Fisher-Rao Residual Flow Matching for Generation from Corrupted Data

We propose Fisher-Rao Residual Flow Matching (FRRF) for learning unconditional generators of clean signals from corrupted observations and a simulatable measurement model. FRRF alternates between two steps: Fisher-Rao feedback decides what the generated population should change, and residual flow matching learns only that change. This "select, then realize" principle requires only forward simulation, avoids per-observation reconstruction, and separates statistical feedback from generator fitting. We establish geometric reduction of observation mismatch up to statistical and computational error, with clean-law recovery under inverse stability. We demonstrate numerical performance on image and scientific problems, and showcase how observation feedback and residual fitting contribute to the resulting updates.