{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,11]],"date-time":"2025-09-11T20:49:57Z","timestamp":1757623797891,"version":"3.44.0"},"reference-count":55,"publisher":"Springer Science and Business Media LLC","issue":"9","license":[{"start":{"date-parts":[[2025,8,20]],"date-time":"2025-08-20T00:00:00Z","timestamp":1755648000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,8,20]],"date-time":"2025-08-20T00:00:00Z","timestamp":1755648000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100015401","name":"Key Research and Development Projects of Shaanxi Province","doi-asserted-by":"publisher","award":["No. 2023GXLH-024"],"award-info":[{"award-number":["No. 2023GXLH-024"]}],"id":[{"id":"10.13039\/501100015401","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"crossref","award":["No.62406242"],"award-info":[{"award-number":["No.62406242"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2025,9]]},"DOI":"10.1007\/s10994-025-06849-3","type":"journal-article","created":{"date-parts":[[2025,8,20]],"date-time":"2025-08-20T14:53:02Z","timestamp":1755701582000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["ALM-PU: positive and unlabeled learning with constrained optimization"],"prefix":"10.1007","volume":"114","author":[{"given":"Jiazhe","family":"Wei","sequence":"first","affiliation":[]},{"given":"Yuefei","family":"Wu","sequence":"additional","affiliation":[]},{"given":"Bin","family":"Shi","sequence":"additional","affiliation":[]},{"given":"Ken","family":"Li","sequence":"additional","affiliation":[]},{"given":"Bo","family":"Dong","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,8,20]]},"reference":[{"key":"6849_CR1","unstructured":"Arpit, D., Jastrz\u0119bski, S., Ballas, N., Krueger, D., Bengio, E., Kanwal, M.S., Maharaj, T., Fischer, A., Courville, A., Bengio, Y., et al.( 2017) A closer look at memorization in deep networks. In: International Conference on Machine Learning, pp. 233\u2013 242 . PMLR"},{"key":"6849_CR2","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1016\/j.neunet.2018.05.001","volume":"105","author":"H Bao","year":"2018","unstructured":"Bao, H., Sakai, T., Sato, I., & Sugiyama, M. (2018). Convex formulation of multiple instance learning from positive and unlabeled bags. Neural Networks, 105, 132\u2013141.","journal-title":"Neural Networks"},{"key":"6849_CR3","volume-title":"Constrained Optimization and Lagrange Multiplier Methods","author":"DP Bertsekas","year":"2014","unstructured":"Bertsekas, D. P. (2014). Constrained Optimization and Lagrange Multiplier Methods. Cambridge: Academic press."},{"key":"6849_CR4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611973365","volume-title":"Practical Augmented Lagrangian Methods for Constrained Optimization","author":"EG Birgin","year":"2014","unstructured":"Birgin, E. G., & Mart\u00ednez, J. M. (2014). Practical Augmented Lagrangian Methods for Constrained Optimization. Philadelphia: SIAM."},{"issue":"1","key":"6849_CR5","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1093\/comjnl\/8.1.42","volume":"8","author":"M Box","year":"1965","unstructured":"Box, M. (1965). A new method of constrained optimization and a comparison with other methods. Computer Journal, 8(1), 42\u201352.","journal-title":"Computer Journal"},{"key":"6849_CR6","first-page":"8246","volume":"38","author":"S Cao","year":"2024","unstructured":"Cao, S., Ruan, J., Dong, B., Shi, B., & Zheng, Q. (2024). Rr-pu: A synergistic two-stage positive and unlabeled learning framework for robust tax evasion detection. In: Proceedings of the AAAI Conference on Artificial Intelligence, 38, 8246\u20138254.","journal-title":"In: Proceedings of the AAAI Conference on Artificial Intelligence"},{"key":"6849_CR7","first-page":"14844","volume":"33","author":"H Chen","year":"2020","unstructured":"Chen, H., Liu, F., Wang, Y., Zhao, L., & Wu, H. (2020). A variational approach for learning from positive and unlabeled data. Advances in Neural