pith. sign in

arxiv: 2605.22639 · v1 · pith:XF2I4NVWnew · submitted 2026-05-21 · 💻 cs.RO

Symmetries Here and There, Combined Everywhere: Cross-space Symmetry Compositions in Robotics

Loizos Hadjiloizou , Rodrigo P\'erez-Dattari , No\'emie Jaquier This is my paper

Pith reviewed 2026-05-22 05:09 UTC · model grok-4.3

classification 💻 cs.RO
keywords symmetry compositionrobot policy learningequivarianceforward kinematicsconfiguration spacetask spacedual-arm robotgeneralization
0
0 comments X

The pith

Cross-space symmetry compositions let robot policies respect multiple symmetries at once by moving them between configuration and task spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to combine symmetries that appear in different parts of robotic systems. Many robots have symmetries in their mechanical structure and in the tasks they perform, but previous methods handled them one at a time. By using the properties of the forward kinematics map, symmetries can be transferred between configuration space and task space to create a single representation that respects all of them together. Tests on a dual-arm robot in simulation and reality indicate that this joint approach leads to policies that generalize more effectively to new conditions.

Core claim

We introduce cross-space symmetry compositions, a framework for learning robot policies that are jointly equivariant to multiple symmetries across configuration and task spaces. Leveraging the differential-geometric structure of the forward kinematics map, we both descend symmetries from configuration to task space and lift symmetries from task to configuration space, enabling their composition within a unified representation space. We validate our framework on simulated and real-world experiments on a dual-arm robot, demonstrating that jointly leveraging multiple symmetries yields improved generalization.

What carries the argument

Cross-space symmetry compositions that descend symmetries from configuration space to task space and lift them in the reverse direction using the forward kinematics map to produce a unified equivariant representation.

If this is right

  • Jointly equivariant policies generalize better than single-symmetry baselines on dual-arm tasks.
  • The unified representation space supports composition of symmetries without separate handling or loss of consistency.
  • Improved performance appears in both simulated environments and physical robot experiments.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same descent-and-lift mechanism could be tested on mobile bases or multi-fingered hands that possess their own symmetry sets.
  • Automatic selection of which symmetries to compose might reduce manual design effort for new robot platforms.
  • The approach may lower sample complexity for policy learning by encoding multiple invariances in one model.

Load-bearing premise

The differential-geometric structure of the forward kinematics map permits clean descent of symmetries from configuration space to task space and clean lifting in the reverse direction without loss of information or introduction of inconsistencies.

What would settle it

A direct comparison experiment in which policies trained with cross-space symmetry compositions show no generalization improvement over policies trained with only single symmetries on the same dual-arm robot tasks.

Figures

Figures reproduced from arXiv: 2605.22639 by Loizos Hadjiloizou, No\'emie Jaquier, Rodrigo P\'erez-Dattari.

Figure 1
Figure 1. Figure 1: Example of manipulation task with several symmetries. We [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Lifted symmetry. The infinitesimal generators of [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: RMSE of policies πGR trained with varying augmentation intervals (5 ◦ ( ), 10◦ ( ), 15◦ ( ), 30◦ ( ), 45◦ ( ), 60◦ ( ), 75◦ ( ), 90◦ ( )) and evaluated on θ-rotated letters [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: Policies π ( ), πGR ( ), πGRT ( ), πGMRT ( ) evaluated on test trajectories ( ) obtained as symmetric transformations of the demonstrations ( ). Trajectories start and end are depicted as · and ×. Diverging trajectories are truncated and indicated by → and are excluded from the visualization of the next composition [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 7
Figure 7. Figure 7: Real-world letter-drawing experiment. The robot draws [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Real-world pan-grasping experiment. The robot grasps the [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
read the original abstract

Robots exhibit a rich variety of symmetries arising from their mechanical structure and the properties of their tasks. Although many robotics problems exhibit several symmetries simultaneously, existing approaches typically treat them in isolation, failing to exploit their combined potential. This paper introduces cross-space symmetry compositions, a framework for learning robot policies that are jointly equivariant to multiple symmetries across configuration and task spaces. Leveraging the differential-geometric structure of the forward kinematics map, we both descend symmetries from configuration to task space and lift symmetries from task to configuration space, enabling their composition within a unified representation space. We validate our framework on simulated and real-world experiments on a dual-arm robot, demonstrating that jointly leveraging multiple symmetries yields improved generalization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces cross-space symmetry compositions, a framework for learning robot policies that are jointly equivariant to multiple symmetries across configuration space Q and task space X. It leverages the differential-geometric structure of the forward kinematics map f: Q → X to descend symmetries from configuration to task space and lift them in the reverse direction, enabling their composition within a unified representation. The approach is validated on simulated and real-world experiments with a dual-arm robot, claiming that jointly leveraging multiple symmetries yields improved generalization.

