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This paper investigates the impact of domain mismatch on the convergence of Plug-and-Play Proximal Gradient Descent (PnP-PGD) for image reconstruction, specifically defining the concept of proximal mismatch between deployed denoisers and target-domain reference maps. The authors derive a stationarity bound that highlights how the average squared proximal mismatch affects convergence rates, leading to a novel adaptation strategy based on proximal matching. Experimental results demonstrate that this adaptation significantly enhances reconstruction quality in scenarios with substantial domain shifts, particularly in few-shot settings, outperforming traditional MSE-based adaptation methods.
Proximal matching adaptation can dramatically improve image reconstruction quality in the presence of domain mismatch, especially when data is scarce.
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.