{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,8]],"date-time":"2026-10-08T16:54:07Z","timestamp":1791478447344,"version":"4.3.4"},"reference-count":78,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","funder":[{"DOI":"10.13039\/100017899","name":"Masason Foundation","doi-asserted-by":"crossref","id":[{"id":"10.13039\/100017899","id-type":"DOI","asserted-by":"crossref"}]},{"name":"GSK.ai"},{"DOI":"10.13039\/501100000275","name":"Leverhulme Trust","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000275","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000288","name":"Royal Society","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000288","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/V029428\/1"],"award-info":[{"award-number":["EP\/V029428\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/S026045\/1"],"award-info":[{"award-number":["EP\/S026045\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/T003553\/1"],"award-info":[{"award-number":["EP\/T003553\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/N014588\/1"],"award-info":[{"award-number":["EP\/N014588\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/T017961\/1"],"award-info":[{"award-number":["EP\/T017961\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010269","name":"Wellcome Trust","doi-asserted-by":"publisher","award":["215733\/Z\/19\/Z"],"award-info":[{"award-number":["215733\/Z\/19\/Z"]}],"id":[{"id":"10.13039\/100010269","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010269","name":"Wellcome Trust","doi-asserted-by":"publisher","award":["221633\/Z\/20\/Z"],"award-info":[{"award-number":["221633\/Z\/20\/Z"]}],"id":[{"id":"10.13039\/100010269","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100010665","name":"H2020 Marie Sk\u0142odowska-Curie Actions","doi-asserted-by":"publisher","award":["777826 NoMADS"],"award-info":[{"award-number":["777826 NoMADS"]}],"id":[{"id":"10.13039\/100010665","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Cantab Capital Institute for the Mathematics of Information"},{"DOI":"10.13039\/100012338","name":"Alan Turing Institute","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100012338","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>Plug-and-Play (PnP) methods are a class of efficient iterative methods that aim to combine data fidelity terms and deep denoisers using classical optimization algorithms, such as ISTA or ADMM, with applications in inverse problems and imaging. Provable PnP methods are a subclass of PnP methods with convergence guarantees, such as fixed point convergence or convergence to critical points of some energy function. Many existing provable PnP methods impose heavy restrictions on the denoiser or fidelity function, such as nonexpansiveness or strict convexity, respectively. In this work, we propose a novel algorithmic approach incorporating quasi-Newton steps into a provable PnP framework based on proximal denoisers, resulting in greatly accelerated convergence while retaining light assumptions on the denoiser. By characterizing the denoiser as the proximal operator of a weakly convex function, we show that the fixed points of the proposed quasi-Newton PnP algorithm are critical points of a weakly convex function. Numerical experiments on image deblurring and super-resolution demonstrate 2\u20138x faster convergence as compared to other provable PnP methods with similar reconstruction quality.<\/jats:p>","DOI":"10.1137\/23m157185x","type":"journal-article","created":{"date-parts":[[2024,4,2]],"date-time":"2024-04-02T04:35:09Z","timestamp":1712032509000},"page":"785-819","source":"Crossref","is-referenced-by-count":26,"title":["Provably Convergent Plug-and-Play Quasi-Newton Methods"],"prefix":"10.1137","volume":"17","author":[{"given":"Hong Ye","family":"Tan","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7957-8758","authenticated-orcid":true,"given":"Subhadip","family":"Mukherjee","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK."},{"name":"Department of Computer Science, University of Bath, Bath, BA2 7AY, UK, and Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology, 721302 Kharagpur, India."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4996-6079","authenticated-orcid":true,"given":"Junqi","family":"Tang","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK."},{"name":"School of Mathematics, University of Birmingham, Birmingham, B15 2TT, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Carola-Bibiane","family":"Sch\u00f6nlieb","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, CB3 0WA, UK."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,4,2]]},"reference":[{"key":"ref1","unstructured":"B. Amos, L. Xu, and J. Z. Kolter, Input convex neural networks, in Proceedings of the International Conference on Machine Learning, PMLR, 2017, pp. 146\u2013155."},{"key":"ref2","unstructured":"S. Arjomand Bigdeli, M. Zwicker, P. Favaro, and M. Jin, Deep mean-shift priors for image restoration, in Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS, 2017."