{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:34:56Z","timestamp":1787330096182,"version":"build-2736575974"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["ERC-StG-2011-277906"],"award-info":[{"award-number":["ERC-StG-2011-277906"]}],"id":[{"id":"10.13039\/100010663","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>In many applications it is useful to replace the Moore--Penrose pseudoinverse (MPP) by a different generalized inverse with more favorable properties. We may want, for example, to have many zero entries, but without giving up too much of the stability of the MPP. One way to quantify stability is by how much the Frobenius norm of a generalized inverse exceeds that of the MPP. In this paper we derive finite-size concentration bounds for the Frobenius norm of $\\ell^p$-minimal general inverses of iid Gaussian matrices, with $1 \\leq p \\leq 2$. For $p = 1$ we prove exponential concentration of the Frobenius norm of the sparse pseudoinverse; for $p = 2$, we get a similar concentration bound for the MPP. Our proof is based on the convex Gaussian min-max theorem, but unlike previous applications which give asymptotic results, we derive finite-size bounds.<\/jats:p>","DOI":"10.1137\/17m1145409","type":"journal-article","created":{"date-parts":[[2019,1,17]],"date-time":"2019-01-17T11:53:35Z","timestamp":1547726015000},"page":"92-121","source":"Crossref","is-referenced-by-count":5,"title":["Concentration of the Frobenius Norm of Generalized Matrix Inverses"],"prefix":"10.1137","volume":"40","author":[{"given":"Ivan","family":"Dokmani\u0107","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R\u00e9mi","family":"Gribonval","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,1,17]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1093\/imaiai\/iau005"},{"key":"atypb2","unstructured":"A. Barvinok,\n                      Math 710: Measure Concentration\n                      , lecture notes, 2005,http:\/\/www.math.lsa.umich.edu\/~barvinok\/total710.pdf."},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"S. Boyd and L. Vandenberghe,\n                      Convex Optimization\n                      , Cambridge University Press, Cambridge, UK, 2004.","DOI":"10.1017\/CBO9780511804441"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2011.2160521"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1137\/100816900"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1109\/TWC.2003.814350"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/BF02678430"},{"key":"atypb8","unstructured":"I. Dokmanic and R. Gribonval,\n                      Beyond Moore-Penrose Part I: Generalized Inverses that Minimize Matrix norms\n                      , CoRR, abs\/1706.08349, 2017,http:\/\/arxiv.org\/abs\/1706.08349."},{"key":"atypb9","unstructured":"I. Dokmanic and R. Gribonval,\n                      Beyond Moore-Penrose Part II: The Sparse Pseudoinverse\n                      , CoRR, 2017,http:\/\/arxiv.org\/abs\/1706.08701."},{"key":"atypb10","first-page":"6526","author":"Dokmani\u0107 I.","year":"2013","journal-title":"Speech and Signal Proceesing, IEEE"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.2009.0152"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2006.871582"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2013.2293654"},{"key":"atypb14","first-page":"84","author":"Gordon Y.","year":"1988","journal-title":"Berlin"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/BF02759761"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1137\/090771806"},{"key":"atypb17","unstructured":"N. L. Hjort and D. Pollard,\n                      Asymptotics for Minimisers of Convex Processes\n                      ,https:\/\/arxiv.org\/abs\/1107.3806v1, 2011."},{"key":"atypb18","first-page":"1","volume":"439","author":"Krahmer F.","year":"2012","journal-title":"Linear Algebra Appl."},{"key":"atypb19","first-page":"120","author":"Ledoux M.","year":"1999","journal-title":"Berlin"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s00041-012-9243-4"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/S0097539792240406"},{"key":"atypb22","unstructured":"S. Oymak, C. Thrampoulidis, and B. Hassibi,\n                      The Squared-Error of Generalized LASSO: A Precise Analysis\n                      ,https:\/\/arxiv.org\/abs\/1311.0830v2, 2013."},{"key":"atypb23","unstructured":"S. Oymak and J. A. Tropp,\n                      Universality laws for randomized dimension reduction, with applications\n                      , Inform. Inference, (2017), pp. 1-110."},{"key":"atypb24","unstructured":"N. Perraudin, N. Holighaus, P. L. S\u00f8ndergaard, and P. Balazs,\n                      Designing Gabor Windows Using Convex Optimization\n                      ,https:\/\/arxiv.org\/abs\/1401.6033, 2014."},{"key":"atypb25","unstructured":"J. Piot, M. Kolund\u017eija, D. Korchagin, I. Dokmani\u0107, M. Vetterli, and O. Drumm,\n                      Optical touch tomography\n                      , July 28, 2015, US Patent 9,092,091."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2008.168.575"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.20227"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1958.8.171"},{"key":"atypb29","unstructured":"M. Stojnic,\n                      Various Thresholds for $\\ell^1$-Optimization in Compressed Sensing\n                      , CoRR,http:\/\/arxiv.org\/abs\/0907.3666, 2009."},{"key":"atypb30","unstructured":"M. Stojnic,\n                      A Framework to Characterize Performance of LASSO Algorithms\n                      , CoRR,http:\/\/arxiv.org\/abs\/1303.7291, 2013."},{"key":"atypb31","unstructured":"C. Thrampoulidis, E. Abbasi, and B. Hassibi,\n                      Precise Error Analysis of Regularized M-estimators in High-Dimensions\n                      ,https:\/\/arxiv.org\/abs\/1601.06233v1, 2016."},{"key":"atypb32","first-page":"1683","author":"Thrampoulidis C.","year":"2015","journal-title":"France"},{"key":"atypb33","first-page":"210","author":"Vershynin R.","year":"2009","journal-title":"UK"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20429"},{"key":"atypb35","unstructured":"M. Wainwright,\n                      Basic tail and concentration bounds,\n                      draft, 2015,https:\/\/www.stat.berkeley.edu\/~mjwain\/stat210b\/Chap2_TailBounds_Jan22_2015.pdf."},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.2307\/2304460"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1145409","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:47:22Z","timestamp":1787327242000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1145409"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":36,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/17M1145409"],"URL":"https:\/\/doi.org\/10.1137\/17m1145409","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}