Information Processing Systems, 33, 14844\u201314854.","journal-title":"Advances in Neural Information Processing Systems"},{"key":"6849_CR8","unstructured":"Chen, X., Chen, W., Chen, T., Yuan, Y., Gong, C., Chen, K., Wang, Z. ( 2020). Self-pu: Self boosted and calibrated positive-unlabeled training. In: International Conference on Machine Learning, pp. 1510\u2013 1519 . PMLR"},{"key":"6849_CR9","unstructured":"Coates, A., Ng, A., Lee, H. ( 2011) An analysis of single-layer networks in unsupervised feature learning. In: Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, pp. 215\u2013 223 . JMLR Workshop and Conference Proceedings"},{"issue":"1","key":"6849_CR10","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1137\/S1052623402417298","volume":"15","author":"R Correa","year":"2004","unstructured":"Correa, R. (2004). A global algorithm for nonlinear semidefinite programming. SIAM Journal on Optimization, 15(1), 303\u2013318.","journal-title":"SIAM Journal on Optimization"},{"issue":"107","key":"6849_CR11","first-page":"1","volume":"24","author":"O Coudray","year":"2023","unstructured":"Coudray, O., Keribin, C., Massart, P., & Pamphile, P. (2023). Risk bounds for positive-unlabeled learning under the selected at random assumption. Journal of Machine Learning Research, 24(107), 1\u201331.","journal-title":"Journal of Machine Learning Research"},{"key":"6849_CR12","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-55404-0","volume-title":"Introduction to Applied Optimization","author":"UM Diwekar","year":"2020","unstructured":"Diwekar, U. M. (2020). Introduction to Applied Optimization (Vol. 22). New York: Springer."},{"key":"6849_CR13","unstructured":"Dong, S.(2006). Methods for constrained optimization. Massachusetts institute of technology, massachusetts"},{"key":"6849_CR14","unstructured":"Du\u00a0Plessis, M.C., Niu, G., Sugiyama, M. (2014) Analysis of learning from positive and unlabeled data. Advances in neural information processing systems 27"},{"key":"6849_CR15","first-page":"973","volume":"17","author":"C Elkan","year":"2001","unstructured":"Elkan, C. (2001). The foundations of cost-sensitive learning. In: International Joint Conference on Artificial Intelligence, 17, 973\u2013978. Lawrence Erlbaum Associates Ltd.","journal-title":"In: International Joint Conference on Artificial Intelligence,"},{"key":"6849_CR16","doi-asserted-by":"crossref","unstructured":"Elkan, C., Noto, K.( 2008) . Learning classifiers from only positive and unlabeled data. In: Proceedings of the 14th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 213\u2013 220","DOI":"10.1145\/1401890.1401920"},{"issue":"1","key":"6849_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.ejor.2023.04.041","volume":"314","author":"AO Fajemisin","year":"2024","unstructured":"Fajemisin, A. O., Maragno, D., & Hertog, D. (2024). Optimization with constraint learning: A framework and survey. European Journal of Operational Research, 314(1), 1\u201314.","journal-title":"European Journal of Operational Research"},{"issue":"2","key":"6849_CR18","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1111\/j.1469-1809.1936.tb02137.x","volume":"7","author":"RA Fisher","year":"1936","unstructured":"Fisher, R. A. (1936). The use of multiple measurements in taxonomic problems. Annals of Eugenics, 7(2), 179\u2013188.","journal-title":"Annals of Eugenics"},{"key":"6849_CR19","volume-title":"Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-value Problems","author":"M Fortin","year":"2000","unstructured":"Fortin, M., & Glowinski, R. (2000). Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-value Problems. Amsterdam: Elsevier."