{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,18]],"date-time":"2025-10-18T15:15:13Z","timestamp":1760800513361},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T00:00:00Z","timestamp":1638316800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,12,1]],"date-time":"2021-12-01T00:00:00Z","timestamp":1638316800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2021,12]]},"DOI":"10.1007\/s10444-021-09914-2","type":"journal-article","created":{"date-parts":[[2021,12,10]],"date-time":"2021-12-10T05:02:48Z","timestamp":1639112568000},"update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Riemannian optimization for phase retrieval from masked Fourier measurements"],"prefix":"10.1007","volume":"47","author":[{"given":"Huiping","family":"Li","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Song","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,12,10]]},"reference":[{"key":"9914_CR1","doi-asserted-by":"publisher","first-page":"5254","DOI":"10.1214\/17-EJS1378SI","volume":"11","author":"S Bahmani","year":"2017","unstructured":"Bahmani, S., Romberg, J.: A flexible convex relaxation for phase retrieval. Electron. J. Stat. 11, 5254\u20135281 (2017)","journal-title":"Electron. J. Stat."},{"key":"9914_CR2","doi-asserted-by":"publisher","first-page":"488","DOI":"10.1007\/s00041-009-9065-1","volume":"15","author":"R Balan","year":"2009","unstructured":"Balan, R., Bodmann, B. G., Cassazza, P. G., Edidin, D.: Painless reconstruction from magnitudes of frame coefficients. J. Fourier Anal. Appl. 15, 488\u2013501 (2009)","journal-title":"J. Fourier Anal. Appl."},{"key":"9914_CR3","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1093\/imaiai\/iau002","volume":"3","author":"AS Bandeira","year":"2014","unstructured":"Bandeira, A. S., Chen, Y., Mixon, D. G.: Phase retrieval from power spectra of masked signals. Inform. Inference, arXiv: Funct. Anal. 3, 83\u2013102 (2014)","journal-title":"Inform. Inference, arXiv: Funct. Anal."},{"key":"9914_CR4","doi-asserted-by":"publisher","first-page":"106","DOI":"10.1016\/j.acha.2013.10.002","volume":"37","author":"AS Bandeira","year":"2014","unstructured":"Bandeira, A. S., Cahill, J., Mixon, D. G., Nelson, A. A.: Saving phase: injectivity and stability for phase retrieval. Appl. Comput. Harmon. Anal. 37, 106\u2013125 (2014)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"9914_CR5","unstructured":"Cai, J. F., Wei, K.: Solving systems of phaseless equations via Riemannian optimization with optimal sampling complexity. arXiv:1809.02773(2018)"},{"key":"9914_CR6","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1137\/110848074","volume":"6","author":"EJ Cand\u00e8s","year":"2013","unstructured":"Cand\u00e8s, E. J., Eldar, Y. C., Strohmer, T., Voroninshi, V.: Phase retrieval via matrix completion. SIAM J. Imaging Sci. 6, 199\u2013225 (2013)","journal-title":"SIAM J. Imaging Sci."},{"key":"9914_CR7","first-page":"1","volume":"149","author":"EJ Cand\u00e8s","year":"2013","unstructured":"Cand\u00e8s, E. J., Li, X.: Solving quadratic equations via phaselift when there are about as many equations as unknowns. Found. Comput. Math. 149, 1\u201310 (2013)","journal-title":"Found. Comput. Math."},{"key":"9914_CR8","doi-asserted-by":"publisher","first-page":"277","DOI":"10.1016\/j.acha.2014.09.004","volume":"39","author":"EJ Cand\u00e8s","year":"2015","unstructured":"Cand\u00e8s, E. J., Li, X., Soltanolkotabi, M.: Phase retrieval from coded diffraction patterns. Appl. Comput. Harmon. Anal. 39, 277\u2013299 (2015)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"9914_CR9","doi-asserted-by":"publisher","first-page":"1985","DOI":"10.1109\/TIT.2015.2399924","volume":"61","author":"EJ Cand\u00e8s","year":"2015","unstructured":"Cand\u00e8s, E. J., Li, X., Soltanolkotabi, M.: Phase retrieval via Wirtinger Flow: theory and algorithms. IEEE Trans. Inform. Theory 61, 1985\u20132007 (2015)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"9914_CR10","doi-asserted-by":"publisher","first-page":"1241","DOI":"10.1002\/cpa.21432","volume":"66","author":"EJ Cand\u00e8s","year":"2013","unstructured":"Cand\u00e8s, E. J., Stromher, T., Voroninshi, V.: Phaselift: exact and stable signal recovery from magnitude measurements via convex programming. Comm. Pure Appl. Math. 66, 1241\u20131274 (2013)","journal-title":"Comm. Pure Appl. Math."