Significance. If the descent and lifting operations produce a consistent group action on the joint representation without kernel or cokernel obstructions, the framework could meaningfully advance equivariant policy learning in robotics by exploiting combined geometric symmetries rather than treating them in isolation, potentially improving sample efficiency and generalization in multi-arm manipulation tasks.

major comments (2)
  1. [Framework construction (post-abstract)] The central construction (described in the framework section following the abstract) assumes that the descended and lifted actions satisfy f(g·q) = ρ(g)·f(q) and the reverse lift without loss of information. However, since f is typically a submersion only away from singularities, this may fail to yield a well-defined group homomorphism near those loci or when arms share task-space coordinates, undermining the joint equivariance claim.
  2. [Experimental validation] The experimental validation on the dual-arm robot (results section) reports improved generalization but provides no explicit analysis or ablation addressing consistency of the composed action at or near kinematic singularities, which is load-bearing for the cross-space composition claim.
minor comments (2)
  1. [Notation and definitions] Notation for the group actions ρ and the representation space could be clarified with an explicit diagram or commutative diagram showing descent and lift.
  2. [Abstract and framework] The abstract claims 'parameter-free' descent via forward kinematics; if any auxiliary choices (e.g., choice of connection or local trivialization) are involved, they should be stated explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the scope and limitations of the cross-space symmetry composition framework. We address each major comment below, proposing targeted revisions where appropriate to strengthen the presentation of the mathematical assumptions and experimental validation.

read point-by-point responses

  1. Referee: The central construction (described in the framework section following the abstract) assumes that the descended and lifted actions satisfy f(g·q) = ρ(g)·f(q) and the reverse lift without loss of information. However, since f is typically a submersion only away from singularities, this may fail to yield a well-defined group homomorphism near those loci or when arms share task-space coordinates, undermining the joint equivariance claim.

    Authors: We agree that the forward kinematics map f: Q → X is a submersion only on the regular set and that singularities (or shared task-space coordinates in dual-arm settings) can obstruct a globally well-defined group homomorphism. The manuscript formulates the descent and lift operations locally in open sets where f is a submersion, following standard differential-geometric practice in robotics. To make this explicit, we will revise the framework section to state the domain of validity, note that joint equivariance holds locally away from singularities, and briefly discuss practical handling via singularity avoidance or local charts. This clarification does not change the core construction but addresses the referee's concern directly. revision: yes

  2. Referee: The experimental validation on the dual-arm robot (results section) reports improved generalization but provides no explicit analysis or ablation addressing consistency of the composed action at or near kinematic singularities, which is load-bearing for the cross-space composition claim.

    Authors: The referee correctly notes the absence of targeted analysis near singularities. Our dual-arm experiments were conducted in task regions deliberately chosen to remain away from kinematic singularities, which is common for demonstrating generalization gains in manipulation. We will add a short discussion subsection in the results to acknowledge this scope, include a qualitative note on expected behavior near singularities based on the local formulation, and, if space permits, report a small additional simulation ablation. This revision will make the experimental claims more precise without requiring entirely new hardware trials. revision: partial

Circularity Check

0 steps flagged

No significant circularity; framework derives from standard differential geometry of forward kinematics

full rationale

The paper grounds its cross-space symmetry compositions in the established differential-geometric properties of the forward kinematics map f: Q → X, using these to descend symmetries from configuration space and lift them from task space for composition. No load-bearing step reduces by construction to a fitted parameter, self-referential definition, or self-citation chain; the central claim of joint equivariance follows from the submersion properties and group action compatibility as an independent geometric fact. Experimental validation on dual-arm robots is presented separately and does not feed back into the derivation. This is a self-contained construction against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the geometric properties of the forward kinematics map as a domain assumption; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption The forward kinematics map possesses a differential-geometric structure that supports both descent of symmetries from configuration to task space and lifting in the reverse direction.
    This assumption is required to enable the cross-space composition of symmetries.