},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1966.16.1"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1100.0449"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-011-0484-9"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-023-01960-6"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9467-7"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974997"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1137\/080716542"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-013-0701-9"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.5201\/ipol.2011.bcm_nlm"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1137\/17M1122451"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1137\/140954362"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-015-0746-4"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2016.2629286"},{"key":"ref16","unstructured":"R. Cohen, Y. Blau, D. Freedman, and E. Rivlin, It has potential: Gradient-driven denoisers for convergent solutions to inverse problems, in Proceedings of the 35th International Conference on Neural Information Processing Systems, NIPS, 2021, pp. 18152\u201318164."},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1137\/20M1337168"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2007.901238"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20042"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1974-0343581-1"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1956-0084194-4"},{"key":"ref22","unstructured":"M. Fazlyab, A. Robey, H. Hassani, M. Morari, and G. Pappas, Efficient and accurate estimation of Lipschitz constants for deep neural networks, in Proceedings of the 33rd International Conference on Neural Information Processing Systems, NIPS, 2019, pp.\u00a011427\u201311438."},{"key":"ref23","doi-asserted-by":"crossref","unstructured":"W. Gan, S. Shoushtari, Y. Hu, J. Liu, H. An, and U. S. Kamilov, Block coordinate plug-and-play methods for blind inverse problems, in Proceedings of the 37th International Conference on Neural Information Processing Systems, NIPS, 2023.","DOI":"10.52202\/075280-3438"},{"key":"ref24","unstructured":"T. Goldstein, C. Studer, and R. Baraniuk, A Field Guide to Forward-Backward Splitting with a FASTA Implementation, preprint, arXiv:1411.3406, 2014."},{"key":"ref25","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2023.3306100"},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1137\/23M1565243"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-020-00951-y"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-6765-7"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2021.09.004"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1137\/140955483"},{"key":"ref31","doi-asserted-by":"crossref","unstructured":"S. Hurault, A. Chambolle, A. Leclaire, and N. Papadakis, A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser, in Proceedings of the International Conference on Scale Space and Variational Methods in Computer Vision, Springer, 2023, pp. 379\u2013392.","DOI":"10.1007\/978-3-031-31975-4_29"},{"key":"ref32","unstructured":"S. Hurault, A. Leclaire, and N. Papadakis, Gradient step denoiser for convergent plug-and-play, in Proceedings of the International Conference on Learning Representations, ICLR, 2022."},{"key":"ref33","unstructured":"S. Hurault, A. Leclaire, and N. Papadakis, Proximal denoiser for convergent plug-and-play optimization with nonconvex regularization, in Proceedings of the International Conference on Machine Learning, PMLR, 2022, pp. 9483\u20139505."},{"key":"ref34","unstructured":"Q. Jin, A. Koppel, K. Rajawat, and A. Mokhtari, Sharpened quasi-Newton methods: Faster superlinear rate and larger local convergence neighborhood, in Proceedings of the International Conference on Machine Learning, PMLR, 2022, pp. 10228\u201310250."},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-022-01887-4"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1007\/b138659"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2022.3199595"},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1109\/LSP.2017.2763583"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1137\/130921428"},{"key":"ref40","doi-asserted-by":"crossref","unstructured":"A. Levin, Y. Weiss, F. Durand, and W. T. Freeman, Understanding and evaluating blind deconvolution algorithms, in Proceedings of the 2009 IEEE Conference on Computer Vision and Pattern Recognition, 2009, pp. 1964\u20131971.","DOI":"10.1109\/CVPR.2009.5206815"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(00)00540-9"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1137\/S1052623499354242"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1007\/BF01589116"},{"key":"ref44","unstructured":"S. Lunz, O. \u00d6ktem, and C.B. Sch\u00f6nlieb, Adversarial regularizers in inverse problems, in Proceedings of the 32nd International Conference on Neural Information Processing Systems, NIPS, 2018, pp.\u00a08516\u20138525."