},{"key":"9914_CR11","doi-asserted-by":"publisher","first-page":"015005","DOI":"10.1088\/0266-5611\/27\/1\/015005","volume":"27","author":"A Chai","year":"2011","unstructured":"Chai, A., Moscoso, M., Papanicolaou, G.: Array imaging using intensity-only measurements. Inverse Probl. 27, 015005\u2013015020 (2011)","journal-title":"Inverse Probl."},{"key":"9914_CR12","first-page":"739","volume":"70","author":"Y Chen","year":"2015","unstructured":"Chen, Y., Cand\u00e8s, E. J.: Solving random quadratic systems of equations is nearly as easy as solving linear systems. Comm. Pure Appl. Math. 70, 739\u2013747 (2015)","journal-title":"Comm. Pure Appl. Math."},{"key":"9914_CR13","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1016\/S0034-4877(06)80008-X","volume":"57","author":"JV Corbett","year":"2006","unstructured":"Corbett, J. V.: The pauli problem, state reconstruction and quantum-real numbers. Rep. Math. Phys. 57, 53\u201368 (2006)","journal-title":"Rep. Math. Phys."},{"key":"9914_CR14","unstructured":"Dainty, J. C., Fienup, J. R.: Array Imaging Using Intensity-Only Measurements, Phase Retrieval and Image Reconstruction for Astronomy. In: Stark, H. (ed.) Image Recovery: Theory and Application, pp 231\u2013275. Academic Press, San Diego (1987)"},{"key":"9914_CR15","doi-asserted-by":"publisher","first-page":"1847","DOI":"10.1364\/JOSAA.29.001847","volume":"29","author":"A Fannjiang","year":"2012","unstructured":"Fannjiang, A., Liao, W.: Phase retrieval with random phase illumination. J. Opt. Soc. Amer. A 29, 1847\u20131859 (2012)","journal-title":"J. Opt. Soc. Amer. A"},{"key":"9914_CR16","doi-asserted-by":"publisher","first-page":"2758","DOI":"10.1364\/AO.21.002758","volume":"21","author":"JR Fienup","year":"1982","unstructured":"Fienup, J. R.: Phase retrieval algorithms: a comparison. Appl. Opt. 21, 2758\u20132769 (1982)","journal-title":"Appl. Opt."},{"key":"9914_CR17","first-page":"237","volume":"35","author":"RW Gerchberg","year":"1972","unstructured":"Gerchberg, R. W., Saxton, W. O.: Practical algorithm for the determination of phase from image and diffraction plane pictures. Optik 35, 237\u2013246 (1972)","journal-title":"Optik"},{"key":"9914_CR18","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s00041-014-9361-2","volume":"21","author":"D Gross","year":"2015","unstructured":"Gross, D., Krahmer, F., Kueng, R.: A partial derandomization of PhaseLift using spherical designs. J. Fourier Anal. Appl. 21, 229\u2013266 (2015)","journal-title":"J. Fourier Anal. Appl."},{"key":"9914_CR19","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1016\/j.acha.2015.05.004","volume":"42","author":"D Gross","year":"2017","unstructured":"Gross, D., Krahmer, F., Kueng, R.: Improved recovery guarantees for phase retrieval from coded diffraction patterns. Appl. Comput. Harmon. Anal. 42, 37\u201364 (2017)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"9914_CR20","doi-asserted-by":"publisher","first-page":"2675","DOI":"10.1109\/TIT.2018.2800768","volume":"64","author":"T Goldstein","year":"2018","unstructured":"Goldstein, T., Studer, C.: Phasemax: convex phase retrieval via basis pursuit. IEEE Trans. Inform. Theory 64, 2675\u20132689 (2018)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"9914_CR21","doi-asserted-by":"publisher","first-page":"2047","DOI":"10.4310\/CMS.2018.v16.n7.a13","volume":"16","author":"P Hand","year":"2018","unstructured":"Hand, P., Voroninski, V.: An elementary proof of convex phase retrieval in the natural parameter space via the linear program phasemax. Commun. Math. Sci. 16, 2047\u20132051 (2018)","journal-title":"Commun. Math. Sci."