# Compositional Symmetry as Compression: Lie Pseudogroup Structure in Algorithmic Agents

arXiv: https://arxiv.org/abs/2510.10586 · HTML: https://mlsanity.com/paper/2510.10586/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 31 references that are in the map.
- First posted on arXiv 12 Oct 2025.

## Builds on (most cited first, 6 of 6)

- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Theoretical Issues in Deep Networks: Approximation, Optimization and Generalization](https://mlsanity.com/paper/1908.09375/) (2019)
- [LieGG: Studying Learned Lie Group Generators](https://mlsanity.com/paper/2210.04345/) (2022)
- [Hardness of Learning Neural Networks under the Manifold Hypothesis](https://mlsanity.com/paper/2406.01461/) (2024)
- [Symmetry Discovery for Different Data Types](https://mlsanity.com/paper/2410.09841/) (2024)
- [Models, networks and algorithmic complexity](https://mlsanity.com/paper/1612.05627/) (2016)

## Abstract (arXiv)

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 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) structural constraints -- equivariance of the agent's constitutive equations and readouts -- and (ii) 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 coarse-grained residual transformations (prediction-error coordinates) along symmetry directions unresolved at lower layers.