Stabilizing Deep Reconstruction Operators with Contractive Anchoring

Trainable denoisers with Lipschitz control have become central to convergent image reconstruction. However, training neural networks that simultaneously offer strong denoising performance and global Lipschitz guarantees is challenging. Existing approaches enforce Lipschitz control only empirically, providing no guarantees beyond the training data. In this work, we show that by exploiting the action of permutations on the image lattice, we can constrain a neural architecture that is globally nonexpansive (Lipschitz bound $\leqslant 1$). We integrate the proposed denoiser with forward imaging operators to develop a reconstruction mechanism that is provably contractive and therefore globally convergent. Experiments on standard inverse problems, such as superresolution and deblurring, demonstrate that our reconstruction performance is competitive with softly constrained baselines while providing Lipschitz guarantees.

Plug-and-play proximal gradient descent (PnP-PGD) enables flexible image reconstruction by using denoisers as implicit priors. In practice, these denoisers are often deployed outside their training domains. Existing analyses establish convergence under structural assumptions on the deployed denoiser, such as requiring it to be a proximal map or a contraction. However, they do not measure how domain mismatch affects convergence of PnP-PGD. We define this effect as \emph{proximal mismatch}: the discrepancy between a deployed denoiser $\widehat{\mathsf D}$ and a target-domain reference map $\mathsf D_\star=\operatorname{prox}_{R_\star}$ associated with the underlying regularizer $R_\star$. Under this mismatch, each denoising update becomes an inexact proximal step for the target objective. We further derive a stationarity bound that decays at a rate of $\mathcal{O}(1/K)$, with an additive term proportional to the average squared proximal mismatch. This result motivates adaptation via proximal matching rather than MSE-based adaptation alone. We study this approach with two established denoiser families: learned proximal networks and gradient-step denoisers. Experiments on Gaussian deblurring and super-resolution under substantial domain shift show that proximal matching adaptation improves reconstruction quality significantly over MSE-based adaptation, yielding the largest numerical gains in the few-shot regime.

We propose a fidelity-constrained framework that anchors the output of a black-box denoiser to its input without retraining and with little additional computation. The method linearly blends the denoised image with the input and selects the maximum blending factor that satisfies a prescribed local fidelity constraint using Peak Signal-to-Noise Ratio (PSNR) or Structural Similarity Index (SSIM). For PSNR control, a closed-form solution is obtained under a local constant-blending assumption. For SSIM control, we derive a tractable formulation based on inverse SSIM under the same assumption and solve it efficiently using iterative root finding. Experiments on DIV2K images with synthetic Gaussian noise and outputs from Real-ESRGAN and a non-local means denoiser show that the proposed anchoring strategy provides effective fidelity control while balancing denoising performance and statistical naturalness, as measured by the excess kurtosis of residual noise. In particular, SSIM-based anchoring yields more consistent behavior across noise levels than PSNR-based anchoring.