{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:33:09Z","timestamp":1787333589543,"version":"build-2736575974"},"reference-count":11,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","award":["U01-HL-114494"],"award-info":[{"award-number":["U01-HL-114494"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004663","name":"Ministry of Science and Technology, Taiwan","doi-asserted-by":"publisher","award":["103-2115-M-005-006-MY2"],"award-info":[{"award-number":["103-2115-M-005-006-MY2"]}],"id":[{"id":"10.13039\/501100004663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1413373"],"award-info":[{"award-number":["DMS-1413373"]}],"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":[[2017,1]]},"abstract":"<jats:p>The null vector method, based on a simple linear algebraic concept, is proposed as an initialization method for nonconvex approaches to the phase retrieval problem. For the stylized measurement with random complex Gaussian matrices, a nonasymptotic error bound is derived, stronger than that of the spectral vector method. Numerical experiments show that the null vector method also has a superior performance for the realistic measurement of coded diffraction patterns in coherent diffractive imaging.<\/jats:p>","DOI":"10.1137\/16m1107747","type":"journal-article","created":{"date-parts":[[2017,8,17]],"date-time":"2017-08-17T15:35:17Z","timestamp":1502984117000},"page":"854-868","source":"Crossref","is-referenced-by-count":10,"title":["Phase Retrieval by Linear Algebra"],"prefix":"10.1137","volume":"38","author":[{"given":"Pengwen","family":"Chen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Albert","family":"Fannjiang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gi-Ren","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,8,17]]},"reference":[{"key":"atypb1","unstructured":"E. J. Cand\u00e8s and Y. Chen,\n                      Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems\n                      , preprint,https:\/\/arxiv.org\/abs\/1505.05114, 2015."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2015.2399924"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"P. Chen and A. Fannjiang,\n                      Phase retrieval with a single mask by the Douglas-Rachford algorithm\n                      , Appl. Comput. Harmonic Anal., 2016,https:\/\/doi.org\/10.1016\/j.acha.2016.07.003.","DOI":"10.1016\/j.acha.2016.07.003"},{"key":"atypb4","doi-asserted-by":"crossref","unstructured":"P. Chen, A. Fannjiang, and G. Liu,\n                      Phase retrieval with one or two coded diffraction patterns by alternating projection with the null initialization\n                      , J. Fourier Anal. Appl., 2017,https:\/\/doi.org\/10.1007\/s00041-017-9536-8.","DOI":"10.1007\/s00041-017-9536-8"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/28\/7\/075008"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1364\/AO.21.002758"},{"key":"atypb7","first-page":"237","volume":"35","author":"Gerchberg R. W.","year":"1972","journal-title":"Optik"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2015.2448516"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2014.2352673"},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"J. A. Tropp,\n                      An Introduction to Matrix Concentration Inequalities\n                      , preprint,https:\/\/arxiv.org\/abs\/1501.01571, 2015.","DOI":"10.1561\/9781601988393"},{"key":"atypb11","first-page":"210","author":"Vershynin R.","year":"2012","journal-title":"UK"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1107747","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:11:31Z","timestamp":1787332291000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1107747"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":11,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1107747"],"URL":"https:\/\/doi.org\/10.1137\/16m1107747","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}