{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T01:32:41Z","timestamp":1772501561041,"version":"3.50.1"},"reference-count":51,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T00:00:00Z","timestamp":1685577600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T00:00:00Z","timestamp":1685577600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2023,11]]},"DOI":"10.1007\/s11075-023-01540-x","type":"journal-article","created":{"date-parts":[[2023,6,1]],"date-time":"2023-06-01T07:02:41Z","timestamp":1685602961000},"page":"1403-1420","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Randomized Kaczmarz method with adaptive stepsizes for inconsistent linear systems"],"prefix":"10.1007","volume":"94","author":[{"given":"Yun","family":"Zeng","sequence":"first","affiliation":[]},{"given":"Deren","family":"Han","sequence":"additional","affiliation":[]},{"given":"Yansheng","family":"Su","sequence":"additional","affiliation":[]},{"given":"Jiaxin","family":"Xie","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,1]]},"reference":[{"issue":"1","key":"1540_CR1","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1088\/0266-5611\/20\/1\/006","volume":"20","author":"C Byrne","year":"2003","unstructured":"Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 20(1), 103\u2013120 (2003)","journal-title":"Inverse Problems"},{"issue":"1","key":"1540_CR2","first-page":"7204","volume":"18","author":"A Patrascu","year":"2017","unstructured":"Patrascu, A., Necoara, I.: Nonasymptotic convergence of stochastic proximal point methods for constrained convex optimization. J. Mach. Learn. Res. 18(1), 7204\u20137245 (2017)","journal-title":"J. Mach. Learn. Res."},{"issue":"7","key":"1540_CR3","first-page":"1369","volume":"9","author":"K-W Chang","year":"2008","unstructured":"Chang, K.-W., Hsieh, C.-J., Lin, C.-J.: Coordinate descent method for large-scale l2-loss linear support vector machines. J. Mach. Learn. Res. 9(7), 1369\u20131398 (2008)","journal-title":"J. Mach. Learn. Res."},{"key":"1540_CR4","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611973464","volume-title":"Iterative Methods for Linear Systems: Theory and Applications","author":"EE Olshanskii","year":"2014","unstructured":"Olshanskii, E.E., Tyrtyshnikov, M.A.: Iterative Methods for Linear Systems: Theory and Applications. SIAM, Philadelphia, PA (2014)"},{"key":"1540_CR5","unstructured":"Karczmarz, S.: Angen\u00e4herte aufl\u00f6sung von systemen linearer glei-chungen. Bull. Int. Acad. Pol. Sic. Let., Cl. Sci. Math. Nat. 355\u2013357 (1937)"},{"issue":"3","key":"1540_CR6","doi-asserted-by":"publisher","first-page":"600","DOI":"10.1109\/42.241889","volume":"12","author":"GT Herman","year":"1993","unstructured":"Herman, G.T., Meyer, L.B.: Algebraic reconstruction techniques can be made computationally efficient (positron emission tomography application). IEEE Trans. Medical Imaging 12(3), 600\u2013609 (1993)","journal-title":"IEEE Trans. Medical Imaging"},{"issue":"3","key":"1540_CR7","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1016\/0022-5193(70)90109-8","volume":"29","author":"R Gordon","year":"1970","unstructured":"Gordon, R., Bender, R., Herman, G.T.: Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography. J. Theor. Biol. 29(3), 471\u2013481 (1970)","journal-title":"J. Theor. Biol."