{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T09:34:08Z","timestamp":1787045648420,"version":"build-2736575974"},"reference-count":73,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T00:00:00Z","timestamp":1772582400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T00:00:00Z","timestamp":1772582400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100004377","name":"The Hong Kong Polytechnic University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100004377","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Prog. Comp."],"published-print":{"date-parts":[[2026,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Support matrix machine (SMM) is a successful supervised classification model for matrix-type samples. Unlike support vector machines, it employs low-rank regularization on the regression matrix to effectively capture the intrinsic structure embedded in each input matrix. When solving a large-scale SMM, a major challenge arises from the potential increase in sample size, leading to substantial computational and storage burdens. To address these issues, we design a semismooth Newton-CG (SNCG) based augmented Lagrangian method (ALM) for solving the SMM. The ALM exhibits an asymptotic R-superlinear convergence if a strict complementarity condition is satisfied. The SNCG method is employed to solve the ALM subproblems, achieving at least a superlinear convergence rate under the nonemptiness of an index set. Furthermore, the sparsity of samples and the low-rank nature of solutions enable us to reduce the computational cost and storage demands for the Newton linear systems. Additionally, we develop an adaptive sieving strategy that generates a solution path for the SMM by exploiting sample sparsity. The finite convergence of this strategy is also demonstrated. Numerical experiments on both large-scale real and synthetic datasets validate the effectiveness of the proposed methods.<\/jats:p>","DOI":"10.1007\/s12532-026-00306-5","type":"journal-article","created":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T07:35:09Z","timestamp":1772609709000},"page":"773-818","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Support matrix machine: exploring sample sparsity, low rank, and adaptive sieving in high-performance computing"],"prefix":"10.1007","volume":"18","author":[{"given":"Can","family":"Wu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dong-Hui","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Defeng","family":"Sun","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,3,4]]},"reference":[{"issue":"2","key":"306_CR1","first-page":"365","volume":"15","author":"FA Artacho","year":"2008","unstructured":"Artacho, F.A., Geoffroy, M.H.: Characterization of metric regularity of subdifferentials. J. Convex Anal. 15(2), 365\u2013380 (2008)","journal-title":"J. Convex Anal."},{"issue":"1","key":"306_CR2","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1007\/s101070050083","volume":"86","author":"HH Bauschke","year":"1999","unstructured":"Bauschke, H.H., Borwein, J.M., Li, W.: Strong conical hull intersection property, bounded linear regularity, Jameson\u2019s property (G), and error bounds in convex optimization. Math. Program. 86(1), 135\u2013160 (1999)","journal-title":"Math. Program."},{"issue":"4","key":"306_CR3","doi-asserted-by":"publisher","first-page":"1956","DOI":"10.1137\/080738970","volume":"20","author":"JF Cai","year":"2010","unstructured":"Cai, J.F., Cand\u00e8s, E.J., Shen, Z.: A singular value thresholding algorithm for matrix completion. SIAM J. Optim. 20(4), 1956\u20131982 (2010)","journal-title":"SIAM J. Optim."},{"issue":"6","key":"306_CR4","doi-asserted-by":"publisher","first-page":"717","DOI":"10.1007\/s10208-009-9045-5","volume":"9","author":"EJ Cand\u00e8s","year":"2009","unstructured":"Cand\u00e8s, E.J., Recht, B.: Exact matrix completion via convex optimization. Found. Comput. Math. 9(6), 717\u2013772 (2009)","journal-title":"Found. Comput. Math."