{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:41:13Z","timestamp":1787330473646,"version":"build-2736575974"},"reference-count":44,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/501100002661","name":"Fonds De La Recherche Scientifique - FNRS","doi-asserted-by":"publisher","award":["PDR T.0173.13"],"award-info":[{"award-number":["PDR T.0173.13"]}],"id":[{"id":"10.13039\/501100002661","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>A framework, PhaseLift, was recently proposed to solve the phase retrieval problem. In this framework, the problem is solved by optimizing a cost function over the set of complex Hermitian positive semidefinite matrices. This approach to phase retrieval motivates a more general consideration of optimizing cost functions on semidefinite Hermitian matrices where the desired minimizers are known to have low rank. This paper considers an approach based on an alternative cost function defined on a union of appropriate manifolds. It is related to the original cost function in a manner that preserves the ability to find a global minimizer and is significantly more efficient computationally. A rank-based optimality condition for stationary points is given and optimization algorithms based on state-of-the-art Riemannian optimization and dynamically reducing rank are proposed. Empirical evaluations are performed using the PhaseLift problem. The new approach is shown to be an effective method of phase retrieval with computational efficiency increased substantially compared to a state-of-the-art algorithm, the Wirtinger flow algorithm. A preliminary version of this paper can be found in [W. Huang, K. A. Gallivan, and X. Zhang, Procedia Comput. Sci., 80 (2016), pp. 1125--1134].<\/jats:p>","DOI":"10.1137\/16m1072838","type":"journal-article","created":{"date-parts":[[2017,9,21]],"date-time":"2017-09-21T11:42:52Z","timestamp":1505994172000},"page":"B840-B859","source":"Crossref","is-referenced-by-count":25,"title":["Solving PhaseLift by Low-Rank Riemannian Optimization Methods for Complex Semidefinite Constraints"],"prefix":"10.1137","volume":"39","author":[{"given":"Wen","family":"Huang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K. A.","family":"Gallivan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiangxiong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,9,21]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1162\/neco.2008.04-08-749"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1515\/9781400830244"},{"key":"atypb3","unstructured":"C. G. Baker,\n                      Riemannian Manifold Trust-Region Methods with Applications to Eigenproblems\n                      . PhD thesis, Department of Computational Science, Florida State University, 2008."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s12532-011-0029-5"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1107\/S0108767307021903"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.69.064108"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-002-0352-8"},{"key":"atypb8","volume-title":"An Introduction to Differentiable Manifolds and Riemannian Geometry","author":"Boothby W. M.","year":"1986","edition":"2"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/0030-4018(79)90358-4"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1137\/080716542"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511804441"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/110848074"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-013-9162-z"},{"key":"atypb14","first-page":"1985","volume":"64","author":"Cand\u00e9s E. J.","year":"2016","journal-title":"IEEE Trans. Inform. Theory"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.76.064113"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.21432"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s00041-013-9305-2"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1107\/S0108767303002812"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1364\/OL.3.000027"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1364\/AO.21.002758"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1109\/JPROC.2004.840301"},{"key":"atypb22","first-page":"237","volume":"35","author":"Gerchberg R. W.","year":"1972","journal-title":"Optik"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-014-0765-1"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-39929-4_60"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-016-0848-4"},{"key":"atypb26","unstructured":"W. Huang, P.A. Absil, K. A. Gallivan, and P. Hand,\n                      ROPTLIB: An Object-Oriented C++ Library for Optimization on Riemannian Manifolds\n                      , Technical Report FSU16-14, Florida State University, 2016."},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1364\/JOSAA.10.001046"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1109\/TASSP.1982.1163863"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1137\/140955483"},{"key":"atypb30","first-page":"01","author":"Huang W.","year":"2015","journal-title":"UCL-INMA-"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1016\/j.procs.2016.05.422"},{"key":"atypb32","unstructured":"W. Huang,\n                      Optimization Algorithms on Riemannian Manifolds with Applications\n                      , PhD thesis, Department of Mathematics, Florida State University, 2013."},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1137\/080731359"},{"key":"atypb34","first-page":"10","volume":"78","author":"Marchesini S.","year":"2007","journal-title":"Rev. Sci. Instruments"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.physchem.59.032607.093642"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-8853-9"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1137\/11082885X"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1137\/0145038"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-015-9801-1"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1080\/02331934.2013.836650"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1080\/10556789908805762"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1080\/713817747"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-013-0738-9"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1016\/j.neucom.2016.02.030"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1072838","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:07:48Z","timestamp":1787328468000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1072838"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":44,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1072838"],"URL":"https:\/\/doi.org\/10.1137\/16m1072838","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}