},{"key":"1540_CR8","doi-asserted-by":"crossref","unstructured":"Natterer, F.: The Mathematics of Computerized Tomography. SIAM (2001)","DOI":"10.1137\/1.9780898719284"},{"key":"1540_CR9","doi-asserted-by":"crossref","unstructured":"Feichtinger, H.G., Cenker, C., Mayer, M., Steier, H., Strohmer, T.: New variants of the POCS method using affine subspaces of finite codimension with applications to irregular sampling. In: Visual Communications and Image Processing\u201992, vol. 1818, pp. 299\u2013310 (1992). SPIE","DOI":"10.1117\/12.131447"},{"issue":"2","key":"1540_CR10","doi-asserted-by":"publisher","first-page":"262","DOI":"10.1007\/s00041-008-9030-4","volume":"15","author":"T Strohmer","year":"2009","unstructured":"Strohmer, T., Vershynin, R.: A randomized Kaczmarz algorithm with exponential convergence. J. Fourier Anal. Appl. 15(2), 262\u2013278 (2009)","journal-title":"J. Fourier Anal. Appl."},{"issue":"297","key":"1540_CR11","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1090\/mcom\/2971","volume":"85","author":"J Liu","year":"2016","unstructured":"Liu, J., Wright, S.: An accelerated randomized Kaczmarz algorithm. Math. Comp. 85(297), 153\u2013178 (2016)","journal-title":"Math. Comp."},{"key":"1540_CR12","unstructured":"Han, D., Xie, J.: On pseudoinverse-free randomized methods for linear systems: Unified framework and acceleration. Preprint at http:\/\/arxiv.org\/abs\/2208.05437 (2022)"},{"issue":"3","key":"1540_CR13","doi-asserted-by":"publisher","first-page":"653","DOI":"10.1007\/s10589-020-00220-z","volume":"77","author":"N Loizou","year":"2020","unstructured":"Loizou, N., Richt\u00e1rik, P.: Momentum and stochastic momentum for stochastic gradient, newton, proximal point and subspace descent methods. Comput. Optim. Appl. 77(3), 653\u2013710 (2020)","journal-title":"Comput. Optim. Appl."},{"key":"1540_CR14","doi-asserted-by":"crossref","unstructured":"Necoara, I.: Faster randomized block Kaczmarz algorithms. SIAM J. Matrix Anal. Appl. 40(4), 1425\u20131452 (2019)","DOI":"10.1137\/19M1251643"},{"key":"1540_CR15","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1016\/j.laa.2012.12.022","volume":"441","author":"D Needell","year":"2014","unstructured":"Needell, D., Tropp, J.A.: Paved with good intentions: analysis of a randomized block kaczmarz method. Linear Algebra Appl. 441, 199\u2013221 (2014)","journal-title":"Linear Algebra Appl."},{"issue":"1","key":"1540_CR16","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/s10543-020-00824-1","volume":"61","author":"JD Moorman","year":"2021","unstructured":"Moorman, J.D., Tu, T.K., Molitor, D., Needell, D.: Randomized Kaczmarz with averaging. BIT. 61(1), 337\u2013359 (2021)","journal-title":"BIT."},{"issue":"4","key":"1540_CR17","doi-asserted-by":"publisher","first-page":"1660","DOI":"10.1137\/15M1025487","volume":"36","author":"RM Gower","year":"2015","unstructured":"Gower, R.M., Richt\u00e1rik, P.: Randomized iterative methods for linear systems. SIAM J. Matrix Anal. Appl. 36(4), 1660\u20131690 (2015)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"1","key":"1540_CR18","doi-asserted-by":"publisher","first-page":"592","DOI":"10.1137\/17M1137747","volume":"40","author":"Z-Z Bai","year":"2018","unstructured":"Bai, Z.-Z., Wu, W.-T.: On greedy randomized Kaczmarz method for solving large sparse linear systems. SIAM J. Sci. Comput. 40(1), 592\u2013606 (2018)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"1540_CR19","doi-asserted-by":"publisher","first-page":"954","DOI":"10.1137\/19M1285846","volume":"42","author":"RM Gower","year":"2021","unstructured":"Gower, R.M., Molitor, D., Moorman, J., Needell, D.: On adaptive sketch-and-project for solving linear systems. SIAM J. Matrix Anal. Appl. 42(2), 954\u2013989 (2021)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"1","key":"1540_CR20","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1007\/s10107-017-1229-1","volume":"173","author":"F Sch\u00f6pfer","year":"2019","unstructured":"Sch\u00f6pfer, F., Lorenz, D.A.: Linear convergence of the randomized sparse Kaczmarz method. Math. Program. 