Vector–Jacobian Endpoint Geometry for Inverse Problems: Exploiting the Local Structure of Flow Predictions

Pretrained flow-matching models provide generative priors for inverse problems, but treating their clean predictions only as point estimates leaves local structure unused during measurement correction. We introduce Vector–Jacobian Endpoint Geometry (\VEG), a clean-image correction that uses both measurements and local flow structure to determine where to move, how far to move, and whether to accept the move. We call the local variation among clean endpoints compatible with the current flow state endpoint geometry. uses a clean-prediction vector–Jacobian product for direction, norm matching for proposal scale, and safeguarded line search for step selection. Our analysis characterizes how measurement evidence should shift the predicted clean image among these plausible endpoints, and shows that this shift is shaped by their local variation. Experiments on nonlinear and linear inverse problems show improved reconstruction quality over ordinary clean-prediction correction with modest additional local computation, while controlled interventions demonstrate that endpoint geometry can guide image changes that the measurement alone cannot determine.