{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:28:46Z","timestamp":1787326126184,"version":"build-2736575974"},"reference-count":80,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["12001286"],"award-info":[{"award-number":["12001286"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002858","name":"China Postdoctoral Science Foundation","doi-asserted-by":"publisher","award":["2022M711672"],"award-info":[{"award-number":["2022M711672"]}],"id":[{"id":"10.13039\/501100002858","id-type":"DOI","asserted-by":"publisher"}]},{"name":"NSFC-RGC","award":["NSFC\/RGC N_CUHK 415\/19"],"award-info":[{"award-number":["NSFC\/RGC N_CUHK 415\/19"]}]},{"DOI":"10.13039\/501100010428","name":"Innovation and Technology Fund","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100010428","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Hong Kong Research Grants Council","award":["14300219"],"award-info":[{"award-number":["14300219"]}]},{"name":"Hong Kong Research Grants Council","award":["14302920"],"award-info":[{"award-number":["14302920"]}]},{"name":"Hong Kong Research Grants Council","award":["14301121"],"award-info":[{"award-number":["14301121"]}]},{"DOI":"10.13039\/501100004853","name":"Chinese University of Hong Kong","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100004853","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2024,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>This paper investigates the convergence properties and applications of the three-operator splitting method, also known as the Davis\u2013Yin splitting (DYS) method, integrated with extrapolation and plug-and-play (PnP) denoiser within a nonconvex framework. We first propose an extrapolated DYS method to effectively solve a class of structural nonconvex optimization problems that involve minimizing the sum of three possibly nonconvex functions. Our approach provides an algorithmic framework that encompasses both extrapolated forward\u2013backward splitting and extrapolated Douglas\u2013Rachford splitting methods. To establish the convergence of the proposed method, we rigorously analyze its behavior based on the Kurdyka\u2013\u0141ojasiewicz property, subject to some tight parameter conditions. Moreover, we introduce two extrapolated PnP-DYS methods with convergence guarantee, where the traditional regularization step is replaced by a gradient step\u2013based denoiser. This denoiser is designed using a differentiable neural network and can be reformulated as the proximal operator of a specific nonconvex functional. We conduct extensive experiments on image deblurring and image superresolution problems, where our numerical results showcase the advantage of the extrapolation strategy and the superior performance of the learning-based model that incorporates the PnP denoiser in terms of achieving high-quality recovery images.<\/jats:p>","DOI":"10.1137\/23m1611166","type":"journal-article","created":{"date-parts":[[2024,6,13]],"date-time":"2024-06-13T04:01:58Z","timestamp":1718251318000},"page":"1145-1181","source":"Crossref","is-referenced-by-count":14,"title":["Extrapolated Plug-and-Play Three-Operator Splitting Methods for Nonconvex Optimization with Applications to Image Restoration"],"prefix":"10.1137","volume":"17","author":[{"given":"Zhongming","family":"Wu","sequence":"first","affiliation":[{"name":"School of Management Science and Engineering, Nanjing University of Information Science and Technology, Nanjing 210044 China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9347-3151","authenticated-orcid":true,"given":"Chaoyan","family":"Huang","sequence":"additional","affiliation":[{"name":"Co-first author. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tieyong","family":"Zeng","sequence":"additional","affiliation":[{"name":"Corresponding author. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,6,13]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1137\/19M1264783"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1100.0449"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-011-0484-9"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1137\/130910294"},{"key":"ref5","unstructured":"A. Barakat and P. Bianchi, Convergence rates of a momentum algorithm with bounded adaptive step size for nonconvex optimization, in Asian Conference on Machine Learning, PMLR, 2020, pp. 225\u2013240."},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/080716542"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1137\/20M1326775"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-013-0701-9"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1007\/s13675-015-0045-8"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1137\/17M1122451"},{"key":"ref11","first-page":"5977","volume":"22","author":"Castera C.","year":"2021","journal-title":"J. Mach. Learn. Res."},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1137\/140980910"},{"key":"ref13","first-page":"18152","volume":"34","author":"Cohen R.","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2021.3069677"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1137\/20M1379344"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1007\/s11228-017-0421-z"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1137\/18M1226361"},{"key":"ref18","first-page":"1048","volume":"33","author":"Dong J.","year":"2020","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2021.3075092"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1137\/080725891"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1007\/s10898-018-0660-z"},{"key":"ref22","doi-asserted-by":"publisher","DOI":"10.1137\/16M1078604"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1007\/s40305-021-00368-3"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2021.09.004"},{"key":"ref25","doi-asserted-by":"crossref","unstructured":"S. Hurault, A. Chambolle, A. Leclaire, and N. Papadakis, Convergent Plug-and-Play with Proximal Denoiser and Unconstrained Regularization Parameter, preprint, arXiv:2311.01216 [math.OC], 2023.","DOI":"10.21203\/rs.3.rs-3462535\/v1"},{"key":"ref26","doi-asserted-by":"crossref","unstructured":"S. Hurault, U. Kamilov, A. Leclaire, and N. Papadakis, Convergent Bregman Plug-and-Play Image Restoration for Poisson Inverse Problems, preprint, arXiv:2306.03466 [eess.IV], 2023.","DOI":"10.52202\/075280-1187"},{"key":"ref27","unstructured":"S. Hurault, A. Leclaire, and N. Papadakis, Gradient step denoiser for convergent plug-and-play, in International Conference on Learning Representations (ICLR\u201922), 2022."