173(1), 509\u2013536 (2019)","journal-title":"Math. Program."},{"issue":"4","key":"1540_CR21","doi-asserted-by":"publisher","first-page":"1439","DOI":"10.1137\/21M1398562","volume":"14","author":"X Chen","year":"2021","unstructured":"Chen, X., Qin, J.: Regularized Kaczmarz algorithms for tensor recovery. SIAM J. Imaging Sci. 14(4), 1439\u20131471 (2021)","journal-title":"SIAM J. Imaging Sci."},{"issue":"2","key":"1540_CR22","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1007\/s10543-010-0265-5","volume":"50","author":"D Needell","year":"2010","unstructured":"Needell, D.: Randomized Kaczmarz solver for noisy linear systems. BIT. 50(2), 395\u2013403 (2010)","journal-title":"BIT."},{"issue":"6","key":"1540_CR23","doi-asserted-by":"publisher","first-page":"3892","DOI":"10.1137\/20M1352235","volume":"43","author":"Z-Z Bai","year":"2021","unstructured":"Bai, Z.-Z., Wu, W.-T.: On greedy randomized augmented Kaczmarz method for solving large sparse inconsistent linear systems. SIAM J. Sci. Comput. 43(6), 3892\u20133911 (2021)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1540_CR24","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/BF01396307","volume":"41","author":"Y Censor","year":"1983","unstructured":"Censor, Y., Eggermont, P.P., Gordon, D.: Strong underrelaxation in Kaczmarz\u2019s method for inconsistent systems. Numer. Math. 41(1), 83\u201392 (1983)","journal-title":"Numer. Math."},{"key":"1540_CR25","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1016\/0024-3795(90)90207-S","volume":"130","author":"M Hanke","year":"1990","unstructured":"Hanke, M., Niethammer, W.: On the acceleration of Kaczmarz\u2019s method for inconsistent linear systems. Linear Algebra Appl. 130, 83\u201398 (1990)","journal-title":"Linear Algebra Appl."},{"issue":"2","key":"1540_CR26","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1137\/120889897","volume":"34","author":"A Zouzias","year":"2013","unstructured":"Zouzias, A., Freris, N.M.: Randomized extended Kaczmarz for solving least squares. SIAM J. Matrix Anal. Appl. 34(2), 773\u2013793 (2013)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"1540_CR27","doi-asserted-by":"publisher","first-page":"2233","DOI":"10.1002\/nla.2233","volume":"26","author":"K Du","year":"2019","unstructured":"Du, K.: Tight upper bounds for the convergence of the randomized extended Kaczmarz and Gauss-Seidel algorithms. Numer. Linear Algebra Appl. 26(3), 2233 (2019)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"1","key":"1540_CR28","doi-asserted-by":"publisher","first-page":"2334","DOI":"10.1002\/nla.2334","volume":"28","author":"N-C Wu","year":"2021","unstructured":"Wu, N.-C., Xiang, H.: Semiconvergence analysis of the randomized row iterative method and its extended variants. Numer. Linear Algebra Appl. 28(1), 2334 (2021)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"2","key":"1540_CR29","doi-asserted-by":"publisher","first-page":"256","DOI":"10.1007\/s00041-012-9248-z","volume":"19","author":"D Needell","year":"2013","unstructured":"Needell, D., Ward, R.: Two-subspace projection method for coherent overdetermined systems. J. Fourier Anal. Appl. 19(2), 256\u2013269 (2013)","journal-title":"J. Fourier Anal. Appl."},{"key":"1540_CR30","unstructured":"Du, K., Sun, X.-H.: Pseudoinverse-free randomized block iterative algorithms for consistent and inconsistent linear systems. Preprint at http:\/\/arxiv.org\/abs\/2011.10353 (2020)"},{"issue":"6","key":"1540_CR31","doi-asserted-by":"publisher","first-page":"3541","DOI":"10.1137\/20M1312629","volume":"42","author":"K Du","year":"2020","unstructured":"Du, K., Si, W.