},{"key":"9914_CR22","unstructured":"Hand, P., Voroninski, V.: Corruption robust phase retireval via linear programming, arXiv:1611.03547 (2016)"},{"key":"9914_CR23","doi-asserted-by":"publisher","first-page":"1046","DOI":"10.1364\/JOSAA.10.001046","volume":"10","author":"RW Harrison","year":"1993","unstructured":"Harrison, R. W.: Phase problem in crystallography. J. Opt. Soc. Amer. A 10, 1046\u20131055 (1993)","journal-title":"J. Opt. Soc. Amer. A"},{"key":"9914_CR24","doi-asserted-by":"publisher","first-page":"730","DOI":"10.1038\/ncomms1733","volume":"3","author":"MJ Humphry","year":"2012","unstructured":"Humphry, M. J., Kraus, B., Hurst, A. C., Maiden, A. M., Rodenburg, J. M.: Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging. Nat. Commun. 3, 730\u2013730 (2012)","journal-title":"Nat. Commun."},{"key":"9914_CR25","doi-asserted-by":"crossref","unstructured":"Jaganathan, K., Eldar, Y., Hassibi, B.: Phase retrieval with masks using convex optimization. IEEE Int. Symp. Inf. Theory, 1655\u20131659 (2015)","DOI":"10.1109\/ISIT.2015.7282737"},{"key":"9914_CR26","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1109\/TIT.2017.2757520","volume":"64","author":"F Kramher","year":"2018","unstructured":"Kramher, F., Liu, Y.: Phase retrieval without small-ball probability assumptions. IEEE Trans. Inform. Theory 64, 485\u2013500 (2018)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"9914_CR27","doi-asserted-by":"publisher","first-page":"88","DOI":"10.1016\/j.acha.2015.07.007","volume":"42","author":"R Kueng","year":"2014","unstructured":"Kueng, R., Rauhut, H., Terstiege, U.: Low rank matrix recovery from rank one measurements. Appl. Comput. Harmon. Anal. 42, 88\u2013116 (2014)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"9914_CR28","doi-asserted-by":"publisher","first-page":"1051","DOI":"10.3934\/ipi.2021028","volume":"5","author":"HP Li","year":"2021","unstructured":"Li, H. P., Li, S.: Phase retrieval from masked Fourier measurements with noise. Inverse Probl. Imag. 5, 1051\u20131075 (2021)","journal-title":"Inverse Probl. Imag."},{"key":"9914_CR29","doi-asserted-by":"publisher","first-page":"861","DOI":"10.1142\/S0219530519400049","volume":"18","author":"HP Li","year":"2020","unstructured":"Li, H. P., Li, S., Xia, Y.: Phasemax: Stable guarantees from noisy sub-gaussian measurements. Anal. Appl. 18, 861\u2013886 (2020)","journal-title":"Anal. Appl."},{"key":"9914_CR30","doi-asserted-by":"publisher","first-page":"394","DOI":"10.1364\/JOSAA.7.000394","volume":"7","author":"R Millane","year":"1990","unstructured":"Millane, R.: Phase retrieval in crystallography and optics. J. Opt. Soc. Amer. A 7, 394\u2013411 (1990)","journal-title":"J. Opt. Soc. Amer. A"},{"key":"9914_CR31","doi-asserted-by":"publisher","first-page":"4814","DOI":"10.1109\/TSP.2015.2448516","volume":"63","author":"P Netrapalli","year":"2015","unstructured":"Netrapalli, P., Jain, P., Sanghavi, S.: Phase retrieval using alternating minimization. IEEE Trans. Signal Process 63, 4814\u20134826 (2015)","journal-title":"IEEE Trans. Signal Process"},{"key":"9914_CR32","doi-asserted-by":"publisher","first-page":"3663","DOI":"10.1109\/TIT.2017.2693287","volume":"63","author":"R Pedarsani","year":"2017","unstructured":"Pedarsani, R., Yin, D., Lee, K., Ramchandran, K.: Phasecode: fast and efficient compressive phase retrieval based on sparse-graph codes. IEEE Trans. Inform. Theory 63, 3663\u20133691 (2017)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"9914_CR33","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1109\/MSP.2014.2352673","volume":"32","author":"Y Shechtman","year":"2015","unstructured":"Shechtman, Y., Eldar, Y. C., Cohen, O., Chapman, H. N., Miao, J., Segev, M.: Phase retrieval with application to optical imaging. IEEE Signal Process. Mag. 32, 87\u2013109 (2015)","journal-title":"IEEE Signal Process. Mag."