{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,21]],"date-time":"2026-03-21T19:28:34Z","timestamp":1774121314282,"version":"3.50.1"},"reference-count":88,"publisher":"Institute of Electrical and Electronics Engineers (IEEE)","issue":"2","license":[{"start":{"date-parts":[[2025,2,1]],"date-time":"2025-02-01T00:00:00Z","timestamp":1738368000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/ieeexplore.ieee.org\/Xplorehelp\/downloads\/license-information\/IEEE.html"},{"start":{"date-parts":[[2025,2,1]],"date-time":"2025-02-01T00:00:00Z","timestamp":1738368000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2025,2,1]],"date-time":"2025-02-01T00:00:00Z","timestamp":1738368000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation","doi-asserted-by":"publisher","award":["61971361"],"award-info":[{"award-number":["61971361"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation","doi-asserted-by":"publisher","award":["62122064"],"award-info":[{"award-number":["62122064"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation","doi-asserted-by":"publisher","award":["62331021"],"award-info":[{"award-number":["62331021"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation","doi-asserted-by":"publisher","award":["62371410"],"award-info":[{"award-number":["62371410"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province of China","doi-asserted-by":"publisher","award":["2023J02005"],"award-info":[{"award-number":["2023J02005"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003392","name":"Natural Science Foundation of Fujian Province of China","doi-asserted-by":"publisher","award":["2021J011184"],"award-info":[{"award-number":["2021J011184"]}],"id":[{"id":"10.13039\/501100003392","id-type":"DOI","asserted-by":"publisher"}]},{"name":"President Fund of Xiamen University","award":["20720220063"],"award-info":[{"award-number":["20720220063"]}]},{"name":"Xiamen University Nanqiang Outstanding Talents Program"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["IEEE Trans. Neural Netw. Learning Syst."],"published-print":{"date-parts":[[2025,2]]},"DOI":"10.1109\/tnnls.2024.3353795","type":"journal-article","created":{"date-parts":[[2024,1,31]],"date-time":"2024-01-31T13:36:33Z","timestamp":1706708193000},"page":"3423-3435","source":"Crossref","is-referenced-by-count":3,"title":["Convex Dual Theory Analysis of Two-Layer Convolutional Neural Networks With Soft-Thresholding"],"prefix":"10.1109","volume":"36","author":[{"given":"Chunyan","family":"Xiong","sequence":"first","affiliation":[{"name":"Institute of Electromagnetics and Acoustics School of Electronic Science and Engineering, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Xiamen University, Xiamen, China"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-0575-2004","authenticated-orcid":false,"given":"Chaoxing","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Electronic Science and Engineering, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Biomedical Intelligent Cloud Research and Development Center, Xiamen University, Xiamen, China"}]},{"given":"Mengli","family":"Lu","sequence":"additional","affiliation":[{"name":"School of Electronic Science and Engineering, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Biomedical Intelligent Cloud Research and Development Center, Xiamen University, Xiamen, China"}]},{"given":"Xiaotong","family":"Yu","sequence":"additional","affiliation":[{"name":"School of Electronic Science and Engineering, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Biomedical Intelligent Cloud Research and Development Center, Xiamen University, Xiamen, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9541-8738","authenticated-orcid":false,"given":"Jian","family":"Cao","sequence":"additional","affiliation":[{"name":"School of Electronic Science and Engineering, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Biomedical Intelligent Cloud Research and Development Center, Xiamen University, Xiamen, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1473-2224","authenticated-orcid":false,"given":"Zhong","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Electronic Science and Engineering, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Biomedical Intelligent Cloud Research and Development Center, Xiamen University, Xiamen, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9910-5720","authenticated-orcid":false,"given":"Di","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Computer and Information Engineering, University of Technology, Xiamen, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8675-5820","authenticated-orcid":false,"given":"Xiaobo","family":"Qu","sequence":"additional","affiliation":[{"name":"School of Electronic Science and Engineering, Fujian Provincial Key Laboratory of Plasma and Magnetic Resonance, Biomedical Intelligent Cloud Research and Development Center, Xiamen University, Xiamen, China"}]}],"member":"263","reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1109\/ICASSP.2013.6638947"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2016.90"},{"key":"ref3","first-page":"1097","article-title":"ImageNet classification with deep convolutional neural networks","volume-title":"Proc. Int. Conf. Neural Inf. Process. Syst.","author":"Krizhevsky"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2022.3183809"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1002\/ange.201908162"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2021.3134717"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2022.3144580"},{"key":"ref8","article-title":"Diminishing