},{"issue":"3","key":"6849_CR20","doi-asserted-by":"publisher","first-page":"807","DOI":"10.1016\/j.ejor.2020.08.045","volume":"290","author":"C Gambella","year":"2021","unstructured":"Gambella, C., Ghaddar, B., & Naoum-Sawaya, J. (2021). Optimization problems for machine learning: A survey. European Journal of Operational Research, 290(3), 807\u2013828.","journal-title":"European Journal of Operational Research"},{"issue":"1","key":"6849_CR21","first-page":"931","volume":"35","author":"Y Gao","year":"2021","unstructured":"Gao, Y., Shi, B., Dong, B., Wang, Y., Mi, L., & Zheng, Q. (2021). Tax evasion detection with fbne-pu algorithm based on pncgcn and pu learning. IEEE Transactions on Knowledge and Data Engineering, 35(1), 931\u2013944.","journal-title":"IEEE Transactions on Knowledge and Data Engineering"},{"key":"6849_CR22","first-page":"8532","volume":"34","author":"S Garg","year":"2021","unstructured":"Garg, S., Wu, Y., Smola, A. J., Balakrishnan, S., & Lipton, Z. (2021). Mixture proportion estimation and pu learning: A modern approach. Advances in Neural Information Processing Systems, 34, 8532\u20138544.","journal-title":"Advances in Neural Information Processing Systems"},{"key":"6849_CR23","doi-asserted-by":"crossref","unstructured":"Gill, P.E., Wong, E. ( 2011) Sequential quadratic programming methods. In: Mixed Integer Nonlinear Programming, pp. 147\u2013 224. Springer, New York","DOI":"10.1007\/978-1-4614-1927-3_6"},{"key":"6849_CR24","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1016\/j.spasta.2016.10.002","volume":"18","author":"JA Gonz\u00e1lez","year":"2016","unstructured":"Gonz\u00e1lez, J. A., Rodr\u00edguez-Cort\u00e9s, F. J., Cronie, O., & Mateu, J. (2016). Spatio-temporal point process statistics: A review. Spatial Statistics, 18, 505\u2013544.","journal-title":"Spatial Statistics"},{"key":"6849_CR25","unstructured":"Gu, Q., Li, Z., Han, J.(2012). Generalized fisher score for feature selection. arXiv preprint arXiv:1202.3725"},{"issue":"11","key":"6849_CR26","doi-asserted-by":"publisher","first-page":"6410","DOI":"10.1109\/TCYB.2024.3393020","volume":"54","author":"J Guo","year":"2024","unstructured":"Guo, J., & Chen, C. P. (2024). A robust semi-supervised broad learning system guided by ensemble-based self-training. IEEE Transactions on Cybernetics, 54(11), 6410\u20136422.","journal-title":"IEEE Transactions on Cybernetics"},{"key":"6849_CR27","first-page":"13088","volume":"33","author":"Z Hammoudeh","year":"2020","unstructured":"Hammoudeh, Z., & Lowd, D. (2020). Learning from positive and unlabeled data with arbitrary positive shift. Advances in Neural Information Processing Systems, 33, 13088\u201313099.","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"5","key":"6849_CR28","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/BF00927673","volume":"4","author":"MR Hestenes","year":"1969","unstructured":"Hestenes, M. R. (1969). Multiplier and gradient methods. Journal of Optimization Theory and Applications, 4(5), 303\u2013320.","journal-title":"Journal of Optimization Theory and Applications"},{"key":"6849_CR29","first-page":"7806","volume":"35","author":"W Hu","year":"2021","unstructured":"Hu, W., Le, R., Liu, B., Ji, F., Ma, J., Zhao, D., & Yan, R. (2021). Predictive adversarial learning from positive and unlabeled data. In: Proceedings of the AAAI Conference on Artificial Intelligence, 35, 7806\u20137814.","journal-title":"In: Proceedings of the AAAI Conference on Artificial Intelligence"},{"key":"6849_CR30","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1016\/j.neucom.2017.02.019","volume":"239","author":"Z Karimi","year":"2017","unstructured":"Karimi, Z., & Ghidary, S. S. (2017). Semi-supervised classification in stratified spaces by considering non-interior points using laplacian behavior. Neurocomputing, 239, 223\u2013231.","journal-title":"Neurocomputing"},{"key":"6849_CR31","unstructured":"Katz-Samuels, J., Nakhleh, J.B., Nowak, R., Li, Y.( 2022). Training ood detectors in their natural habitats. In: International Conference on Machine Learning, pp. 10848\u2013 10865 . PMLR"},{"key":"6849_CR32","unstructured":"Kiryo, R., Niu, G., Du\u00a0Plessis, M.C., Sugiyama, M. (2017) Positive-unlabeled learning with non-negative risk estimator. Advances in neural information processing systems 30"},{"key":"6849_CR33","doi-asserted-by":"crossref","unstructured":"Kotary, J., Fioretto, F., Van\u00a0Hentenryck, P., Wilder, B.