# Fisher-Rao Metric

This work models iterative generative-model training as a closed-loop stochastic process on the probability simplex and analyzes its dynamics under the Fisher-Rao metric instead of the Euclidean metric, deriving contraction and invariance bounds that remain meaningful as dimension grows and giving a human-to-synthetic data ratio threshold that guarantees convergence to a Fisher-Rao ball, concluding that the effective required human-data ratio is higher than previously implied.

This work models iterative generative-model training as a closed-loop stochastic process on the probability simplex and analyzes its dynamics under the Fisher-Rao metric instead of the Euclidean metric, deriving contraction and invariance bounds that remain meaningful as dimension grows and giving a human-to-synthetic data ratio threshold that guarantees convergence to a Fisher-Rao ball, concluding that the effective required human-data ratio is higher than previously implied.