},{"issue":"1","key":"306_CR5","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/s10107-016-1007-5","volume":"161","author":"L Chen","year":"2017","unstructured":"Chen, L., Sun, D.F., Toh, K.C.: An efficient inexact symmetric Gauss-Seidel based majorized ADMM for high-dimensional convex composite conic programming. Math. Program. 161(1), 237\u2013270 (2017)","journal-title":"Math. Program."},{"key":"306_CR6","doi-asserted-by":"publisher","DOI":"10.3389\/fnins.2020.606949","volume":"14","author":"Y Chen","year":"2020","unstructured":"Chen, Y., Hang, W., Liang, S., Liu, X., Li, G., Wang, Q., Qin, J., Choi, K.S.: A novel transfer support matrix machine for motor imagery-based brain computer interface. Front. Neurosci. 14, 606949 (2020)","journal-title":"Front. Neurosci."},{"key":"306_CR7","volume-title":"Optimization and Nonsmooth Analysis","author":"FH Clarke","year":"1983","unstructured":"Clarke, F.H.: Optimization and Nonsmooth Analysis. John Wiley and Sons, New York (1983)"},{"key":"306_CR8","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1023\/A:1022627411411","volume":"20","author":"C Cortes","year":"1995","unstructured":"Cortes, C., Vapnik, V.: Support-vector networks. Mach. Learn. 20, 273\u2013297 (1995)","journal-title":"Mach. Learn."},{"issue":"4","key":"306_CR9","doi-asserted-by":"publisher","first-page":"2332","DOI":"10.1137\/17M1116325","volume":"27","author":"Y Cui","year":"2017","unstructured":"Cui, Y., Ding, C., Zhao, X.: Quadratic growth conditions for convex matrix optimization problems associated with spectral functions. SIAM J. Optim. 27(4), 2332\u20132355 (2017)","journal-title":"SIAM J. Optim."},{"key":"306_CR10","unstructured":"Cui, Y., Sun, D.F., Toh, K.C.: On the asymptotic superlinear convergence of the augmented Lagrangian method for semidefinite programming with multiple solutions. arXiv preprint arXiv:1610.00875 (2016)"},{"issue":"1","key":"306_CR11","doi-asserted-by":"publisher","first-page":"381","DOI":"10.1007\/s10107-018-1300-6","volume":"178","author":"Y Cui","year":"2019","unstructured":"Cui, Y., Sun, D.F., Toh, K.C.: On the R-superlinear convergence of the KKT residuals generated by the augmented Lagrangian method for convex composite conic programming. Math. Program. 178(1), 381\u2013415 (2019)","journal-title":"Math. Program."},{"issue":"3","key":"306_CR12","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevA.96.032301","volume":"96","author":"B Duan","year":"2017","unstructured":"Duan, B., Yuan, J., Liu, Y., Li, D.: Quantum algorithm for support matrix machines. Phys. Rev. A 96(3), 032301 (2017)","journal-title":"Phys. Rev. A"},{"key":"306_CR13","volume-title":"Finite-Dimensional Variational Inequalities and Complementarity Problems","author":"F Facchinei","year":"2007","unstructured":"Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Science & Business Media, New York (2007)"},{"key":"306_CR14","doi-asserted-by":"crossref","unstructured":"Fazel, M., Hindi, H., Boyd, S.P.: A rank minimization heuristic with application to minimum order system approximation. In: Proceedings of the 2001 American Control Conference. (Cat. No. 01CH37148), vol. 6, pp. 4734\u20134739. IEEE (2001)","DOI":"10.1109\/ACC.2001.945730"},{"key":"306_CR15","doi-asserted-by":"publisher","first-page":"18643","DOI":"10.1007\/s00521-022-07460-6","volume":"34","author":"R Feng","year":"2022","unstructured":"Feng, R., Xu, Y.: Support matrix machine with pinball loss for classification. Neural Comput. Appl. 34, 18643\u201318661 (2022)","journal-title":"Neural Comput. Appl."},{"issue":"1","key":"306_CR16","first-page":"155","volume":"18","author":"R Feng","year":"2022","unstructured":"Feng, R., Zhong, P., Xu, Y.: A subspace elimination strategy for accelerating support matrix machine. Pac. J. Optim. 18(1), 155\u2013176 (2022)","journal-title":"Pac. J. Optim."