Where Priors Leave Fingerprints: Identifying Learned Regularizers from Reconstruction Response

Many learned priors for inverse problems reconstruct equally well, so the reconstruction alone reveals little about which prior produced it. Model fingerprints read the outputs of a model, and the classical analysis of how a solver reacts to its data assumes a known regularizer. In this paper, we ask what the behavior of a solver reveals about its learned prior beyond the reconstruction, and when it suffices to distinguish two priors. We study the response of a solver, namely how its reconstruction changes when the measurement is randomly perturbed. For solvers that combine a least-squares data fit with a learned regularizer, we prove that, when the operator and the perturbation level are known, the covariance of random re-solves determines the Jacobian of the solver in closed form, which identifies the curvature of the regularizer on the subspace that the solver can reach. Guided by this result, we introduce the response fingerprint, built from the centered energies of the response in frequency bands, which applies to any differentiable solver. Across denoising, deblurring, inpainting, super-resolution, MRI and CT, it separates learned priors that are hard to tell apart from their outputs, and it transfers without retraining to external priors, which played no role in its design. It also provides a label-free reliability indicator that flags the priors it separates poorly.