batch normalization","author":"Ma","year":"2022","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref9","article-title":"A convergence path to deep learning on noisy labels","author":"Liu","year":"2022","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2022.3146062"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1109\/TNNLS.2020.2979228"},{"key":"ref12","article-title":"On weight initialization in deep neural networks","author":"Krishna Kumar","year":"2017","journal-title":"arXiv:1704.08863"},{"key":"ref13","first-page":"16410","article-title":"Gradinit: Learning to initialize neural networks for stable and efficient training","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Zhu"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1109\/TVT.2013.2251920"},{"key":"ref15","first-page":"249","article-title":"Understanding the difficulty of training deep feedforward neural networks","volume-title":"Proc. Int. Conf. Aquatic Invasive. Species","author":"Glorot"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1109\/ICCV.2015.123"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.48550\/ARXIV.1706.03762"},{"key":"ref18","first-page":"10524","article-title":"On layer normalization in the transformer architecture","volume-title":"Proc. Int. Conf. Mach. Learn. (ICML)","author":"Xiong"},{"key":"ref19","first-page":"4475","article-title":"Improving transformer optimization through better initialization","volume-title":"Proc. Int. Conf. Mach. Learn. (ICML)","author":"Huang"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.18653\/v1\/2020.emnlp-main.463"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.48550\/ARXIV.1907.11692"},{"key":"ref22","first-page":"1877","article-title":"Language models are few-shot learners","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Brown"},{"key":"ref23","article-title":"Adam: A method for stochastic optimization","author":"Kingma","year":"2014","journal-title":"arXiv:1412.6980"},{"key":"ref24","article-title":"Fixup initialization: Residual learning without normalization","author":"Zhang","year":"2019","journal-title":"arXiv:1901.09321"},{"key":"ref25","first-page":"19964","article-title":"Batch normalization biases residual blocks towards the identity function in deep networks","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"De"},{"key":"ref26","article-title":"Characterizing signal propagation to close the performance gap in unnormalized ResNets","author":"Brock","year":"2021","journal-title":"arXiv:2101.08692"},{"key":"ref27","first-page":"1059","article-title":"High-performance large-scale image recognition without normalization","volume-title":"Proc. Int. Conf. Mach. Learn. (ICML)","author":"Brock"},{"key":"ref28","first-page":"123","article-title":"Convex neural networks","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Bengio"},{"key":"ref29","first-page":"146","article-title":"Input convex neural networks","volume-title":"Proc. Int. Conf. Mach. Learn. (ICML)","author":"Amos"},{"key":"ref30","first-page":"4024","article-title":"Convex geometry of two-layer ReLU networks: Implicit autoencoding and interpretable models","volume-title":"Proc. Int. Conf. Artif. Intell. Statist.","author":"Ergen"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1109\/ALLERTON.2019.8919769"},{"key":"ref32","first-page":"7695","article-title":"Neural networks are convex regularizers: Exact polynomial-time convex optimization formulations for two-layer networks","volume-title":"Proc. Int. Conf. Mach. Learn. (ICML)","author":"Pilanci"},{"key":"ref33","article-title":"The convex geometry of backpropagation: Neural network gradient flows converge to extreme points of the dual convex program","author":"Wang","year":"2021","journal-title":"arXiv:2110.06488"},{"key":"ref34","first-page":"15770","article-title":"Fast convex optimization for two-layer relu networks: Equivalent model classes and cone decompositions","volume-title":"Proc. Int. Conf. Mach. Learn. (ICML)","author":"Mishkin"},{"key":"ref35","article-title":"Vector-output ReLU neural network problems are copositive programs: Convex analysis of two layer networks and polynomial-time algorithms","author":"Sahiner","year":"2020","journal-title":"arXiv:2012.13329"},{"key":"ref36","first-page":"2993","article-title":"Global optimality beyond two layers: Training deep ReLU networks via convex programs","volume-title":"Proc. Int. Conf. Mach. Learn. (ICML)","author":"Ergen"},{"issue":"1","key":"ref37","first-page":"9646","article-title":"Convex geometry and duality of over-parameterized neural networks","volume":"22","author":"Ergen","year":"2021","journal-title":"J. Mach. Learn. Res."},{"key":"ref38","article-title":"Convex regularization behind neural reconstruction","author":"Sahiner","year":"2020","journal-title":"arXiv:2012.05169"},{"key":"ref39","article-title":"Training convolutional relu neural networks in polynomial time: Exact convex optimization formulations","author":"Ergen","year":"2020","journal-title":"arXiv:2006.14798"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1109\/5.58357"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1109\/29.21701"},{"key":"ref42","first-page":"2147","article-title":"Shift-invariant pattern recognition neural network and its optical architecture","volume-title":"Proc. Annu. Conf. Jpn. Soc. Appl. Phys.","author":"Zhang"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1162\/neco.1989.1.4.541"},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1109\/LSP.2018.2850222"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1109\/72.554195"},{"key":"ref46","doi-asserted-by":"publisher","DOI":"10.1109\/CVPR.2015.7298594"},{"key":"ref47","first-page":"82","article-title":"Recurrent convolutional neural networks for scene labeling","volume-title":"Proc. Int. Conf. Mach. Learn.","author":"Pinheiro"},{"key":"ref48","doi-asserted-by":"publisher","DOI":"10.1109\/18.382009"},{"issue":"3","key":"ref49","first-page":"1027","article-title":"Denoising using soft thresholding","volume":"2","author":"Joy","year":"2013","journal-title":"Int. J. Adv. Res. Elect., Electron. Instrum. Eng."