},{"issue":"22","key":"306_CR17","doi-asserted-by":"publisher","first-page":"12375","DOI":"10.3390\/app132212375","volume":"13","author":"M Geng","year":"2023","unstructured":"Geng, M., Xu, Z., Mei, M.: Fault diagnosis method for railway turnout with pinball loss-based multiclass support matrix machine. Appl. Sci. 13(22), 12375 (2023)","journal-title":"Appl. Sci."},{"issue":"3","key":"306_CR18","doi-asserted-by":"publisher","first-page":"1588","DOI":"10.1137\/120896219","volume":"7","author":"T Goldstein","year":"2014","unstructured":"Goldstein, T., O\u2019Donoghue, B., Setzer, S., Baraniuk, R.: Fast alternating direction optimization methods. SIAM J. Imaging Sci. 7(3), 1588\u20131623 (2014)","journal-title":"SIAM J. Imaging Sci."},{"key":"306_CR19","doi-asserted-by":"publisher","DOI":"10.1016\/j.cmpb.2020.105466","volume":"193","author":"W Hang","year":"2020","unstructured":"Hang, W., Feng, W., Liang, S., Wang, Q., Liu, X., Choi, K.S.: Deep stacked support matrix machine based representation learning for motor imagery eeg classification. Comput. Meth. Programs Biomed. 193, 105466 (2020)","journal-title":"Comput. Meth. Programs Biomed."},{"key":"306_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.bspc.2023.104579","volume":"82","author":"W Hang","year":"2023","unstructured":"Hang, W., Li, Z., Yin, M., Liang, S., Shen, H., Wang, Q., Qin, J., Choi, K.S.: Deep stacked least square support matrix machine with adaptive multi-layer transfer for EEG classification. Biomed. Signal Process. Control 82, 104579 (2023)","journal-title":"Biomed. Signal Process. Control"},{"issue":"15","key":"306_CR21","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.103.150502","volume":"103","author":"AW Harrow","year":"2009","unstructured":"Harrow, A.W., Hassidim, A., Lloyd, S.: Quantum algorithm for linear systems of equations. Phys. Rev. Lett. 103(15), 150502 (2009)","journal-title":"Phys. Rev. Lett."},{"key":"306_CR22","doi-asserted-by":"crossref","unstructured":"Hastie, T., Tibshirani, R., Friedman, J.H., Friedman, J.H.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction, vol. 2. Springer (2009)","DOI":"10.1007\/978-0-387-84858-7"},{"key":"306_CR23","unstructured":"Hsieh, C.J., Olsen, P.: Nuclear norm minimization via active subspace selection. In: International Conference on Machine Learning, pp. 575\u2013583. PMLR (2014)"},{"key":"306_CR24","unstructured":"Jiang, K.: Algorithms for Large Scale Nuclear Norm Minimization and Convex Quadratic Semidefinite Programming Problems. Ph.D. thesis (2011)"},{"key":"306_CR25","doi-asserted-by":"crossref","unstructured":"Jiang, K., Sun, D.F., Toh, K.C.: Solving nuclear norm regularized and semidefinite matrix least squares problems with linear equality constraints. In: Discrete Geometry and Optimization, pp. 133\u2013162. Springer (2013)","DOI":"10.1007\/978-3-319-00200-2_9"},{"key":"306_CR26","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2024.106767","volume":"181","author":"A Kumari","year":"2025","unstructured":"Kumari, A., Akhtar, M., Shah, R., Tanveer, M.: Support matrix machine: A review. Neural Netw. 181, 106767 (2025)","journal-title":"Neural Netw."},{"issue":"4","key":"306_CR27","first-page":"667","volume":"8","author":"EG Laurent","year":"2012","unstructured":"Laurent, E.G., Vivian, V., Tarek, R.: Safe feature elimination in sparse supervised learning. Pac. J. Optim. 8(4), 667\u2013698 (2012)","journal-title":"Pac. J. Optim."},{"key":"306_CR28","first-page":"1","volume":"24","author":"Q Li","year":"2023","unstructured":"Li, Q., Jiang, B., Sun, D.F.: MARS: A second-order reduction algorithm for high-dimensional sparse precision matrices estimation. J. Mach. Learn. Res. 24, 1\u201344 (2023)","journal-title":"J. Mach. Learn. Res."},{"issue":"8","key":"306_CR29","doi-asserted-by":"publisher","first-page":"5180","DOI":"10.1109\/TII.2021.3125385","volume":"18","author":"X Li","year":"2022","unstructured":"Li, X., Cheng, J., Shao, H., Liu, K., Cai, B.: A fusion cwsmm-based framework for rotating machinery fault diagnosis under strong interference and imbalanced case. IEEE Trans. Industr. Inform. 18(8), 5180\u20135189 (2022)","journal-title":"IEEE Trans. Industr. Inform."