-T., Sun, X.-H.: Randomized extended average block Kaczmarz for solving least squares. SIAM J. Sci. Comput. 42(6), 3541\u20133559 (2020)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"1540_CR32","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11075-021-01104-x","volume":"89","author":"W-T Wu","year":"2022","unstructured":"Wu, W.-T.: On two-subspace randomized extended Kaczmarz method for solving large linear least-squares problems. Numer. Algorithms 89(1), 1\u201331 (2022)","journal-title":"Numer. Algorithms"},{"key":"1540_CR33","unstructured":"Wu, N.-C., Liu, C., Wang, Y., Zuo, Q.: On the extended randomized multiple row method for solving linear least-squares problems. Preprint at http:\/\/arxiv.org\/abs\/2210.03478 (2022)"},{"issue":"1","key":"1540_CR34","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1007\/BF02510922","volume":"38","author":"C Popa","year":"1998","unstructured":"Popa, C.: Extensions of block-projections methods with relaxation parameters to inconsistent and rank-deficient least-squares problems. BIT. 38(1), 151\u2013176 (1998)","journal-title":"BIT."},{"issue":"1","key":"1540_CR35","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1007\/BF02941906","volume":"6","author":"C Popa","year":"1999","unstructured":"Popa, C.: Characterization of the solutions set of inconsistent least-squares problems by an extended Kaczmarz algorithm. Korean J. Comput. Appl. Math. 6(1), 51\u201364 (1999)","journal-title":"Korean J. Comput. Appl. Math."},{"key":"1540_CR36","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1016\/j.laa.2019.05.005","volume":"578","author":"Z-Z Bai","year":"2019","unstructured":"Bai, Z.-Z., Wu, W.-T.: On partially randomized extended Kaczmarz method for solving large sparse overdetermined inconsistent linear systems. Linear Algebra Appl. 578, 225\u2013250 (2019)","journal-title":"Linear Algebra Appl."},{"issue":"4","key":"1540_CR37","doi-asserted-by":"publisher","first-page":"1590","DOI":"10.1137\/15M1014425","volume":"36","author":"A Ma","year":"2015","unstructured":"Ma, A., Needell, D., Ramdas, A.: Convergence properties of the randomized extended Gauss-Seidel and Kaczmarz methods. SIAM J. Matrix Anal. Appl. 36(4), 1590\u20131604 (2015)","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"3","key":"1540_CR38","doi-asserted-by":"publisher","first-page":"641","DOI":"10.1287\/moor.1100.0456","volume":"35","author":"D Leventhal","year":"2010","unstructured":"Leventhal, D., Lewis, A.S.: Randomized methods for linear constraints: convergence rates and conditioning. Math. Oper. Res. 35(3), 641\u2013654 (2010)","journal-title":"Math. Oper. Res."},{"key":"1540_CR39","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1016\/j.laa.2020.10.028","volume":"611","author":"Z-Z Bai","year":"2021","unstructured":"Bai, Z.-Z., Wang, L., Wu, W.-T.: On convergence rate of the randomized Gauss-Seidel method. Linear Algebra Appl. 611, 237\u2013252 (2021)","journal-title":"Linear Algebra Appl."},{"key":"1540_CR40","doi-asserted-by":"crossref","unstructured":"Golub, G.H., Van\u00a0Loan, C.F.: Matrix Computations. JHU Press (2013)","DOI":"10.56021\/9781421407944"},{"key":"1540_CR41","unstructured":"Ben-Israel, A., Greville, T.N.: Generalized Inverses: Theory and Applications vol. 15. Springer (2003)"},{"key":"1540_CR42","doi-asserted-by":"publisher","first-page":"549","DOI":"10.1007\/s10107-015-0864-7","volume":"155","author":"D Needell","year":"2016","unstructured":"Needell, D., Srebro, N., Ward, R.: Stochastic gradient descent, weighted sampling, and the randomized Kaczmarz algorithm. Math. Program. 