},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1109\/97.720560"},{"key":"ref51","doi-asserted-by":"publisher","DOI":"10.1109\/83.862633"},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2005.858979"},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2010.936023"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2016.2550080"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1016\/j.media.2021.101987"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2022.3164472"},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.1137\/080738970"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1002\/anie.201409291"},{"key":"ref59","doi-asserted-by":"publisher","DOI":"10.1002\/mrc.5082"},{"issue":"2","key":"ref60","first-page":"114","article-title":"An efficient adaptive thresholding technique for wavelet based image denoising","volume":"2","author":"Gnanadurai","year":"2006","journal-title":"Int. J. Signal Process."},{"key":"ref61","doi-asserted-by":"publisher","DOI":"10.1109\/TII.2019.2943898"},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1109\/72.925559"},{"key":"ref63","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2022.3203312"},{"key":"ref64","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmr.2020.106790"},{"key":"ref65","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2022.3184845"},{"key":"ref66","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804441"},{"key":"ref67","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1958.8.171"},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.2005.11920204"},{"key":"ref69","doi-asserted-by":"publisher","DOI":"10.1080\/02331930902730070"},{"key":"ref70","first-page":"4558","article-title":"Bounding and counting linear regions of deep neural networks","volume-title":"Proc. Int. Conf. Mach. Learn. (ICML)","author":"Serra"},{"key":"ref71","doi-asserted-by":"publisher","DOI":"10.1109\/72.857765"},{"key":"ref72","doi-asserted-by":"publisher","DOI":"10.1109\/PGEC.1965.264137"},{"key":"ref73","doi-asserted-by":"publisher","DOI":"10.1016\/S0893-6080(05)80010-3"},{"key":"ref74","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-35289-8_26"},{"key":"ref75","first-page":"797","article-title":"Escaping from saddle points\u2014Online stochastic gradient for tensor decomposition","volume-title":"Proc. Conf. Learn. Theory","author":"Ge"},{"key":"ref76","volume-title":"Deep Learning","author":"Goodfellow","year":"2016"},{"key":"ref77","article-title":"Geometry of optimization and implicit regularization in deep learning","author":"Neyshabur","year":"2017","journal-title":"arXiv:1705.03071"},{"key":"ref78","doi-asserted-by":"publisher","DOI":"10.1609\/aaai.v32i1.11694"},{"key":"ref79","article-title":"On the reproducibility of neural network predictions","author":"Bhojanapalli","year":"2021","journal-title":"arXiv:2102.03349"},{"key":"ref80","doi-asserted-by":"publisher","DOI":"10.1214\/aoms\/1177729586"},{"key":"ref81","first-page":"315","article-title":"Accelerating stochastic gradient descent using predictive variance reduction","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Johnson"},{"key":"ref82","doi-asserted-by":"publisher","DOI":"10.1126\/science.1127647"},{"issue":"7","key":"ref83","first-page":"2121","article-title":"Adaptive subgradient methods for online learning and stochastic optimization","volume":"12","author":"Duchi","year":"2011","journal-title":"J. Mach. Learn. Res."},{"issue":"8","key":"ref84","first-page":"2","article-title":"Neural networks for machine learning lecture 6a overview of mini-batch gradient descent","volume":"14","author":"Hinton","year":"2012"},{"key":"ref85","first-page":"288","article-title":"On the convergence of a class of adam-type algorithms for non-convex optimization","volume-title":"Proc. AISTATS","volume":"84","author":"Chen"},{"key":"ref86","first-page":"8026","article-title":"Pytorch: An imperative style, high-performance deep learning library","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","author":"Paszke"},{"key":"ref87","doi-asserted-by":"publisher","DOI":"10.1109\/5.726791"},{"key":"ref88","first-page":"1097","article-title":"ImageNet classification with deep convolutional neural networks","volume-title":"Proc. Adv. Neural Inf. Process. Syst.","volume":"25","author":"Krizhevsky"}],"container-title":["IEEE Transactions on Neural Networks and Learning Systems"],"original-title":[],"link":[{"URL":"http:\/\/xplorestaging.ieee.org\/ielx7\/5962385\/10877690\/10417866.pdf?arnumber=10417866","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T18:39:21Z","timestamp":1764959961000},"score":1,"resource":{"primary":{"URL":"https:\/\/ieeexplore.ieee.org\/document\/10417866\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2]]},"references-count":88,"journal-issue":{"issue":"2"},"URL":"https:\/\/doi.org\/10.1109\/tnnls.2024.3353795","relation":{},"ISSN":["2162-237X","2162-2388"],"issn-type":[{"value":"2162-237X","type":"print"},{"value":"2162-2388","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2]]}}}