Geometric Prediction Error: Free Energy on Lie Groups
Zenodo (CERN European Organization for Nuclear Research) March 16, 2026 Walter Henrique Alves da Silva
Prediction error, the core quantity of the Free Energy Principle, is reformulated using compact Lie groups with bi-invariant metrics. A system composes sequential inputs into a running product on a Lie group, and prediction error becomes the geodesic distance from this product to the group identity. This geometric prediction error propagates exactly—a perturbation of any size at any position produces the same magnitude in the error, with no distortion regardless of sequence length—and is uniformly sensitive to every input. The reformulation eliminates the need for a generative model, replacing variational inference with a single group multiplication per timestep.