Lipschitz-Controlled Flow Matching for Robust Image Restoration

Sedjro Salomon Hotegni, Konstantin Sonntag, Paul-Christian Bürkner, Sebastian Peitz

Flow-matching models can produce high-quality image restorations, but their learned dynamics may amplify small input perturbations and make reconstructions unstable. We introduce LipsFlower to reduce this amplification. We derive stability bounds for Flow-from-$Y$, which initializes the ordinary differential equation from a noise-perturbed observation, as well as for observation-conditioned posterior sampling, which conditions the velocity field on the observation throughout integration. Based on this analysis, we develop LipHDiT, a Lipschitz-aware hourglass diffusion transformer that serves as the vector-field architecture of LipsFlower. LipHDiT incorporates stabilized approximate spectral normalization, bounded modulation, and convex residual mixing. We also use adversarial finite-radius gain regularization at sampled interpolation states to penalize excessive sensitivity to the state and, for conditional sampling, the observation. We test both formulations on blind face restoration, measuring adversarial robustness and the perceptual quality of the reconstructions. LipsFlower reduces perturbation amplification while keeping restoration quality competitive with the evaluated…