},{"key":"ref28","unstructured":"S. Hurault, A. Leclaire, and N. Papadakis, Proximal denoiser for convergent plug-and-play optimization with nonconvex regularization, in International Conference on Machine Learning, PMLR, 2022, pp. 9483\u20139505."},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1561\/2200000058"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2021.3136623"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-8134-0"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-017-9909-6"},{"key":"ref33","unstructured":"H. Le, N. Gillis, and P. Patrinos, Inertial block proximal methods for non-convex non-smooth optimization, in International Conference on Machine Learning, PMLR, 2020, pp. 5671\u20135681."},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-015-0963-5"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1137\/22M152462X"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-019-01564-1"},{"key":"ref37","volume":"29","author":"Liang J.","year":"2016","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1137\/16M106340X"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788719000570"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1137\/21M143772X"},{"key":"ref41","first-page":"5921","volume":"34","author":"Liu J.","year":"2021","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-019-01477-z"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-014-0523-2"},{"key":"ref44","series-title":"Appl. Optim. 87","volume-title":"Introductory Lectures on Convex Optimization: A Basic Course","author":"Nesterov Y.","year":"2003"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1137\/130942954"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1109\/LSP.2017.2710233"},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1137\/20M1387961"},{"key":"ref48","first-page":"1","volume":"24","author":"Phan D. N.","year":"2023","journal-title":"J. Mach. Learn. Res."},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1137\/16M1064064"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(64)90137-5"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1137\/120872802"},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2018.2880326"},{"key":"ref53","series-title":"Grundlehren Math. Wiss. 317","volume-title":"Variational Analysis","author":"Rockafellar R. T.","year":"2009"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(92)90242-F"},{"key":"ref55","unstructured":"E. Ryu, J. Liu, S. Wang, X. Chen, Z. Wang, and W. Yin, Plug-and-play methods provably converge with properly trained denoisers, Int. Conf. Mach. Learn. (2019), pp. 5546\u20135557."},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1137\/19M1304854"},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-022-02061-8"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1007\/s11263-010-0357-3"},{"key":"ref59","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2016.2599778"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2019.2893568"},{"key":"ref61","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2021.3094062"},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-022-01958-w"},{"key":"ref63","doi-asserted-by":"publisher","DOI":"10.1137\/18M1163993"},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1137\/16M1080240"},{"key":"ref65","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-022-00366-y"},{"key":"ref66","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2018.2875569"},{"key":"ref67","doi-asserted-by":"crossref","unstructured":"S. V. Venkatakrishnan, C. A. Bouman, and B. Wohlberg, Plug-and-play priors for model based reconstruction, in 2013 IEEE Global Conference on Signal and Information Processing, IEEE, 2013, pp. 945\u2013948.","DOI":"10.1109\/GlobalSIP.2013.6737048"},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.1137\/110844805"},{"key":"ref69","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2023.03.014"},{"key":"ref70","first-page":"699","volume":"23","author":"Wei K.","year":"2022","journal-title":"J. Mach. Learn. Res."},{"key":"ref71","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2023.3339786"},{"key":"ref72","doi-asserted-by":"publisher","DOI":"10.1007\/s10898-020-00943-7"},{"key":"ref73","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-019-00073-1"},{"key":"ref74","doi-asserted-by":"publisher","DOI":"10.1137\/090777761"},{"key":"ref75","doi-asserted-by":"publisher","DOI":"10.1137\/140952363"},{"key":"ref76","unstructured":"A. Yurtsever, V. Mangalick, and S. Sra, Three operator splitting with a nonconvex loss function, in International Conference on Machine Learning, PMLR, 2021, pp. 12267\u201312277."},{"key":"ref77","unstructured":"J. Zeng, T. T.K. Lau, S. Lin, and Y. Yao, Global convergence of block coordinate descent in deep learning, in International Conference on Machine Learning, PMLR, 2019, pp. 7313\u20137323."},{"key":"ref78","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2021.3088914"},{"key":"ref79","doi-asserted-by":"crossref","unstructured":"K. Zhang, L. Van Gool, and R. Timofte, Deep unfolding network for image super-resolution, in IEEE Conference on Computer Vision and Pattern Recognition, 2020, pp. 3217\u20133226.","DOI":"10.1109\/CVPR42600.2020.00328"},{"key":"ref80","doi-asserted-by":"crossref","unstructured":"K. Zhang, W. Zuo, S. Gu, and L. Zhang, Learning deep CNN denoiser prior for image restoration, in IEEE Conference on Computer Vision and Pattern Recognition, 2017, pp. 3929\u20133938.","DOI":"10.1109\/CVPR.2017.300"}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/23M1611166","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:27:13Z","timestamp":1787322433000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23M1611166"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,13]]},"references-count":80,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,6,30]]}},"alternative-id":["10.1137\/23M1611166"],"URL":"https:\/\/doi.org\/10.1137\/23m1611166","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,13]]}}}