Norm-Controlled Likelihood Guidance for Diffusion-based Inverse Solver

Diffusion-based inverse solvers approximate the posterior by combining a pretrained diffusion prior with an approximate likelihood guidance term. For Tweedie plug-in likelihood-guided solvers, which evaluate measurement likelihoods on a Tweedie posterior-mean estimate ${\hat{x}}_0(x_t)$, we identify a clear diagnostic signal in the evaluated settings: for the same image and inverse task, different noise realizations lead to different reconstructions, and larger low-noise likelihood-guidance norms correlate with worse perceptual quality across the tested hosts and runs. This makes the likelihood-score norm an actionable empirical diagnostic for this solver family. To leverage this signal, we propose Norm-Controlled Likelihood Guidance, a training-free wrapper composed of three modules that steer the sampling trajectory toward smaller likelihood norms while retaining the pretrained diffusion prior term. Theoretically, we establish a low-noise certificate showing that, under a tractable multi-modal Tweedie plug-in model, smaller likelihood norms imply smaller score approximation errors up to an explicit floor. Experimentally, NCLG improves the host solvers across all reported rows in our FFHQ and ImageNet-256 evaluations, with ablation studies and cost analyses further supporting the design and quantifying representative quality–compute trade-offs.