},{"key":"306_CR30","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2023.109882","volume":"243","author":"X Li","year":"2024","unstructured":"Li, X., Li, S., Wei, D., Si, L., Yu, K., Yan, K.: Dynamics simulation-driven fault diagnosis of rolling bearings using security transfer support matrix machine. Reliab. Eng. Syst. Saf. 243, 109882 (2024)","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"306_CR31","doi-asserted-by":"publisher","DOI":"10.1016\/j.ress.2022.108921","volume":"230","author":"X Li","year":"2023","unstructured":"Li, X., Li, Y., Yan, K., Shao, H., Lin, J.J.: Intelligent fault diagnosis of bevel gearboxes using semi-supervised probability support matrix machine and infrared imaging. Reliab. Eng. Syst. Saf. 230, 108921 (2023)","journal-title":"Reliab. Eng. Syst. Saf."},{"issue":"1","key":"306_CR32","doi-asserted-by":"publisher","first-page":"388","DOI":"10.1109\/TMECH.2023.3278710","volume":"29","author":"X Li","year":"2024","unstructured":"Li, X., Li, Y., Yan, K., Si, L., Shao, H.: An intelligent fault detection method of industrial gearboxes with robustness one-class support matrix machine toward multisource nonideal data. IEEE ASME Trans. Mechatron. 29(1), 388\u2013399 (2024)","journal-title":"IEEE ASME Trans. Mechatron."},{"issue":"12","key":"306_CR33","doi-asserted-by":"publisher","first-page":"7328","DOI":"10.1109\/TSMC.2022.3151185","volume":"52","author":"X Li","year":"2022","unstructured":"Li, X., Shao, H., Lu, S., Xiang, J., Cai, B.: Highly efficient fault diagnosis of rotating machinery under time-varying speeds using LSISMM and small infrared thermal images. IEEE Trans. Syst. Man Cybern. Syst. 52(12), 7328\u20137340 (2022)","journal-title":"IEEE Trans. Syst. Man Cybern. Syst."},{"key":"306_CR34","doi-asserted-by":"publisher","first-page":"703","DOI":"10.1007\/s12532-018-0137-6","volume":"10","author":"X Li","year":"2018","unstructured":"Li, X., Sun, D.F., Toh, K.C.: QSDPNAL: A two-phase augmented Lagrangian method for convex quadratic semidefinite programming. Math. Program. Comput. 10, 703\u2013743 (2018)","journal-title":"Math. Program. Comput."},{"key":"306_CR35","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechmachtheory.2019.103676","volume":"145","author":"X Li","year":"2020","unstructured":"Li, X., Yang, Y., Pan, H., Cheng, J., Cheng, J.: Non-parallel least squares support matrix machine for rolling bearing fault diagnosis. Mech. Mach. Theory 145, 103676 (2020)","journal-title":"Mech. Mach. Theory"},{"issue":"6","key":"306_CR36","doi-asserted-by":"publisher","first-page":"7726","DOI":"10.1109\/TNNLS.2022.3220551","volume":"35","author":"S Liang","year":"2024","unstructured":"Liang, S., Hang, W., Lei, B., Wang, J., Qin, J., Choi, K.S., Zhang, Y.: Adaptive multimodel knowledge transfer matrix machine for EEG classification. IEEE Trans. Neural Netw. Learn. Syst. 35(6), 7726\u20137739 (2024)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"306_CR37","doi-asserted-by":"publisher","DOI":"10.1016\/j.bspc.2022.103531","volume":"74","author":"S Liang","year":"2022","unstructured":"Liang, S., Hang, W., Yin, M., Shen, H., Wang, Q., Qin, J., Choi, K.S., Zhang, Y.: Deep EEG feature learning via stacking common spatial pattern and support matrix machine. Biomed. Signal Process. Control 74, 103531 (2022)","journal-title":"Biomed. Signal Process. Control"},{"key":"306_CR38","unstructured":"Lin, M., Yuan, Y., Sun, D.F., Toh, K.C.: Adaptive sieving with PPDNA: Generating solution paths of exclusive lasso models. arXiv preprint arXiv:2009.08719 (2020)"},{"key":"306_CR39","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1007\/s10107-010-0437-8","volume":"133","author":"YJ Liu","year":"2012","unstructured":"Liu, Y.J., Sun, D., Toh, K.C.: An implementable proximal point algorithmic framework for nuclear norm minimization. Math. Program. 133, 399\u2013436 (2012)","journal-title":"Math. Program."