{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:35:05Z","timestamp":1787330105284,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["IIS-1302231"],"award-info":[{"award-number":["IIS-1302231"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["IIS-1447283"],"award-info":[{"award-number":["IIS-1447283"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>Low-rank approximations to a real matrix $\\mathbf{A}$ can be computed from $\\mathbf{Z}\\mathbf{Z}^T\\mathbf{A}$, where $\\mathbf{Z}$ is a matrix with orthonormal columns, and the accuracy of the approximation can be estimated from some norm of $\\mathbf{A}-\\mathbf{Z}\\mathbf{Z}^T\\mathbf{A}$. We show that computing $\\mathbf{A}-\\mathbf{Z}\\mathbf{Z}^T\\mathbf{A}$ in the two-norm, Frobenius norms, and more generally any Schatten $p$-norm is a well-posed mathematical problem; and, in contrast to dominant subspace computations, it does not require a singular value gap. We also show that this problem is well-conditioned (insensitive) to additive perturbations in $\\mathbf{A}$ and $\\mathbf{Z}$, and to dimension-changing or multiplicative perturbations in $\\mathbf{A}$---regardless of the accuracy of the approximation. For the special case when $\\mathbf{A}$ does indeed have a singular values gap, connections are established between low-rank approximations and subspace angles.<\/jats:p>","DOI":"10.1137\/18m1163658","type":"journal-article","created":{"date-parts":[[2019,2,26]],"date-time":"2019-02-26T15:27:05Z","timestamp":1551194825000},"page":"299-319","source":"Crossref","is-referenced-by-count":20,"title":["Low-Rank Matrix Approximations Do Not Need a Singular Value Gap"],"prefix":"10.1137","volume":"40","author":[{"given":"Petros","family":"Drineas","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5645-5854","authenticated-orcid":true,"given":"Ilse C. F.","family":"Ipsen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,2,26]]},"reference":[{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1145\/1219092.1219097"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"R. Bhatia,\n                      Matrix analysis\n                      , Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997.","DOI":"10.1007\/978-1-4612-0653-8"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1969-12330-X"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/0707001"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/16M1091745"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539704442684"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539704442696"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1145\/2842602"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.ipl.2011.01.010"},{"key":"atypb11","unstructured":"G. H. Golub and C. F. Van Loan,\n                      Matrix Computations\n                      , 4th ed., The Johns Hopkins University Press, Baltimore, 2013."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/130938700"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-015-0716-7"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"N. J. Higham,\n                      Accuracy and Stability of Numerical Algorithms\n                      , 2nd ed., SIAM, Philadelphia, 2002.","DOI":"10.1137\/1.9780898718027"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/130940116"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"R. A. Horn and C. R. Johnson,\n                      Topics in Matrix Analysis\n                      , Cambridge University Press, Cambridge, 1991.","DOI":"10.1017\/CBO9780511840371"},{"key":"atypb17","unstructured":"R. A. Horn and C. R. Johnson,\n                      Matrix Analysis\n                      , 2nd ed., Cambridge University Press, Cambridge, 2013."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(00)00404-0"},{"key":"atypb19","unstructured":"C. Musco and C. Musco,\n                      Randomized block krylov methods for stronger and faster approximate singular value decomposition\n                      , in Advances in Neural Information Processing Systems 28, C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett, eds., Curran Associates, Red Hook, NY, 2015, pp. 1396-1404."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(94)90446-4"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/0717059"},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"Y. Saad,\n                      Numerical Methods for Large Eigenvalue Problems\n                      , revised ed., Classics in Applied Mathematics, SIAM, Philadelphia, 2011.","DOI":"10.1137\/1.9781611970739"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1137\/1015095"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/100792093"},{"key":"atypb25","unstructured":"G. W. Stewart and J.G. Sun,\n                      Matrix Perturbation Theory\n                      , Academic Press, San Diego, 1990."},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"P. \\AA. Wedin,\n                      Perturbation bounds in connection with singular value decomposition\n                      , Nordisk Tidskr. Informationsbehandling (BIT), 12 (1972), pp. 99-111.","DOI":"10.1007\/BF01932678"},{"key":"atypb27","first-page":"263","author":"P.","year":"1983","journal-title":"New York"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1561\/0400000060"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1515\/jnum-2013-0013"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1163658","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:47:45Z","timestamp":1787327265000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1163658"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1163658"],"URL":"https:\/\/doi.org\/10.1137\/18m1163658","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}