{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,17]],"date-time":"2026-10-17T02:45:46Z","timestamp":1792205146870,"version":"4.4.0"},"reference-count":21,"publisher":"Wiley","issue":"8","license":[{"start":{"date-parts":[[2012,11,14]],"date-time":"2012-11-14T00:00:00Z","timestamp":1352851200000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Comm Pure Appl Math"],"published-print":{"date-parts":[[2013,8]]},"abstract":"Abstract<\/jats:title>\n \n Suppose we wish to recover a signal\n \\input amssym $\\font\\abc=cmmib10\\def\\bi#1{\\hbox{\\abc#1}} {\\bi x} \\in {\\Bbb C}^n$<\/jats:styled-content>\n from\n m<\/jats:italic>\n intensity measurements of the form\n \n $\\font\\abc=cmmib10\\def\\bi#1{\\hbox{\\abc#1}} |\\langle \\bi x,\\bi z_i \\rangle|^2$\n \n <\/jats:styled-content>\n ,\n \n $i = 1, 2, \\ldots, m$\n \n <\/jats:styled-content>\n ; that is, from data in which phase information is missing. We prove that if the vectors\n \n $\\font\\abc=cmmib10\\def\\bi#1{\\hbox{\\abc#1}}{\\bi z}_i$\n \n <\/jats:styled-content>\n are sampled independently and uniformly at random on the unit sphere, then the signal\n \n x<\/jats:bold>\n <\/jats:italic>\n can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program\u2013\u2010a trace\u2010norm minimization problem; this holds with large probability provided that\n m<\/jats:italic>\n is on the order of\n \n $n {\\log n}$\n \n <\/jats:styled-content>\n , and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis\u2010\u00e0\u2010vis additive noise. \u00a9 2012 Wiley Periodicals, Inc.\n <\/jats:p>","DOI":"10.1002\/cpa.21432","type":"journal-article","created":{"date-parts":[[2012,11,15]],"date-time":"2012-11-15T10:08:08Z","timestamp":1352974088000},"page":"1241-1274","update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":890,"title":["PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming"],"prefix":"10.1002","volume":"66","author":[{"given":"Emmanuel J.","family":"Cand\u00e8s","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas","family":"Strohmer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vladislav","family":"Voroninski","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2012,11,14]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"crossref","DOI":"10.1117\/12.731117","article-title":"Fast algorithms for signal reconstruction without phase","volume":"6701","author":"R. 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Chai","year":"2011","journal-title":"Inverse Problems"},{"key":"e_1_2_1_12_2","doi-asserted-by":"publisher","DOI":"10.1137\/0707001"},{"issue":"5","key":"e_1_2_1_13_2","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevA.70.052107","article-title":"Pure\u2010state informationally complete and \u201creally\u201d complete measurements","volume":"3","author":"J. Finkelstein","year":"2004","journal-title":"Phys. Rev. A"},{"key":"e_1_2_1_14_2","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2011.2104999"},{"issue":"15","key":"e_1_2_1_15_2","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1103\/PhysRevLett.105.150401","article-title":"Quantum\u2010state tomography via compressed sensing","volume":"105","author":"D. Gross","year":"2010","journal-title":"Phys. Rev. Lett."},{"key":"e_1_2_1_16_2","doi-asserted-by":"crossref","DOI":"10.1007\/978-94-010-9608-9","volume-title":"Phase retrieval and zero crossings","author":"N. E. 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Vershynin","year":"2010"}],"container-title":["Communications on Pure and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fcpa.21432","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/cpa.21432","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,18]],"date-time":"2023-10-18T00:15:45Z","timestamp":1697588145000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/cpa.21432"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,11,14]]},"references-count":21,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2013,8]]}},"alternative-id":["10.1002\/cpa.21432"],"URL":"https:\/\/doi.org\/10.1002\/cpa.21432","archive":["Portico"],"relation":{},"ISSN":["0010-3640","1097-0312"],"issn-type":[{"value":"0010-3640","type":"print"},{"value":"1097-0312","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,11,14]]},"assertion":[{"value":"2011-09-01","order":0,"name":"received","label":"Received","group":{"name":"publication_history","label":"Publication History"}},{"value":"2012-11-14","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}