Recovering a signal from indirect, noisy measurements is especially hard when the forward operator smooths the underlying object: the modes that are hardest to recover are exactly those the data reveal least. Diffusion priors are now a standard tool for such ill-posed inverse problems, but existing theory has not said clearly when a learned score remains reliable under a smoothing operator.
The paper studies posterior diffusion for linear inverse problems on Hilbert scales, a graded family of function spaces indexed by regularity. Under a smoothing condition on the forward map, a source condition on the true signal, and weak-norm control of score error, the authors prove non-asymptotic recovery bounds in (H^s) norms. The bounds separate the contributions of observation noise, score approximation, and reverse-time discretization, and they identify a recoverable regularity threshold determined jointly by operator smoothing, truth regularity, score regularity, and the posterior correction.
Above that threshold, the inverse geometry magnifies score error and uniform stability is lost. The authors also show that a posterior correction adapted to the forward operator improves stability relative to isotropic guidance in the regularity ranges where the smoothing action matters most. The result is a regularity-sensitive stability theory for posterior diffusion, clarifying when learned score error stays controlled after propagation through the reverse dynamics.