},{"key":"306_CR40","unstructured":"Luo, L., Xie, Y., Zhang, Z., Li, W.J.: Support matrix machines. In: International Conference on Machine Learning, pp. 938\u2013947. PMLR (2015)"},{"key":"306_CR41","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/0167-6377(88)90047-8","volume":"7","author":"OL Mangasarian","year":"1988","unstructured":"Mangasarian, O.L.: A simple characterization of solution sets of convex programs. Oper. Res. Lett. 7, 21\u201326 (1988)","journal-title":"Oper. Res. Lett."},{"key":"306_CR42","doi-asserted-by":"publisher","first-page":"561","DOI":"10.1007\/s10107-005-0629-9","volume":"104","author":"F Meng","year":"2005","unstructured":"Meng, F., Sun, D.F., Zhao, G.: Semismoothness of solutions to generalized equations and the Moreau-Yosida regularization. Math. Program. 104, 561\u2013581 (2005)","journal-title":"Math. Program."},{"key":"306_CR43","unstructured":"Ogawa, K., Suzuki, Y., Takeuchi, I.: Safe screening of non-support vectors in pathwise SVM computation. In: International Conference on Machine Learning, pp. 1382\u20131390. PMLR (2013)"},{"key":"306_CR44","doi-asserted-by":"publisher","DOI":"10.1016\/j.asoc.2022.109209","volume":"125","author":"H Pan","year":"2022","unstructured":"Pan, H., Sheng, L., Xu, H., Tong, J., Zheng, J., Liu, Q.: Pinball transfer support matrix machine for roller bearing fault diagnosis under limited annotation data. Appl. Soft Comput. 125, 109209 (2022)","journal-title":"Appl. Soft Comput."},{"key":"306_CR45","doi-asserted-by":"publisher","DOI":"10.1016\/j.engappai.2023.105991","volume":"121","author":"H Pan","year":"2023","unstructured":"Pan, H., Sheng, L., Xu, H., Zheng, J., Tong, J., Niu, L.: Deep stacked pinball transfer matrix machine with its application in roller bearing fault diagnosis. Eng. Appl. Artif. Intell. 121, 105991 (2023)","journal-title":"Eng. Appl. Artif. Intell."},{"issue":"1","key":"306_CR46","doi-asserted-by":"publisher","first-page":"854","DOI":"10.1109\/TII.2023.3265525","volume":"20","author":"H Pan","year":"2024","unstructured":"Pan, H., Xu, H., Zheng, J., Shao, H., Tong, J.: A semi-supervised matrixized graph embedding machine for roller bearing fault diagnosis under few-labeled samples. IEEE Trans. Industr. Inform. 20(1), 854\u2013863 (2024)","journal-title":"IEEE Trans. Industr. Inform."},{"key":"306_CR47","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1016\/j.mechmachtheory.2019.05.019","volume":"140","author":"H Pan","year":"2019","unstructured":"Pan, H., Yang, Y., Zheng, J., Li, X., Cheng, J.: A fault diagnosis approach for roller bearing based on symplectic geometry matrix machine. Mech. Mach. Theory 140, 31\u201343 (2019)","journal-title":"Mech. Mach. Theory"},{"issue":"8","key":"306_CR48","doi-asserted-by":"publisher","first-page":"2263","DOI":"10.1109\/TNNLS.2018.2879800","volume":"30","author":"X Pan","year":"2019","unstructured":"Pan, X., Xu, Y.: A novel and safe two-stage screening method for support vector machine. IEEE Trans. Neural Netw. Learn. Syst. 30(8), 2263\u20132274 (2019)","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"issue":"6","key":"306_CR49","doi-asserted-by":"publisher","first-page":"3465","DOI":"10.1137\/090763184","volume":"20","author":"TK Pong","year":"2010","unstructured":"Pong, T.K., Tseng, P., Ji, S., Ye, J.: Trace norm regularization: Reformulations, algorithms, and multi-task learning. SIAM J. Optim. 20(6), 3465\u20133489 (2010)","journal-title":"SIAM J. Optim."},{"issue":"2","key":"306_CR50","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1137\/050624509","volume":"28","author":"H Qi","year":"2006","unstructured":"Qi, H., Sun, D.F.: A quadratically convergent Newton method for computing the nearest correlation matrix. SIAM J. Matrix Anal. Appl. 28(2), 360\u2013385 (2006)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"306_CR51","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1007\/BF01581275","volume":"58","author":"L Qi","year":"1993","unstructured":"Qi, L., Sun, J.: A nonsmooth version of Newton\u2019s method. Math. Program. 