Compositional Symmetry as Compression: Lie Pseudogroup Structure in Algorithmic Agents
This work provides a theoretical framework for improving the efficiency and interpretability of algorithmic agents in AI and machine learning, though it appears incremental by building on existing symmetry and predictive coding concepts.
The paper tackles the problem of understanding how agents can efficiently compress sensory streams by proposing that compositional symmetry in Lie pseudogroups serves as a structural prior, leading to constraints that reduce the dimensionality of agent dynamics and align with compositional factorization in deep models.
In the algorithmic (Kolmogorov) view, agents are programs that track and compress sensory streams using generative programs. We propose a framework where the relevant structural prior is simplicity (Solomonoff) understood as \emph{compositional symmetry}: natural streams are well described by (local) actions of finite-parameter Lie pseudogroups on geometrically and topologically complex low-dimensional configuration manifolds (latent spaces). Modeling the agent as a generic neural dynamical system coupled to such streams, we show that accurate world-tracking imposes (i) \emph{structural constraints} -- equivariance of the agent's constitutive equations and readouts -- and (ii) \emph{dynamical constraints}: under static inputs, symmetry induces conserved quantities (Noether-style labels) in the agent dynamics and confines trajectories to reduced invariant manifolds; under slow drift, these manifolds move but remain low-dimensional. This yields a hierarchy of reduced manifolds aligned with the compositional factorization of the pseudogroup, providing a geometric account of the ``blessing of compositionality'' in deep models. We connect these ideas to the Spencer formalism for Lie pseudogroups and formulate a symmetry-based, self-contained version of predictive coding in which higher layers receive only \emph{coarse-grained residual transformations} (prediction-error coordinates) along symmetry directions unresolved at lower layers.