},{"key":"ref45","doi-asserted-by":"crossref","unstructured":"D. Martin, C. Fowlkes, D. Tal, and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, in Proceedings of the Eighth IEEE International Conference on Computer Vision, ICCV, Vol. 2, 2001, pp. 416\u2013423.","DOI":"10.1109\/ICCV.2001.937655"},{"key":"ref46","unstructured":"T. Miyato, T. Kataoka, M. Koyama, and Y. Yoshida, Spectral normalization for generative adversarial networks, in Proceedings of the International Conference on Learning Representations, ICLR, 2018."},{"key":"ref47","first-page":"3151","volume":"16","author":"Mokhtari A.","year":"2015","journal-title":"J. Mach. Learn. Res."},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.24033\/bsmf.1625"},{"key":"ref49","unstructured":"S. Mukherjee, S. Dittmer, Z. Shumaylov, S. Lunz, O. \u00d6ktem, and C.B. Sch\u00f6nlieb, Learned Convex Regularizers for Inverse Problems, manuscript, 2020."},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2022.3207451"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2021.3066053"},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1137\/22M1504573"},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2017.2777407"},{"key":"ref54","volume-title":"Numerical Optimization","author":"Nocedal J.","year":"2006","edition":"2"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1137\/17M1156678"},{"key":"ref56","unstructured":"A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, PyTorch: An imperative style, high-performance deep learning library, in Proceedings of the 33rd International Conference on Neural Information Processing Systems, NIPS, H. Wallach, H. Larochelle, A. Beygelzimer, F. d\u2019Alch\u00e9-Buc, E. Fox, and R. Garnett, eds. Curran Associates, Red Hook, NY, 2019, pp. 8024\u20138035, http:\/\/papers.neurips.cc\/paper\/9015-pytorch-an-imperative-style-high-performance-deep-learning-library.pdf."},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.1137\/20M1387961"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2018.2880326"},{"key":"ref59","series-title":"Princeton Math. Ser. 28","volume-title":"Convex Analysis","author":"Rockafellar R.","year":"1972"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.1137\/20M1320651"},{"key":"ref61","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-021-01622-5"},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1088\/0954-898X_5_4_006"},{"key":"ref63","doi-asserted-by":"publisher","DOI":"10.1016\/0167-2789(92)90242-F"},{"key":"ref64","unstructured":"E. Ryu, J. Liu, S. Wang, X. Chen, Z. Wang, and W. Yin, Plug-and-play methods provably converge with properly trained denoisers, in Proceedings of the International Conference on Machine Learning, PMLR, 2019, pp. 5546\u20135557."},{"key":"ref65","unstructured":"N. N. Schraudolph, J. Yu, and S. G\u00fcnter, A stochastic quasi-Newton method for online convex optimization, in Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, PMLR, 2007, pp. 436\u2013443."},{"key":"ref66","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2016.2599778"},{"key":"ref67","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-017-9912-y"},{"key":"ref68","unstructured":"Y. Sun, J. Liu, Y. Sun, B. Wohlberg, and U. S. Kamilov, Async-RED: A provably convergent asynchronous block parallel stochastic method using deep denoising priors, in Proceedings of the International Conference on Learning Representations, ICLR, 2021."},{"key":"ref69","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2019.2893568"},{"key":"ref70","doi-asserted-by":"publisher","DOI":"10.1109\/TCI.2021.3094062"},{"key":"ref71","unstructured":"J. Tang, Accelerating Plug-and-Play Image Reconstruction via Multi-stage Sketched Gradients, preprint, arXiv:2203.07308, 2022."},{"key":"ref72","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2020.3027314"},{"key":"ref73","doi-asserted-by":"publisher","DOI":"10.1007\/s00521-019-04051-w"},{"key":"ref74","doi-asserted-by":"crossref","unstructured":"S. V. Venkatakrishnan, C. A. Bouman, and B. Wohlberg, Plug-and-Play priors for model based reconstruction, in Proceedings of the 2013 IEEE Global Conference on Signal and Information Processing, 2013, pp. 945\u2013948.","DOI":"10.1109\/GlobalSIP.2013.6737048"},{"key":"ref75","doi-asserted-by":"publisher","DOI":"10.1137\/15M1053141"},{"key":"ref76","doi-asserted-by":"publisher","DOI":"10.1109\/ACCESS.2018.2880454"},{"key":"ref77","doi-asserted-by":"publisher","DOI":"10.1109\/TPAMI.2021.3088914"},{"key":"ref78","doi-asserted-by":"crossref","unstructured":"K. Zhang, W. Zuo, S. Gu, and L. Zhang, Learning deep CNN denoiser prior for image restoration, in Proceedings of the 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\/23M157185X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:26:10Z","timestamp":1787322370000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/23M157185X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,2]]},"references-count":78,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,6,30]]}},"alternative-id":["10.1137\/23M157185X"],"URL":"https:\/\/doi.org\/10.1137\/23m157185x","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,2]]}}}