58, 353\u2013367 (1993)","journal-title":"Math. Program."},{"key":"306_CR52","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1007\/s10107-019-01386-z","volume":"176","author":"C Qian","year":"2019","unstructured":"Qian, C., Tran-Dinh, Q., Fu, S., Zou, C., Liu, Y.: Robust multicategory support matrix machines. Math. Program. 176, 429\u2013463 (2019)","journal-title":"Math. Program."},{"issue":"6","key":"306_CR53","doi-asserted-by":"publisher","first-page":"1117","DOI":"10.1109\/TNSRE.2019.2913142","volume":"27","author":"I Razzak","year":"2019","unstructured":"Razzak, I., Blumenstein, M., Xu, G.: Multiclass support matrix machines by maximizing the inter-class margin for single trial EEG classification. IEEE Trans. Neural Syst. Rehabilitation Eng. 27(6), 1117\u20131127 (2019)","journal-title":"IEEE Trans. Neural Syst. Rehabilitation Eng."},{"key":"306_CR54","doi-asserted-by":"crossref","unstructured":"Robinson, S.M.: Some continuity properties of polyhedral multifunctions. in Mathematical Programming at Oberwolfach, Math. Program. Stud. pp. 206\u2013214 (1981)","DOI":"10.1007\/BFb0120929"},{"key":"306_CR55","doi-asserted-by":"publisher","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"RT Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)"},{"issue":"2","key":"306_CR56","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1287\/moor.1.2.97","volume":"1","author":"RT Rockafellar","year":"1976","unstructured":"Rockafellar, R.T.: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Math. Oper. Res. 1(2), 97\u2013116 (1976)","journal-title":"Math. Oper. Res."},{"key":"306_CR57","unstructured":"Rockafellar, R.T., Wets, R.J.B.: Variational Analysis, vol. 317. Springer Science & Business Media (2009)"},{"key":"306_CR58","unstructured":"Sanei, S., Chambers, J.A.: EEG Signal Processing. John Wiley & Sons (2013)"},{"issue":"1","key":"306_CR59","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/s10107-010-0420-4","volume":"127","author":"S Shalev-Shwartz","year":"2011","unstructured":"Shalev-Shwartz, S., Singer, Y., Srebro, N., Cotter, A.: Pegasos: primal estimated sub-gradient solver for SVM. Math. Program. 127(1), 3\u201330 (2011)","journal-title":"Math. Program."},{"issue":"4","key":"306_CR60","doi-asserted-by":"publisher","first-page":"2554","DOI":"10.1023\/A:1022940300114","volume":"115","author":"A Shapiro","year":"2003","unstructured":"Shapiro, A.: Sensitivity analysis of generalized equations. J. Math. Sci. 115(4), 2554\u20132565 (2003)","journal-title":"J. Math. Sci."},{"key":"306_CR61","unstructured":"Wang, J., Wonka, P., Ye, J.: Scaling SVM and least absolute deviations via exact data reduction. In: International Conference on Machine Learning, pp. 523\u2013531. PMLR (2014)"},{"key":"306_CR62","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1016\/0024-3795(92)90407-2","volume":"170","author":"GA Watson","year":"1992","unstructured":"Watson, G.A.: Characterization of the subdifferential of some matrix norms. Linear Alg. Appl. 170, 33\u201345 (1992)","journal-title":"Linear Alg. Appl."},{"issue":"4","key":"306_CR63","doi-asserted-by":"publisher","first-page":"797","DOI":"10.1287\/ijoc.2023.1282","volume":"35","author":"C Wu","year":"2023","unstructured":"Wu, C., Cui, Y., Li, D., Sun, D.F.: Convex and nonconvex risk-based linear regression at scale. INFORMS J. Comput. 35(4), 797\u2013816 (2023)","journal-title":"INFORMS J. Comput."},{"key":"306_CR64","doi-asserted-by":"publisher","DOI":"10.1016\/j.asoc.2024.111229","volume":"152","author":"H Xu","year":"2024","unstructured":"Xu, H., Pan, H., Zheng, J., Tong, J., Zhang, F., Chu, F.: Intelligent fault identification in sample imbalance scenarios using robust low-rank matrix classifier with fuzzy weighting factor. Appl. Soft Comput. 152, 111229 (2024)","journal-title":"Appl. Soft Comput."},{"key":"306_CR65","doi-asserted-by":"crossref","unstructured":"Yang, J., Zhang, D., Frangi, A.F., Yang, J.y.: Two-dimensional PCA: a new approach to appearance-based face representation and recognition. IEEE Trans. Pattern Anal. Mach. Intell. 