{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T04:46:37Z","timestamp":1764996397563,"version":"3.37.3"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T00:00:00Z","timestamp":1651708800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971177","11701186"],"award-info":[{"award-number":["11971177","11701186"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Basic and Applied Basic Research of Guangzhou","award":["202102020283"],"award-info":[{"award-number":["202102020283"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Optim Lett"],"published-print":{"date-parts":[[2023,1]]},"DOI":"10.1007\/s11590-022-01882-w","type":"journal-article","created":{"date-parts":[[2022,5,5]],"date-time":"2022-05-05T12:11:12Z","timestamp":1651752672000},"page":"143-162","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["A low-rank spectral method for learning Markov models"],"prefix":"10.1007","volume":"17","author":[{"given":"Shujun","family":"Bi","sequence":"first","affiliation":[]},{"given":"Zhen","family":"Yin","sequence":"additional","affiliation":[]},{"given":"Yihong","family":"Weng","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,5,5]]},"reference":[{"key":"1882_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1051\/proc:2003004","volume":"13","author":"D Az\u00e9","year":"2003","unstructured":"Az\u00e9, D.: A survey on error bounds for lower semicontinuous functions. Esaim Proc. 13, 1\u201317 (2003)","journal-title":"Esaim Proc."},{"key":"1882_CR2","doi-asserted-by":"crossref","unstructured":"Bencz\u00far, A.A., Csalog\u00e1ny, K., Sarl\u00f3s, T.: On the feasibility of low-rank approximation for personalized pagerank. In: Special Interest Tracks and Posters of the 14th international conference on World Wide Web, pp. 972\u2013973 (2005)","DOI":"10.1145\/1062745.1062824"},{"key":"1882_CR3","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1137\/16M1074023","volume":"59","author":"AR Benson","year":"2017","unstructured":"Benson, A.R., Gleich, D.F., Lim, L.-H.: The spacey random walk: a stochastic process for higher-order data. SIAM Rev. 59, 321\u2013345 (2017)","journal-title":"SIAM Rev."},{"key":"1882_CR4","doi-asserted-by":"publisher","first-page":"336","DOI":"10.1016\/j.orl.2016.03.002","volume":"44","author":"SJ Bi","year":"2016","unstructured":"Bi, S.J., Pan, S.H.: Error bounds for rank constrained optimization problems and applications. Oper. Res. Lett. 44, 336\u2013341 (2016)","journal-title":"Oper. Res. Lett."},{"key":"1882_CR5","doi-asserted-by":"publisher","first-page":"1340","DOI":"10.1137\/0331063","volume":"31","author":"JV Burke","year":"1993","unstructured":"Burke, J.V., Ferris, M.C.: Weak sharp minima in mathematical programming. SIAM J. Control Optim. 31, 1340\u20131359 (1993)","journal-title":"SIAM J. Control Optim."},{"key":"1882_CR6","doi-asserted-by":"publisher","first-page":"1505","DOI":"10.1007\/s10208-020-09490-9","volume":"21","author":"V Charisopoulos","year":"2021","unstructured":"Charisopoulos, V., Chen, Y., Davis, D., D\u00edaz, M., Ding, L., Drusvyatskiy, D.: Low-rank matrix recovery with composite optimization: good conditioning and rapid convergence. Found. Comput. Math. 21, 1505\u20131593 (2021)","journal-title":"Found. Comput. Math."},{"key":"1882_CR7","doi-asserted-by":"publisher","first-page":"2204","DOI":"10.1214\/18-AOS1745","volume":"47","author":"Y Chen","year":"2019","unstructured":"Chen, Y., Fan, J., Ma, C., Wang, K.: Spectral method and regularized MLE are both optimal for top-$$k$$ ranking. Ann. Stat. 47, 2204\u20132235 (2019)","journal-title":"Ann. Stat."},{"key":"1882_CR8","doi-asserted-by":"publisher","first-page":"244","DOI":"10.1137\/18M1220790","volume":"41","author":"YQ Duan","year":"2020","unstructured":"Duan, Y.Q., Wang, M.D., Wen, Z.W., Yuan, Y.X.: Adaptive low-nonnegative-rank approximation for state aggregation of markov chains. SIAM J. Matrix Anal. Appl. 41, 244\u2013278 (2020)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"1882_CR9","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/s10107-017-1159-y","volume":"167","author":"EX Fang","year":"2018","unstructured":"Fang, E.X., Liu, H., Toh, K.-C., Zhou, W.X.: Max-norm optimization for robust matrix recovery. Math. Program. 167, 5\u201335 (2018)","journal-title":"Math. Program."},{"key":"1882_CR10","unstructured":"Hoekstra, A.: On Markov chains of finite rank. PhD thesis, Stichting Mathematisch Centrum, Amsterdam (1983)"},{"key":"1882_CR11","unstructured":"Huang, Q.Q., Kakade, S.M., Kong, W.H., Valiant, G.: Recovering structured probability matrices. arXiv:1602.06586 (2016)"},{"key":"1882_CR12","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1007\/s12532-014-0069-8","volume":"6","author":"KF Jiang","year":"2012","unstructured":"Jiang, K.F., Sun, D.F., Toh, K.-C.: A partial proximal point algorithm for nuclear norm regularized matrix least squares problems with polyhedral constraints. Math. Program. Comput. 6, 281\u2013325 (2012)","journal-title":"Math. Program. Comput."