},{"key":"9914_CR34","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1093\/imaiai\/iay005","volume":"8","author":"Y Tan","year":"2019","unstructured":"Tan, Y., Vershynin, R.: Phase retrieval via randomized Kaczmarz: theoretical guarantees. Inf. Inference: J. IMA 8, 97\u2013123 (2019)","journal-title":"Inf. Inference: J. IMA"},{"key":"9914_CR35","doi-asserted-by":"publisher","first-page":"1214","DOI":"10.1137\/110845768","volume":"23","author":"B Vandereycken","year":"2013","unstructured":"Vandereycken, B.: Low rank matrix completion by Riemannian optimization. SIAM J. Optim. 23, 1214\u20131236 (2013)","journal-title":"SIAM J. Optim."},{"key":"9914_CR36","doi-asserted-by":"publisher","first-page":"773","DOI":"10.1109\/TIT.2017.2756858","volume":"64","author":"G Wang","year":"2018","unstructured":"Wang, G., Giannakis, G. B., Eldar, Y. C.: Solving systems of random quadratic equations via truncated amplitude flow. IEEE Trans. Inform. Theory 64, 773\u2013794 (2018)","journal-title":"IEEE Trans. Inform. Theory"},{"key":"9914_CR37","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1016\/j.acha.2014.04.001","volume":"37","author":"Y Wang","year":"2014","unstructured":"Wang, Y., Xu, Z. Q.: Phase retrieval for sparse signals. Appl. Comput. Harmon. Anal. 37, 531\u2013544 (2014)","journal-title":"Appl. Comput. Harmon. Anal."},{"key":"9914_CR38","doi-asserted-by":"crossref","unstructured":"Wei, K., Cai, J. F., Chan, T. F., Leung, S.: Guarantees of Riemannian optimization for low rank matrix completion. arXiv:1603.06610(2016)","DOI":"10.1137\/15M1050525"},{"key":"9914_CR39","doi-asserted-by":"publisher","first-page":"1198","DOI":"10.1137\/15M1050525","volume":"37","author":"K Wei","year":"2016","unstructured":"Wei, K., Cai, J. F., Chan, T. F., Leung, S.: Guarantees of Riemannian optimization for low rank matrix recovery. SIAM J. Matrix Anal. Appl 37, 1198\u20131222 (2016)","journal-title":"SIAM J. Matrix Anal. Appl"},{"key":"9914_CR40","unstructured":"Wei, K.: Efficient algorithms for compressed sensing and matrix completion. Doctoral thesis, University of Oxford (2014)"},{"key":"9914_CR41","doi-asserted-by":"publisher","first-page":"125008","DOI":"10.1088\/0266-5611\/31\/12\/125008","volume":"31","author":"K Wei","year":"2015","unstructured":"Wei, K.: Solving systems of phaseless equations via Kaczmarz methods: a proof of concept study. Inverse Probl. 31, 125008\u2013125030 (2015)","journal-title":"Inverse Probl."},{"key":"9914_CR42","doi-asserted-by":"crossref","unstructured":"Zheng, G., Horstmeyer, R., Yang, C.: Wide-field, high-resolution Fourier pty-chographic microscopy. Nat. Photon. 7, 739\u2013745. liographystyleieee (2013)","DOI":"10.1038\/nphoton.2013.187"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-021-09914-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-021-09914-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-021-09914-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,22]],"date-time":"2021-12-22T04:26:18Z","timestamp":1640147178000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-021-09914-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,12]]},"references-count":42,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2021,12]]}},"alternative-id":["9914"],"URL":"https:\/\/doi.org\/10.1007\/s10444-021-09914-2","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"type":"print","value":"1019-7168"},{"type":"electronic","value":"1572-9044"}],"subject":[],"published":{"date-parts":[[2021,12]]},"assertion":[{"value":"12 January 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 November 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 December 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}],"article-number":"88"}}