GSDSplit+ is introduced, a splitting method built upon PnPSplit+, in which the generic denoising block is replaced by a relaxed Gradient Step Denoiser, which retains explicit Poisson fidelity updates.
Abstract
Plug and Play methods combine classical variational models with learned denoisers and have achieved strong results in imaging inverse problems. Their convergence has been widely studied for Gaussian data, whereas Poisson models require additional care because of the nonquadratic Kullback-Leibler fidelity. This work introduces GSDSplit+ , a splitting method built upon PnPSplit+, in which the generic denoising block is replaced by a relaxed Gradient Step Denoiser. The resulting method retains explicit Poisson fidelity updates. The denoiser is defined through the gradient of a learned convex potential parameterized by an Input Convex Neural Network. Its architecture and training promote both blind denoising capability and a smoothness condition sufficient for firm nonexpansiveness. Empirical smoothness estimates are used to select a relaxation parameter compatible with the convergence assumptions of PnPSplit+. Numerical experiments provide evidence that the trained denoiser operates in an FNE-compatible regime on the considered validation data. GSDSplit+ also yields stable reconstructions outside the empirically supported range and shows substantially lower sensitivity to the ADMM parameter, while remaining effective for different blur operators and Poisson noise levels.
A model-based deep denoising framework that integrates three tightly coupled components, and proves that every limit point of the L-ADMM sequence is a stationary point of the nonconvex objective, and that the iterate sequence itself converges globally.
Plug-and-Play (PnP) methods have emerged as a highly effective paradigm for solving imaging inverse problems by replacing traditional proximity operators of regularization terms with highly expressive deep denoisers. While empirically successful, establishing rigorous convergence guarantees for PnP algorithms remains a major challenge. Existing provable approaches based on the Gradient-Step (GS) denoiser suffer from theoretical and practical limitations, such as restrictive bounds on the regularization parameter, rigid step-size rules, and the inability to handle nonconvex data-fidelity terms. In this paper, we introduce PnP-IPA (Plug-and-Play Inexact Proximal Algorithm), a novel optimization scheme that overcomes these bottlenecks. We propose a new splitting strategy that evaluates the proximal operator of the scaled implicit regularizer inexactly. To enable adaptive step-size selection without exact objective evaluations, we design a novel surrogate merit function that successfully drives an Armijo-like backtracking line-search. Relying on the Kurdyka-Lojasiewicz property, we establish global convergence to a stationary point of the nonconvex objective without imposing any assumption on the regularization parameter. Extensive numerical experiments on image deblurring under both Gaussian and Cauchy noise demonstrate the practical advantages of PnP-IPA. By effectively lifting previous theoretical constraints, our method allows for optimal parameter tuning, yielding state-of-the-art restoration quality and robust convergence even in nonconvex regimes.
Cristiano Parenti, S. Bonettini, M. Prato· 0 citations
Sampling from high-dimensional posterior distributions is a central challenge in Bayesian inference and noisy inverse problems. Standard first-order Langevin-based methods often suffer from slow convergence and sensitivity to step-size hyperparameters, particularly in annealed score-based inverse imaging pipelines. We propose Adaptive Momentum Langevin Dynamics (AMLD), a practical stochastic correction kernel that introduces a momentum variable into the annealed posterior sampling framework and equips it with an annealing-aware momentum retention schedule. The method is fully compatible with the SNIPS framework and retains its coordinate-wise adaptive step structure, acting as a lightweight drop-in replacement for the conventional first-order Langevin correction step. Extensive experiments on three representative image inverse problems—Gaussian deblurring, inpainting, and 4× super-resolution—demonstrate that AMLD consistently achieves strong PSNR and LPIPS performance, with competitive FID in most settings, compared to three state-of-the-art baselines (DDRM, DPS, SNIPS) under both nearly noiseless and noisy measurement conditions, while reaching target reconstruction quality using fewer sampling iterations. The proposed momentum-based sampler provides empirically improved exploration and robustness across evolving posterior landscapes, offering a practical and computationally efficient alternative to first-order annealed Langevin samplers in high-dimensional Bayesian inverse problems.
This paper proposes a novel learned spherical alternating direction method of multipliers (LSADMMs) for effective Rician noise removal under spherical constraints, and incorporates a noise-level estimation prefix that provides adaptive guidance across noise levels.
Jun Shi, Zhifang Liu, Chunlin Wu et al.· Inverse Problems· 0 citations
This study presents a novel Deep Proximal Gradient Descent framework for ill-posed problems by employing a tailored second-order differentiable Input-Convex Neural Networks (ICNNs) as a learned regularizer. A key contribution is the design of convex residual mapping, which preserves the convexity of the regularized objective, thereby enhancing the interpretability of the deep network without sacrificing its expressive power. Based on this framework, we develop two types of algorithms. For linear problems, the ICNN-based regularizer is embedded into the standard proximal gradient structure. For nonlinear problems, we introduce an innovative formulation that employs the learned residual to guide gradient descent, while using the traditional data misfit as a proximal regularizer to avoid network-dominated spurious solutions. Building on this iterative scheme, we establish groundbreaking convergence results for both algorithms, complete with rigorous proofs. Extensive numerical experiments, particularly on real low-dose Computed Tomography data, validate the superior imaging quality and high computational efficiency of our algorithms.
T. Ye, Guangyu Gao, Yang Li et al.· Inverse Problems· 0 citations
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.
Guixian Xu, Jinglai Li, Junqi Tang· 0 citations
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