[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85853-en":3,"doc-seo-85853-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},85853,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","PnP-IPA: A Provably Convergent Plug-and-Play Inexact Proximal Algorithm for Nonconvex Imaging Problems","PnP-IPA (Plug-and-Play Inexact Proximal Algorithm) addresses nonconvex imaging inverse problems solved via Plug-and-Play methods that use deep denoisers in place of traditional proximal operators. The work targets shortcomings of existing provable PnP approaches, including restrictive regularization assumptions, rigid step-size rules, and limited ability to treat nonconvex data fidelities. A new inexact proximal splitting and an Armijo-like line-search based on a surrogate merit function enable adaptive step sizes. Using Kurdyka–Łojasiewicz analysis, the method guarantees global convergence to a stationary point without assuming constraints on the regularization parameter, supported by image deblurring experiments under Gaussian and Cauchy noise.","arXiv :2607 . 10223v1 [math .NA] 11 Jul 2026  \nPnP-IPA: A Provably Convergent Plug-and-Play Inexact Proximal Algorithm for Nonconvex Imaging Problems  \nCristiano Parenti 1 ,∗, Silvia Bonettini 1 and Marco Prato 1  \n1 Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università di Modena e Reggio Emilia, Via Campi 213/b, 41125 Modena, Italy  \n∗ Author to whom any correspondence should be addressed.  \nE-mail: [cristiano.parenti@unimore.it](cristiano.parenti@unimore.it)  \nKeywords: Nonconvex optimization, inverse problems, Plug-and-play, inexact methods, line–search  \nAbstract  \nPlug-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.  \nExisting 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–Łojasiewicz 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.  \n1 Introduction  \nThe restoration of high-quality images from corrupted or incomplete measurements is a fundamental task in modern computational imaging, encompassing applications from medical reconstruction to microscopic and astronomical imaging [4] . Mathematically, the observation process is typically modeled as a linear or nonlinear forward process subject to noise:  \ny = N (A(x)),  \nwhere x ∈ Rn is the unknown underlying image, y ∈ Rm represents the noisy observation as a realization of the noise model, A : Rn → Rm is the forward operator (e.g. , a blurring kernel, subsampled Fourier transform, or Radon transform), and N denotes the noise model.  \nSince the operator A is often ill-conditioned or possesses a nontrivial null space, recovering x from y is an ill-posed inverse problem. To overcome this, the classical framework relies on  \nBayesian inference. By Bayes’ theorem, the posterior probability of the image given the observation is:  \nP (y|x)P (x)  \nP (x|y) = ∝ P(y|x)P (x) .  \nP (y)  \nThe most common approach to estimate x is to find the Maximum A Posteriori (MAP) estimator, which maximizes P (x|y), or equivalently, minimizes its negative logarithm. This yields the composite optimization problem:  \nmin F(x) ≡ fdata (x) + λR(x), (1)  \nx∈Rn  \nwhere fdata (x) = −log P(y|x) ensures data fidelity (e.g., the least-squares ~~1~~2 ∥Ax − y∥2 under Gaussian noise), and R(x) = −log P(x) is a regularization term encoding prior knowledge about the statistical distribution of images we wish to recover. The parameter λ > 0 controls the trade-off between fidelity to the measurements and the prior. Historically, analytical priors were widely used to address this problem, with milestones such as Tikhonov regularization [18] and the Total Variation (TV) model [36] .  \nWhen the regularizer R is convex but","cbCaijkmGKMZuIuV","https://ap.wps.com/l/cbCaijkmGKMZuIuV","pdf",2872930,5,1,30,"English","en",105,"# Introduction\n## Imaging inverse problems and MAP formulation\n## Proximal gradient and denoising interpretation\n## Plug-and-Play and the proposed contribution","[{\"question\":\"What problem does PnP-IPA target?\",\"answer\":\"It targets imaging inverse problems where the objective is nonconvex, solved within the Plug-and-Play framework that replaces regularization proximal operators with deep denoisers.\"},{\"question\":\"What limitations in existing provable PnP methods does the paper address?\",\"answer\":\"It addresses restrictive bounds on the regularization parameter, inflexible step-size rules from gradient-step denoisers, and difficulty handling nonconvex data-fidelity terms.\"},{\"question\":\"How does PnP-IPA achieve adaptive step sizes without exact objective evaluations?\",\"answer\":\"It introduces a surrogate merit function that drives an Armijo-like backtracking line search while using inexact evaluation of the scaled implicit regularizer’s proximal operator.\"}]",1784206713,76,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"pnp-ipa-a-provably-convergent-plug-and-play-inexact-proximal-algorithm-for-nonconvex-imaging-problems","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/pnp-ipa-a-provably-convergent-plug-and-play-inexact-proximal-algorithm-for-nonconvex-imaging-problems/85853/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What problem does PnP-IPA target?","Question",{"text":76,"@type":77},"It targets imaging inverse problems where the objective is nonconvex, solved within the Plug-and-Play framework that replaces regularization proximal operators with deep denoisers.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What limitations in existing provable PnP methods does the paper address?",{"text":81,"@type":77},"It addresses restrictive bounds on the regularization parameter, inflexible step-size rules from gradient-step denoisers, and difficulty handling nonconvex data-fidelity terms.",{"name":83,"@type":74,"acceptedAnswer":84},"How does PnP-IPA achieve adaptive step sizes without exact objective evaluations?",{"text":85,"@type":77},"It introduces a surrogate merit function that drives an Armijo-like backtracking line search while using inexact evaluation of the scaled implicit regularizer’s proximal 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