26(1), 131\u2013137 (2004)","DOI":"10.1109\/TPAMI.2004.1261097"},{"issue":"3","key":"306_CR66","doi-asserted-by":"publisher","first-page":"2294","DOI":"10.1137\/21M1441080","volume":"32","author":"Y Yuan","year":"2022","unstructured":"Yuan, Y., Chang, T.H., Sun, D.F., Toh, K.C.: A dimension reduction technique for large-scale structured sparse optimization problems with application to convex clustering. SIAM J. Optim. 32(3), 2294\u20132318 (2022)","journal-title":"SIAM J. Optim."},{"key":"306_CR67","doi-asserted-by":"crossref","unstructured":"Yuan, Y., Lin, M., Sun, D.F., Toh, K.C.: Adaptive sieving: A dimension reduction technique for sparse optimization problems. Math. Program. Comput. 17, 585\u2013616 (2025)","DOI":"10.1007\/s12532-025-00282-2"},{"issue":"4","key":"306_CR68","doi-asserted-by":"publisher","first-page":"1737","DOI":"10.1137\/080718206","volume":"20","author":"XY Zhao","year":"2010","unstructured":"Zhao, X.Y., Sun, D.F., Toh, K.C.: A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20(4), 1737\u20131765 (2010)","journal-title":"SIAM J. Optim."},{"issue":"3","key":"306_CR69","doi-asserted-by":"publisher","first-page":"551","DOI":"10.1109\/TNSRE.2018.2794534","volume":"26","author":"Q Zheng","year":"2018","unstructured":"Zheng, Q., Zhu, F., Heng, P.A.: Robust support matrix machine for single trial EEG classification. IEEE Trans. Neural Syst. Rehabilitation Eng. 26(3), 551\u2013562 (2018)","journal-title":"IEEE Trans. Neural Syst. Rehabilitation Eng."},{"key":"306_CR70","doi-asserted-by":"publisher","first-page":"869","DOI":"10.1016\/j.neucom.2017.09.030","volume":"275","author":"Q Zheng","year":"2018","unstructured":"Zheng, Q., Zhu, F., Qin, J., Heng, P.A.: Multiclass support matrix machine for single trial EEG classification. Neurocomputing 275, 869\u2013880 (2018)","journal-title":"Neurocomputing"},{"issue":"2","key":"306_CR71","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1111\/rssb.12031","volume":"76","author":"H Zhou","year":"2014","unstructured":"Zhou, H., Li, L.: Regularized matrix regression. J. R. Stat. Soc. Ser. B Stat Methodol. 76(2), 463\u2013483 (2014)","journal-title":"J. R. Stat. Soc. Ser. B Stat Methodol."},{"key":"306_CR72","doi-asserted-by":"publisher","first-page":"689","DOI":"10.1007\/s10107-016-1100-9","volume":"165","author":"Z Zhou","year":"2017","unstructured":"Zhou, Z., So, A.M.C.: A unified approach to error bounds for structured convex optimization problems. Math. Program. 165, 689\u2013728 (2017)","journal-title":"Math. Program."},{"key":"306_CR73","unstructured":"Zimmert, J., de Witt, C.S., Kerg, G., Kloft, M.: Safe screening for support vector machines. In: NIPS Workshop on Optimization in Machine Learning (OPT) (2015)"}],"container-title":["Mathematical Programming Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12532-026-00306-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12532-026-00306-5","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12532-026-00306-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,18]],"date-time":"2026-08-18T09:02:43Z","timestamp":1787043763000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12532-026-00306-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,4]]},"references-count":73,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,9]]}},"alternative-id":["306"],"URL":"https:\/\/doi.org\/10.1007\/s12532-026-00306-5","relation":{},"ISSN":["1867-2949","1867-2957"],"issn-type":[{"value":"1867-2949","type":"print"},{"value":"1867-2957","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,4]]},"assertion":[{"value":"9 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 February 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 March 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare that they have no conflict of interest.","order":1,"name":"Ethics","label":"Conflicts of Interest","group":{"name":"EthicsHeading","label":"Declarations"}}]}}