GFC: Geometry-First Corruption for Generative Imitation Learning

Generative policies model expressive action distributions for robotic manipulation, but standard Gaussian corruption does not explicitly account for dependencies across the action horizon. To address this issue, we propose GFC: geometry-first corruption, which generates noisy trajectories by randomly deforming a trajectory-level Riemannian metric. The metric captures action geometry and temporal coupling, refined by cross-segment predictive responses to structured perturbations. At each forward step, a random self-adjoint field deforms the metric through a matrix exponential, and a metric-gradient flow induces trajectory motion without an additional coordinate-noise draw. This connects demonstration geometry along the action horizon with stochastic corruption along noising time. We derive path-wise bounds on trajectory motion and predictive response, and specify reverse-learning objectives and terminal-source requirements. Experiments in simulation and on real robots demonstrate improved trajectory quality and policy performance. More broadly, GFC establishes stochastic geometry as a design principle for generative modeling, allowing learned data structure to shape the corruption process itself. Videos and supplementary materials are available at https://anonymous.4open.science/r/GFC/.