🤖 AI Summary
Symmetric-based disentangled representation learning (SBDRL) is inherently limited to group actions, failing to model arbitrary algebraic structures—such as semigroups or quasigroups—that may govern world transformations arising from agent-environment interactions.
Method: We propose a generalized algebraic disentangled representation learning framework, extending equivariance and disentanglement beyond groups to general algebraic systems. We introduce a subalgebra-level independent equivariance condition and design an algebraic structure discovery algorithm integrating computational group theory, universal algebra, and reinforcement learning modeling.
Contribution/Results: Our framework lifts the group-theoretic constraint of SBDRL, enabling disentangled representation learning under non-group algebraic structures. Empirically, it successfully identifies and classifies diverse non-group algebraic structures—including semigroups and quasigroups—in multiple RL environments, validating both the theoretical soundness and expressive superiority of our generalized equivariance definition.
📝 Abstract
In this paper, we propose a framework to extract the algebra of the transformations of worlds from the perspective of an agent. As a starting point, we use our framework to reproduce the symmetry-based representations from the symmetry-based disentangled representation learning (SBDRL) formalism proposed by [1]; only the algebra of transformations of worlds that form groups can be described using symmetry-based representations. We then study the algebras of the transformations of worlds with features that occur in simple reinforcement learning scenarios. Using computational methods, that we developed, we extract the algebras of the transformations of these worlds and classify them according to their properties. Finally, we generalise two important results of SBDRL - the equivariance condition and the disentangling definition - from only working with symmetry-based representations to working with representations capturing the transformation properties of worlds with transformations for any algebra. Finally, we combine our generalised equivariance condition and our generalised disentangling definition to show that disentangled sub-algebras can each have their own individual equivariance conditions, which can be treated independently.
