Symmetry-based disentangled representation learning leverages the group structure of environment transformations to uncover the latent factors of variation. Prior approaches to symmetry-based disentanglement have required strong prior knowledge of the symmetry group's structure, or restrictive assumptions about the subgroup properties. In this work, we remove these constraints by proposing a method whereby an embodied agent autonomously discovers the group structure of its action space through unsupervised interaction with the environment. We prove the identifiability of the true symmetry group decomposition under minimal assumptions, and derive two algorithms: one for discovering the group decomposition from interaction data, and another for learning Linear Symmetry-Based Disentangled (LSBD) representations without assuming specific subgroup properties. Our method is validated on three environments exhibiting different group decompositions, where it outperforms existing LSBD approaches.
Figures & tables
Figure 1
Figure 3: Graphical model
Figure 4: Two isomorphic group actions satisfying Assumption 2 .
Figure 5: Masking used to build disentangled action matrices
Figure 6: Median of disentanglement metrics. Vertical lines indicates the 25th to 75th percentile
Figure 7: Median of long-term prediction error, the shaded area indicates the 25th to 75th percentile
Figure 8: Median of disentanglement metrics on MPI3D
Appendix figures & tables9 assets
Supplementary material from the paper’s appendix.
Appendix
Symbol
Meaning
Symbol
Meaning
W
World space
Z1⊕Z2
Direct sum of vector spaces
X
Observation space
A↪B
Injective map from A to B
X
Observation random variable
\vvvertA\vvvert
Spectral norm of matrix A
Z
Latent space
Z/nZ
Cyclic group of order n
Z
Latent random variable
Sn
Symmetric group on n elements
G
Action Group
GL(V)
General linear group on V
Appendix
Table 2: Table of symbols
Figure 9
Figure 9: Matrix representation of ρ(g)zk=ρ(gk)zk∈Zk , here the two first dimensions corresponds to Z1 and the remaining two to Z2 .
Figure 10: Metric of interest as a function of the prediction error
Figure 11: Disentanglement according to action entanglement
Figure 12: Decoding mixed latent representations of 3DShapes images
Figure 13: Decoding rotated latent representations of MPI3D image
Figure 14: dG matrices averaged over 5 seeds, normalized so that the clustering threshold is 1.
Symmetries play a central role in reducing the complexity of reinforcement learning problems, yet most existing approaches rely on fixed group actions or predefined state abstractions. Classical reinforcement learning algorithms typically assume a globally structured Markov decision process with uniformly applicable actions and transitions, an assumption that limits their ability to exploit modularity and local, context-dependent regularities present in many realistic environments. We propose a reinforcement learning framework using groupoids to capture local, state-dependent symmetries and support the dy- namic discovery of equivalence structures during interaction. The agent maintains orbit representatives together with transporters that map raw states to canonical forms, enabling learning and decision-making to be performed in a symmetry-reduced space while preserving local distinctions. Empirical results demonstrate that the proposed groupoid-based approach improves sample efficiency and convergence in dense and large-scale environments exhibiting strong partial symmetries, yielding substantial performance gains over standard Q-learning. These findings show that dynamically exploiting local symmetry provides a practical and mathematically principled route to scalable and generalisable reinforcement learning.
Ben Opperman, Eduardo Alonso, Esther Mondragón
CitAI, AI Research Centre, City St George’s, University of London, London, United Kingdom
Representation learning seeks meaningful sensory representations without supervision and can model aspects of human development. Although many neural networks empirically learn useful features, a principled account of what makes a representation "good" remains elusive. We study unsupervised categorization of transformations between pairs of inputs under algebraic constraints. Classical disentanglement favors mutually independent factors and fails when factors are coupled. Our prior Galois-theoretic approach decomposes a group via normal subgroups by learning a product of two transformations with one factor constrained to a normal subgroup, covering both commutative and non-commutative cases. That method, however, relied on auxiliary assumptions (e.g., motion and isometry restrictions) not required by decomposition theory, and ablations did not separate theory-based from auxiliary effects. We propose parameter division for a single transformation: we split its parameter into components, impose homomorphism constraints mapping the full transformation to one component, and identify the normal subgroup as the set of transformations when that component is fixed to the identity. This formulation drops the previous auxiliary assumptions and applies more broadly. We evaluate on image pairs involving rotation, translation, and scale; ablations show that group-decomposition constraints drive appropriate categorization.
Graduate School of Information Science and Technology, The University of Tokyo, Tokyo, Japan. · Next Generation Artificial Intelligence Research Center (AI Center), The University of Tokyo, Tokyo, Japan.
Symmetry-aware robot policy learning requires coupled observation-action transformations. For pixel observations, even simple physical symmetries can induce complex, viewpoint-dependent image transformations that are difficult to specify by hand. Existing symmetry-aware methods often rely on representations with predefined transformations, requiring preprocessing tailored to the sensing setup and task. We introduce MIRROR, a framework that learns how a known symmetry group operates jointly on observations and actions in partially observable Markov decision processes (POMDPs) from offline interaction data. MIRROR couples these transformations through adversarial tuple matching over one-step transition tuples, while explicit group-consistency regularization penalizes violations of the group-operation axioms. On four Robosuite tabletop manipulation tasks, MIRROR achieves the lowest aggregate error on three of four symmetry-quality metrics, including a 61% reduction in compatibility error (which measures consistency under transformation composition) relative to an adapted LaLiGAN baseline. Using the learned transformations in frame-averaged behavior cloning (BC) yields higher mean success rates than vanilla BC on all four tasks, increasing average success from 80.48% to 85.12%, with per-task gains up to 9.66 percentage points. We also repeat symmetry discovery and policy training in simulation with heightmap and segmentation-mask observations, then deploy the resulting policy zero-shot on a real Kinova GEN3 arm. Project page: https://pomdp-mirror.github.io.