(2021). End-to-end constrained optimization learning: A survey. arXiv preprint arXiv:2103.16378","DOI":"10.24963\/ijcai.2021\/610"},{"key":"6849_CR34","unstructured":"Krizhevsky, A., Hinton, G. (2009) Learning multiple layers of features from tiny images. Technical Report TR-2009, University of Toronto, Toronto, Ontario, Canada"},{"key":"6849_CR35","unstructured":"Li, C., Li, X., Feng, L., Ouyang, J. ( 2022). Who is your right mixup partner in positive and unlabeled learning. In: International Conference on Learning Representations"},{"key":"6849_CR36","doi-asserted-by":"crossref","unstructured":"Li, X.-L., Liu, B.( 2005). Learning from positive and unlabeled examples with different data distributions. In: Machine Learning: ECML 2005: 16th European Conference on Machine Learning, Porto, Portugal, October 3-7, 2005. Proceedings 16, pp. 218\u2013 229 . Springer","DOI":"10.1007\/11564096_24"},{"issue":"5","key":"6849_CR37","doi-asserted-by":"publisher","first-page":"2326","DOI":"10.1214\/009053606000000786","volume":"34","author":"P Massart","year":"2006","unstructured":"Massart, P., & N\u00e9d\u00e9lec, \u00c9. (2006). Risk bounds for statistical learning. Annals of Statistics, 34(5), 2326\u20132366.","journal-title":"Annals of Statistics"},{"key":"6849_CR38","doi-asserted-by":"publisher","DOI":"10.1007\/b98874","volume-title":"Numerical Optimization","author":"J Nocedal","year":"1999","unstructured":"Nocedal, J., & Wright, S. J. (1999). Numerical Optimization. New York: Springer."},{"key":"6849_CR39","unstructured":"Northcutt, C.G., Wu, T., Chuang, I.L.(2017). Learning with confident examples: Rank pruning for robust classification with noisy labels. arXiv preprint arXiv:1705.01936"},{"key":"6849_CR40","first-page":"136","volume":"6","author":"BR Ott","year":"2017","unstructured":"Ott, B. R., Jones, R. N., Noto, R. B., Yoo, D. C., Snyder, P. J., Bernier, J. N., Carr, D. B., & Roe, C. M. (2017). Brain amyloid in preclinical alzheimer\u2019s disease is associated with increased driving risk. Alzheimer\u2019s & Dementia: Diagnosis, Assessment & Disease Monitoring, 6, 136\u2013142.","journal-title":"Alzheimer\u2019s & Dementia: Diagnosis, Assessment & Disease Monitoring"},{"key":"6849_CR41","unstructured":"Padesky, C.A. ( 1993) Socratic questioning: Changing minds or guiding discovery. In: A Keynote Address Delivered at the European Congress of Behavioural and Cognitive Therapies, London, vol. 24"},{"key":"6849_CR42","doi-asserted-by":"publisher","first-page":"116409","DOI":"10.1016\/j.eswa.2021.116409","volume":"193","author":"M Savi\u0107","year":"2022","unstructured":"Savi\u0107, M., Atanasijevi\u0107, J., Jakoveti\u0107, D., & Kreji\u0107, N. (2022). Tax evasion risk management using a hybrid unsupervised outlier detection method. Expert Systems with Applications, 193, 116409.","journal-title":"Expert Systems with Applications"},{"key":"6849_CR43","doi-asserted-by":"crossref","unstructured":"Shchur, O., T\u00fcrkmen, A.C., Januschowski, T., G\u00fcnnemann, S. (2021) Neural temporal point processes: A review. arXiv preprint arXiv:2104.03528","DOI":"10.24963\/ijcai.2021\/623"},{"key":"6849_CR44","first-page":"596","volume":"33","author":"K Sohn","year":"2020","unstructured":"Sohn, K., Berthelot, D., Carlini, N., Zhang, Z., Zhang, H., Raffel, C. A., Cubuk, E. D., Kurakin, A., & Li, C.