Homomorphism Autoencoder -- Learning Group Structured Representations from Observed Transitions
Hamza Keurti, Hsiao-Ru Pan, Michel Besserve, Benjamin F. Grewe, Bernhard Schölkopf
TL;DR
The paper introduces the Homomorphism Autoencoder (HAE), a self-supervised architecture that learns group-structured latent representations from observed state-action transitions without prior knowledge of the underlying group. By enforcing equivariance between environment transformations and latent encodings through a learned ρ = \exp ∘ φ, and using a two-step latent prediction plus reconstruction objective, HAEs recover symmetry-based representations and disentangle latent factors aligned with group decompositions. The framework is validated across multiple experiments, demonstrating toroidal latent structures, disentangled subspaces, Lie-algebra-based latent traversals, and informative long-horizon rollouts for groups such as SO(2)×SO(2)×SO(2) and SO(3). This approach advances interventional world modeling by linking actions to latent dynamics in a group-theoretic, unsupervised manner, with potential implications for neuroscience-inspired representation learning and robust model-based planning.
Abstract
How can agents learn internal models that veridically represent interactions with the real world is a largely open question. As machine learning is moving towards representations containing not just observational but also interventional knowledge, we study this problem using tools from representation learning and group theory. We propose methods enabling an agent acting upon the world to learn internal representations of sensory information that are consistent with actions that modify it. We use an autoencoder equipped with a group representation acting on its latent space, trained using an equivariance-derived loss in order to enforce a suitable homomorphism property on the group representation. In contrast to existing work, our approach does not require prior knowledge of the group and does not restrict the set of actions the agent can perform. We motivate our method theoretically, and show empirically that it can learn a group representation of the actions, thereby capturing the structure of the set of transformations applied to the environment. We further show that this allows agents to predict the effect of sequences of future actions with improved accuracy.