},{"key":"1882_CR13","doi-asserted-by":"publisher","first-page":"282","DOI":"10.3150\/12-BEJ486","volume":"20","author":"O Klopp","year":"2014","unstructured":"Klopp, O.: Noisy low-rank matrix completion with general sampling distribution. Bernoulli 20, 282\u2013303 (2014)","journal-title":"Bernoulli"},{"key":"1882_CR14","doi-asserted-by":"crossref","unstructured":"Levin, D.A., Peres, Y.: Markov Chains and Mixing Times. American Mathematical Society (2017)","DOI":"10.1090\/mbk\/107"},{"key":"1882_CR15","unstructured":"Lewis, A.S., Pang, J.S.: Error Bounds for Convex Inequality Systems. Springer (1996)"},{"issue":"1","key":"1882_CR16","doi-asserted-by":"publisher","first-page":"660","DOI":"10.1137\/18M1224738","volume":"30","author":"X Li","year":"2020","unstructured":"Li, X., Zhu, Z., So, A.M.C., Vidal, R.: Nonconvex robust low-rank matrix recovery. SIAM J. Optim. 30(1), 660\u2013686 (2020)","journal-title":"SIAM J. Optim."},{"key":"1882_CR17","unstructured":"Li, X.D., Wang, M.D., Zhang, A.R.: Estimation of Markov chain via rank-constrained likelihood. In: Proceedings of the 35th International Conference on Machine Learning, pp. 3033\u20133042 (2018)"},{"key":"1882_CR18","doi-asserted-by":"publisher","first-page":"463","DOI":"10.1007\/s10109-012-0166-z","volume":"14","author":"Y Liu","year":"2012","unstructured":"Liu, Y., Kang, C., Gao, S., Xiao, Y., Tian, Y.: Understanding intra-urban trip patterns from taxi trajectory data. J. Geogr. Syst. 14, 463\u2013483 (2012)","journal-title":"J. Geogr. Syst."},{"key":"1882_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01582210","volume":"67","author":"ZQ Luo","year":"1994","unstructured":"Luo, Z.Q., Pang, J.S.: Error bounds for analytic systems and their applications. Math. Program. 67, 1\u201328 (1994)","journal-title":"Math. Program."},{"key":"1882_CR20","doi-asserted-by":"publisher","first-page":"266","DOI":"10.1287\/opre.2016.1534","volume":"65","author":"S Negahban","year":"2016","unstructured":"Negahban, S., Oh, S., Shah, D.: Rank centrality: ranking from pair-wise comparisons. Oper. Res. 65, 266\u2013287 (2016)","journal-title":"Oper. Res."},{"key":"1882_CR21","doi-asserted-by":"crossref","unstructured":"Norris, J.R.: Markov Chains. Cambridge university press (1998)","DOI":"10.1017\/CBO9780511810633"},{"key":"1882_CR22","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1007\/BF02614322","volume":"79","author":"JS Pang","year":"1997","unstructured":"Pang, J.S.: Error bounds in mathematical programming. Math. Program. 79, 299\u2013332 (1997)","journal-title":"Math. Program."},{"key":"1882_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-02431-3","volume-title":"Variational Analysis","author":"RT Rockafellar","year":"1998","unstructured":"Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)"},{"key":"1882_CR24","doi-asserted-by":"publisher","first-page":"3488","DOI":"10.1214\/19-AOS1939","volume":"48","author":"J Sanders","year":"2020","unstructured":"Sanders, J., Proutiere, A., Yun, S.-Y.: Clustering in block markov chains. Ann. Stat. 48, 3488\u20133512 (2020)","journal-title":"Ann. Stat."},{"key":"1882_CR25","unstructured":"Yang, L., Braverman, V., Zhao, T., Wang, M.: Dynamic factorization and partition of complex networks from random walks. arXiv:1705.07881 (2017)"},{"key":"1882_CR26","doi-asserted-by":"publisher","first-page":"3202","DOI":"10.1109\/TIT.2019.2956737","volume":"66","author":"AR Zhang","year":"2020","unstructured":"Zhang, A.R., Wang, M.D.: Spectral state compression of markov processes. IEEE Trans. Inf. Theory 66, 3202\u20133231 (2020)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1882_CR27","doi-asserted-by":"publisher","first-page":"836","DOI":"10.1137\/S105262349834429X","volume":"10","author":"SZ Zhang","year":"2000","unstructured":"Zhang, S.Z.: Global error bounds for convex conic problems. SIAM J. Optim. 10, 836\u2013851 (2000)","journal-title":"SIAM J. Optim."},{"key":"1882_CR28","doi-asserted-by":"publisher","DOI":"10.1287\/opre.2021.2115","author":"ZW Zhu","year":"2021","unstructured":"Zhu, Z.W., Li, X.D., Wang, M.D., Zhang, A.R.: Learning markov models via low-rank optimization. Oper. Res. (2021). https:\/\/doi.org\/10.1287\/opre.2021.2115","journal-title":"Oper. Res."}],"container-title":["Optimization Letters"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-022-01882-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11590-022-01882-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11590-022-01882-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,4]],"date-time":"2023-01-04T10:57:39Z","timestamp":1672829859000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11590-022-01882-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,5]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2023,1]]}},"alternative-id":["1882"],"URL":"https:\/\/doi.org\/10.1007\/s11590-022-01882-w","relation":{},"ISSN":["1862-4472","1862-4480"],"issn-type":[{"type":"print","value":"1862-4472"},{"type":"electronic","value":"1862-4480"}],"subject":[],"published":{"date-parts":[[2022,5,5]]},"assertion":[{"value":"29 May 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 April 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2022","order":3,"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 that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}