pith-pipeline@v0.9.0 · 5655 in / 1110 out tokens · 38660 ms · 2026-05-22T05:09:21.003138+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

  1. [1]

    A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning,

    S. E. Otto, N. Zolman, J. N. Kutz, and S. L. Brunton, “A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning,”Journal of Machine Learning Research, vol. 26, no. 248, pp. 1–83, 2025

    work page 2025

  2. [2]

    J. E. Marsden and T. S. Ratiu,Introduction to Mechanics and Sym- metry. Springer, 1999, vol. 2

    work page 1999

  3. [3]

    Symmetry Discovery with Deep Learning,

    K. Desai, B. Nachman, and J. Thaler, “Symmetry Discovery with Deep Learning,”Physical Review D, vol. 105, no. 9, p. 096031, 2022

    work page 2022

  4. [4]

    Computing reduced equations for robotic systems with constraints and symmetries,

    J. Ostrowski, “Computing reduced equations for robotic systems with constraints and symmetries,”IEEE Transactions on Robotics and Automation, vol. 15, no. 1, pp. 111–123, 1999

    work page 1999

  5. [5]

    Symmetry reduction of optimal control systems and principal connections,

    T. Ohsawa, “Symmetry reduction of optimal control systems and principal connections,”SIAM Journal on Control and Optimization, vol. 51, no. 1, pp. 96–120, 2013

    work page 2013

  6. [6]

    Reduced Euler-Lagrange Equations of Floating-Base Robots: Computation, Properties, & Applications,

    H. Mishra, G. Garofalo, A. M. Giordano, M. De Stefano, C. Ott, and A. Kugi, “Reduced Euler-Lagrange Equations of Floating-Base Robots: Computation, Properties, & Applications,”IEEE Trans. on Robotics, vol. 39, no. 2, pp. 1439–1457, 2023

    work page 2023

  7. [7]

    Leverag- ing symmetry to accelerate learning of trajectory tracking controllers for free-flying robotic systems,

    J. Welde, N. Rao, P. Kunapuli, D. Jayaraman, and V . Kumar, “Leverag- ing symmetry to accelerate learning of trajectory tracking controllers for free-flying robotic systems,” inIEEE Intl. Conf. on Robotics and Automation (ICRA), 2025, pp. 15 121–15 127

    work page 2025

  8. [8]

    Morphological Sym- metries in Robotics,

    D. O. Apraez, G. Turrisi, V . Kostic, M. Martin, A. Agudo, F. Moreno- Noguer, M. Pontil, C. Semini, and C. Mastalli, “Morphological Sym- metries in Robotics,”Intl. Journal of Robotics Research, vol. 44, no. 10-11, pp. 1743–1766, 2025

    work page 2025

  9. [9]

    Morphologically symmetric reinforcement learning for ambidextrous bimanual manipulation,

    Z. Li, Y . Jin, D. O. Apraez, C. Semini, P. Liu, and G. Chalvatzaki, “Morphologically symmetric reinforcement learning for ambidextrous bimanual manipulation,” inConference on Robot Learning (CoRL), 2025

    work page 2025

  10. [10]

    Sampling-based motion planning with discrete configuration-space symmetries,

    T. Cohn and R. Tedrake, “Sampling-based motion planning with discrete configuration-space symmetries,” inIEEE/RSJ Intl. Conf. on Intelligent Robots and Systems (IROS), 2025, pp. 13 119–13 126

    work page 2025

  11. [11]

    Transporter Networks: Rearranging the Visual World for Robotic Manipulation,

    A. Zeng, P. Florence, J. Tompson, S. Welker, J. Chien, M. Attarian, T. Armstrong, I. Krasin, D. Duong, V . Sindhwaniet al., “Transporter Networks: Rearranging the Visual World for Robotic Manipulation,” inConference on Robot Learning (CoRL), 2021

    work page 2021

  12. [12]