155, 549\u2013573 (2016)","journal-title":"Math. Program."},{"key":"1540_CR43","doi-asserted-by":"publisher","first-page":"2815","DOI":"10.1090\/mcom\/3644","volume":"90","author":"S Steinerberger","year":"2021","unstructured":"Steinerberger, S.: A weighted randomized Kaczmarz method for solving linear systems. Math. Comp. 90, 2815\u20132826 (2021)","journal-title":"Math. Comp."},{"key":"1540_CR44","unstructured":"Deren, H., Su, Y., Xie, J.: Randomized Douglas-Rachford method for linear systems: Improved accuracy and efficiency. Preprint at http:\/\/arxiv.org\/abs\/2207.04291 (2022)"},{"issue":"35","key":"1540_CR45","doi-asserted-by":"publisher","first-page":"1244","DOI":"10.21105\/joss.01244","volume":"4","author":"SP Kolodziej","year":"2019","unstructured":"Kolodziej, S.P., Aznaveh, M., Bullock, M., David, J., Davis, T.A., Henderson, M., Hu, Y., Sandstrom, R.: The suitesparse matrix collection website interface. J. Open Source Softw. 4(35), 1244 (2019)","journal-title":"J. Open Source Softw."},{"key":"1540_CR46","unstructured":"Hardt, M., Recht, B., Singer, Y.: Train faster, generalize better: Stability of stochastic gradient descent. In: Proceedings of the 33rd International Conference on Machine Learning, pp. 1225\u20131234. PMLR (2016)"},{"key":"1540_CR47","doi-asserted-by":"crossref","unstructured":"Robbins, H., Monro, S.: A stochastic approximation method. Ann. Math. Statistics. 400\u2013407 (1951)","DOI":"10.1214\/aoms\/1177729586"},{"issue":"1","key":"1540_CR48","doi-asserted-by":"publisher","first-page":"1","DOI":"10.4208\/nmtma.OA-2018-0066","volume":"12","author":"A Ma","year":"2019","unstructured":"Ma, A., Needell, D.: Stochastic gradient descent for linear systems with missing data. Numer. Math. Theory Methods Appl. 12(1), 1\u201320 (2019)","journal-title":"Numer. Math. Theory Methods Appl."},{"key":"1540_CR49","unstructured":"Nguyen, L.M., Liu, J., Scheinberg, K., Tak\u00e1\u010d, M.: SARAH: A novel method for machine learning problems using stochastic recursive gradient. In: Proceedings of the 34th International Conference on Machine Learning, pp. 2613\u20132621. PMLR (2017)"},{"key":"1540_CR50","unstructured":"Johnson, R., Zhang, T.: Accelerating stochastic gradient descent using predictive variance reduction. In: Proceedings of the Advances in Neural Information Processing Systems, pp. 315\u2013323. (2013)"},{"issue":"4","key":"1540_CR51","doi-asserted-by":"publisher","first-page":"1574","DOI":"10.1137\/070704277","volume":"19","author":"A Nemirovski","year":"2009","unstructured":"Nemirovski, A., Juditsky, A., Lan, G., Shapiro, A.: Robust stochastic approximation approach to stochastic programming. SIAM J. Optim. 19(4), 1574\u20131609 (2009)","journal-title":"SIAM J. Optim."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01540-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-023-01540-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01540-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,10]],"date-time":"2023-10-10T08:10:11Z","timestamp":1696925411000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-023-01540-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,1]]},"references-count":51,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,11]]}},"alternative-id":["1540"],"URL":"https:\/\/doi.org\/10.1007\/s11075-023-01540-x","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,1]]},"assertion":[{"value":"17 February 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 March 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 June 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval and consent to participate"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}