Developmental Symmetry-Loss: A Free-Energy Perspective on Brain-Inspired Invariance Learning

We propose Symmetry-Loss, a brain-inspired algorithmic principle that enforces invariance and equivariance through a differentiable constraint derived from environmental symmetries. The framework models learning as the iterative refinement of an effective symmetry group, paralleling developmental processes in which cortical representations align with the world's structure. By minimizing structural surprise, i.e. deviations from symmetry consistency, Symmetry-Loss operationalizes a Free-Energy--like objective for representation learning. This formulation bridges predictive-coding and group-theoretic perspectives, showing how efficient, stable, and compositional representations can emerge from symmetry-based self-organization. The result is a general computational mechanism linking developmental learning in the brain with principled representation learning in artificial systems.