not studied in detail. 

tbqU notes "I would gladly give this paper high marks if it were at a conference dedicated to robotics.". I conclude that this paper makes modest ML/equivariance-related contributions, and additionally makes a robotics-specific contribution, both of which are relevant to ICLR.

Since this papers is cross-disciplinary and does a lot of things at once, it is a bit hard to pinpoint exactly what its key contribution is, and requires an above-average cognitive effort to read. Some methodological contributions are made that could be turned into a paper on their own, if they were unpacked / explained in more detail, analyzed, and evaluated by rigorous experiments. Instead they are mentioned as an aside, without in-depth analysis and evaluation, which diminishes their value to the research community. The main analysis of DMSs will be straightforward for roboticists or physicists familiar with Langrangians and their symmetries, but may be hard to follow for ML researchers. Finally, the contribution is not so much a learning algorithm or architecture, but rather an analysis that requires explicit calculations by hand for each robot it is to be applied to, which will not seem very appealing to most ML researchers. Some of these issues are inherent to cross-disciplinary work, and in my view should not be reasons for rejection. Other issues could be better addressed by writing a longer journal or tutorial version (or splitting the paper), and I encourage the authors to do so.

In summary: several reviewer comments have been addressed. In my view the paper topic is relevant to ICLR. However, the paper is lacking in focus, and its main contribution is a bit hard to pinpoint, and the paper may be hard to read and in some ways unsatisfying to both roboticists and machine learning researchers. Still it's an insightful analysis with some promising early results, so I encourage the authors to continue to improve the research and publish it, perhaps in a longer format, at another venue.
justification_for_why_not_higher_score: The paper lacks focus
justification_for_why_not_lower_score: n/a
Decision: Reject