-L. (2020). Fixmatch: Simplifying semi-supervised learning with consistency and confidence. Advances in Neural Information Processing Systems, 33, 596\u2013608.","journal-title":"Advances in Neural Information Processing Systems"},{"issue":"8","key":"6849_CR45","doi-asserted-by":"publisher","first-page":"3668","DOI":"10.1109\/TCYB.2019.2950779","volume":"50","author":"S Sun","year":"2019","unstructured":"Sun, S., Cao, Z., Zhu, H., & Zhao, J. (2019). A survey of optimization methods from a machine learning perspective. IEEE Transactions on Cybernetics, 50(8), 3668\u20133681.","journal-title":"IEEE Transactions on Cybernetics"},{"key":"6849_CR46","unstructured":"Tanaka, D., Ikami, D., Aizawa, K. (2021). A novel perspective for positive-unlabeled learning via noisy labels. arXiv preprint arXiv:2103.04685"},{"key":"6849_CR47","doi-asserted-by":"crossref","unstructured":"Vapnik, V.N. ( 1995) The Nature of Statistical Learning Theory. Information Science and Statistics, p. 188. Springer, New York","DOI":"10.1007\/978-1-4757-2440-0"},{"key":"6849_CR48","unstructured":"Wei, T., Shi, F., Wang, H., Li, W.-W.T.Y.-F. (2020) Mixpul: consistency-based augmentation for positive and unlabeled learning. arXiv preprint arXiv:2004.09388"},{"key":"6849_CR49","unstructured":"Xia, X., Liu, T., Han, B., Gong, M., Yu, J., Niu, G., Sugiyama, M. (2021) Sample selection with uncertainty of losses for learning with noisy labels. arXiv preprint arXiv:2106.00445"},{"key":"6849_CR50","unstructured":"Xiao, H., Rasul, K., Vollgraf, R. (2017) Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms. arXiv preprint arXiv:1708.07747"},{"key":"6849_CR51","unstructured":"Xinrui, W., Wan, W., Geng, C., Li, S.-Y., Chen, S.(2024). Beyond myopia: Learning from positive and unlabeled data through holistic predictive trends. Advances in Neural Information Processing Systems 36"},{"issue":"1","key":"6849_CR52","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1287\/ijoo.2019.0033","volume":"3","author":"Y Xu","year":"2021","unstructured":"Xu, Y. (2021). First-order methods for constrained convex programming based on linearized augmented lagrangian function. INFORMS Journal on Optimization, 3(1), 89\u2013117.","journal-title":"INFORMS Journal on Optimization"},{"key":"6849_CR53","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/s10107-019-01425-9","volume":"185","author":"Y Xu","year":"2021","unstructured":"Xu, Y. (2021). Iteration complexity of inexact augmented lagrangian methods for constrained convex programming. Mathematical Programming, 185, 199\u2013244.","journal-title":"Mathematical Programming"},{"key":"6849_CR54","doi-asserted-by":"crossref","unstructured":"Zhao, Y., Xu, Q., Jiang, Y., Wen, P., Huang, Q.: Dist-pu: Positive-unlabeled learning from a label distribution perspective. In: Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition, pp. 14461\u2013 14470 ( 2022)","DOI":"10.1109\/CVPR52688.2022.01406"},{"issue":"1","key":"6849_CR55","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1093\/nsr\/nwx106","volume":"5","author":"Z-H Zhou","year":"2018","unstructured":"Zhou, Z.-H. (2018). A brief introduction to weakly supervised learning. National Science Review, 5(1), 44\u201353.","journal-title":"National Science Review"}],"container-title":["Machine Learning"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06849-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10994-025-06849-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10994-025-06849-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,9]],"date-time":"2025-09-09T09:06:56Z","timestamp":1757408816000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10994-025-06849-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,20]]},"references-count":55,"journal-issue":{"issue":"9","published-print":{"date-parts":[[2025,9]]}},"alternative-id":["6849"],"URL":"https:\/\/doi.org\/10.1007\/s10994-025-06849-3","relation":{},"ISSN":["0885-6125","1573-0565"],"issn-type":[{"type":"print","value":"0885-6125"},{"type":"electronic","value":"1573-0565"}],"subject":[],"published":{"date-parts":[[2025,8,20]]},"assertion":[{"value":"6 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 July 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 July 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 August 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"210"}}