    Sample efficient grasp learning using equivariant models,

    X. Zhu, D. Wang, O. Biza, G. Su, R. Walters, and R. Platt, “Sample efficient grasp learning using equivariant models,” inRobotics: Science and Systems (R:SS), 2022

    work page 2022

  13. [13]

    SO(2)-equivariant reinforcement learning,

    D. Wang, R. Walters, and R. Platt, “SO(2)-equivariant reinforcement learning,” inIntl. Conf. on Learning Representations (ICLR), 2022

    work page 2022

  14. [14]

    Equiv- ariant Descriptor Fields: SE(3)-Equivariant Energy-Based Models for End-to-End Visual Robotic Manipulation Learning,

    R. Hyunwoo, L. Hong-in, L. Jeong-Hoon, and C. Jongeun, “Equiv- ariant Descriptor Fields: SE(3)-Equivariant Energy-Based Models for End-to-End Visual Robotic Manipulation Learning,” inIntl. Conf. on Learning Representations (ICLR), 2023

    work page 2023

  15. [15]

    Leveraging Symmetries in Pick and Place,

    H. Huang, D. Wang, A. Tangri, R. Walters, and R. Platt, “Leveraging Symmetries in Pick and Place,”Intl. Journal of Robotics Research, vol. 43, no. 4, pp. 550–571, 2024

    work page 2024

  16. [16]

    Task-Induced Symmetry and Reduction with Application to Needle Steering,

    V . Kallem, D. E. Chang, and N. J. Cowan, “Task-Induced Symmetry and Reduction with Application to Needle Steering,”IEEE Transac- tions on Automatic Control, vol. 55, no. 3, pp. 664–673, 2010

    work page 2010

  17. [17]

    Regularizing Towards Soft Equivariance Under Mixed Symmetries,

    H. Kim, H. Lee, H. Yang, and J. Lee, “Regularizing Towards Soft Equivariance Under Mixed Symmetries,” inIntl. Conf. on Machine Learning (ICML), 2023

    work page 2023

  18. [18]

    Latent Mixture of Symme- tries for Sample-Efficient Dynamic Learning,

    H. Li, C. Xiao, M. Guo, and Y . Weng, “Latent Mixture of Symme- tries for Sample-Efficient Dynamic Learning,” inNeural Information Processing Systems (NeurIPS), 2025

    work page 2025

  19. [19]

    Redundant Robotic Chains on Riemannian Submersions,

    C. Altafini, “Redundant Robotic Chains on Riemannian Submersions,” IEEE Transactions on Robotics and Automation, vol. 20, no. 2, pp. 335–340, 2004

    work page 2004

  20. [20]

    B. C. Hall,Lie Groups, Lie Algebras, and Representations, 2nd ed. Springer, 2015, vol. 2

    work page 2015

  21. [21]

    Reinforcement Learning with Euclidean Data Augmentation for State-Based Continuous Control,

    J. Luo, D. Chen, and Q. Zhang, “Reinforcement Learning with Euclidean Data Augmentation for State-Based Continuous Control,” inNeural Information Processing Systems (NeurIPS), 2024

    work page 2024

  22. [22]

    Approximately Equivariant Networks for Imperfectly Symmetric Dynamics,

    R. Wang, R. Walters, and R. Yu, “Approximately Equivariant Networks for Imperfectly Symmetric Dynamics,” inIntl. Conf. on Machine Learning (ICML), 2022

    work page 2022

  23. [23]

    A practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups,

    M. Finzi, M. Welling, and A. G. Wilson, “A practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups,” inIntl. Conf. on Machine Learning (ICML), 2021

    work page 2021

  24. [24]

    J. M. Lee,Introduction to Smooth Manifolds. Springer, 2012, vol. 218

    work page 2012

  25. [25]

    Rainbow Robotics,

    Rainbow Robotics, “Rainbow Robotics,” https://www. rainbow-robotics.com/, 2026

    work page 2026

  26. [26]

    Stable Motion Primitives via Imitation and Contrastive Learning,

    R. P ´erez-Dattari and J. Kober, “Stable Motion Primitives via Imitation and Contrastive Learning,”IEEE Transactions on Robotics, vol. 39, no. 5, pp. 3909–3928, 2023

    work page 2023