{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:38:27Z","timestamp":1760240307099,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,6]],"date-time":"2019-05-06T00:00:00Z","timestamp":1557100800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11401076, 61473328"],"award-info":[{"award-number":["11401076, 61473328"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Random matrices have played an important role in many fields including machine learning, quantum information theory, and optimization. One of the main research focuses is on the deviation inequalities for eigenvalues of random matrices. Although there are intensive studies on the large-deviation inequalities for random matrices, only a few works discuss the small-deviation behavior of random matrices. In this paper, we present the small-deviation inequalities for the largest eigenvalues of sums of random matrices. Since the resulting inequalities are independent of the matrix dimension, they are applicable to high-dimensional and even the infinite-dimensional cases.<\/jats:p>","DOI":"10.3390\/sym11050638","type":"journal-article","created":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T11:22:35Z","timestamp":1557400955000},"page":"638","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Small-Deviation Inequalities for Sums of Random Matrices"],"prefix":"10.3390","volume":"11","author":[{"given":"Xianjie","family":"Gao","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China"}]},{"given":"Chao","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China"}]},{"given":"Hongwei","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,6]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"805","DOI":"10.1007\/s10208-012-9135-7","article-title":"The convex geometry of linear inverse problems","volume":"12","author":"Chandrasekaran","year":"2012","journal-title":"Found. Comput. Math."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"B\u00fchlmann, P., and van de Geer, S. (2011). Statistics for High-Dimensional Data: Methods, Theory and Applications, Springer.","DOI":"10.1007\/978-3-642-20192-9"},{"key":"ref_3","first-page":"3977","article-title":"Revisiting the nystr\u00f6m method for improved large-scale machine learning","volume":"17","author":"Gittens","year":"2016","journal-title":"J. Mach. Learn. Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1137\/090771806","article-title":"Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions","volume":"53","author":"Halko","year":"2011","journal-title":"SIAM Rev."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Clarkson, K.L., and Woodruff, D.P. (2013, January 1\u20134). Low rank approximation and regression in input sparsity time. Proceedings of the forty-fifth annual ACM symposium on Theory of computing, Palo Alto, CA, USA.","DOI":"10.1145\/2488608.2488620"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1109\/18.985947","article-title":"Strong converse for identification via quantum channels","volume":"48","author":"Ahlswede","year":"2002","journal-title":"IEEE Trans. Inf. Theory."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1007\/s10208-011-9099-z","article-title":"User-friendly tail bounds for sums of random matrices","volume":"12","author":"Tropp","year":"2012","journal-title":"Found. Comput. Math."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/ECP.v17-1869","article-title":"Tail inequalities for sums of random matrices that depend on the intrinsic dimension","volume":"17","author":"Hsu","year":"2012","journal-title":"Electron. Commun. Prob."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1016\/j.spl.2017.03.020","article-title":"On some extensions of bernstein\u2019s inequality for self-adjoint operators","volume":"127","author":"Minsker","year":"2017","journal-title":"Stat. Probabil. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1162\/NECO_a_00901","article-title":"Lsv-based tail inequalities for sums of random matrices","volume":"29","author":"Zhang","year":"2017","journal-title":"Neural. Comput."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Milman, V.D., and Schechtman, G. (2007). Deviation inequalities on largest eigenvalues. Geometric Aspects of Functional Analysis, Springer.","DOI":"10.1007\/978-3-540-72053-9"},{"key":"ref_12","unstructured":"Vershynin, R. (2010). Introduction to the non-asymptotic analysis of random matrices. arXiv."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1556","DOI":"10.1214\/aop\/1022677459","article-title":"Approximation, metric entropy and small ball estimates for gaussian measures","volume":"27","author":"Li","year":"1999","journal-title":"Ann. Probab."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1007\/PL00008796","article-title":"Capture time of brownian pursuits","volume":"121","author":"Li","year":"2001","journal-title":"Probab. Theory. Rel."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1023\/A:1022242924198","article-title":"On the link between small ball probabilities and the quantization problem for gaussian measures on banach spaces","volume":"16","author":"Dereich","year":"2003","journal-title":"J. Theor. Probab."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/s11856-006-0007-1","article-title":"Small ball probability and dvoretzky\u2019s theorem","volume":"157","author":"Klartag","year":"2007","journal-title":"ISR J. Math."},{"key":"ref_17","unstructured":"Lifshits, M. (2016, December 15). Bibliography of Small Deviation Probabilities. Available online: https:\/\/www.lpsm.paris\/pageperso\/smalldev\/biblio.pdf."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Aubrun, G. (2005). A sharp small deviation inequality for the largest eigenvalue of a random matrix. S\u00e9minaire de Probabilit\u00e9s XXXVIII, Springer.","DOI":"10.1007\/978-3-540-31449-3_22"},{"key":"ref_19","unstructured":"Rudelson, M., and Vershynin, R. (2010). Non-asymptotic theory of random matrices: extreme singular values. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.spl.2013.09.019","article-title":"Small deviations of the determinants of random matrices with gaussian entries","volume":"84","author":"Volodko","year":"2014","journal-title":"Stat Probabil. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"543","DOI":"10.1137\/0609045","article-title":"Eigenvalues and condition numbers of random matrices","volume":"9","author":"Edelman","year":"1988","journal-title":"SIAM J. Matrix Anal. A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1016\/j.aim.2008.01.010","article-title":"The littlewood\u2013offord problem and invertibility of random matrices","volume":"218","author":"Rudelson","year":"2008","journal-title":"Adv. Math."},{"key":"ref_23","unstructured":"Li, W.V. (2012, March 15). Small Value Probabilities in Analysis and Mathematical Physics. Available online: https:\/\/www.math.arizona.edu\/~mathphys\/school_2012\/\/WenboLi.pdf."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/638\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:49:32Z","timestamp":1760186972000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/11\/5\/638"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,6]]},"references-count":23,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["sym11050638"],"URL":"https:\/\/doi.org\/10.3390\/sym11050638","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2019,5,6]]}}}