Sample-Efficient Low Rank Phase Retrieval
This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an <inline-formula> <tex-math notation="LaTeX">n \times q </tex-math></inline-formula> rank-<inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formul...
Saved in:
| Published in: | IEEE transactions on information theory Vol. 67; no. 12; pp. 8190 - 8206 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
New York
IEEE
01.12.2021
The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Subjects: | |
| ISSN: | 0018-9448, 1557-9654 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an <inline-formula> <tex-math notation="LaTeX">n \times q </tex-math></inline-formula> rank-<inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula> matrix <inline-formula> <tex-math notation="LaTeX">{ \boldsymbol {X}^{\ast}} </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">\boldsymbol {y}_{k} = | \boldsymbol {A}_{k}^\top \boldsymbol {x}^{\ast} _{k}| </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">k=1, 2,\ldots, q </tex-math></inline-formula>, when each <inline-formula> <tex-math notation="LaTeX">\boldsymbol {y}_{k} </tex-math></inline-formula> is an m-length vector containing independent phaseless linear projections of <inline-formula> <tex-math notation="LaTeX">\boldsymbol {x}^{\ast}_{k} </tex-math></inline-formula>. Here <inline-formula> <tex-math notation="LaTeX">|.| </tex-math></inline-formula> takes element-wise magnitudes of a vector. The different matrices <inline-formula> <tex-math notation="LaTeX">\boldsymbol {A}_{k} </tex-math></inline-formula> are i.i.d. and each contains i.i.d. standard Gaussian entries. We obtain an improved guarantee for AltMinLowRaP, which is an Alternating Minimization solution to LRPR that was introduced and studied in our recent work. As long as the right singular vectors of <inline-formula> <tex-math notation="LaTeX">{ \boldsymbol {X}^{\ast}} </tex-math></inline-formula> satisfy the incoherence assumption, we can show that the AltMinLowRaP estimate converges geometrically to <inline-formula> <tex-math notation="LaTeX">{ \boldsymbol {X}^{\ast}} </tex-math></inline-formula> if the total number of measurements <inline-formula> <tex-math notation="LaTeX">mq \gtrsim nr^{2} (r + \log (1/\epsilon)) </tex-math></inline-formula>. In addition, we also need <inline-formula> <tex-math notation="LaTeX">m \gtrsim max(r, \log q, \log n) </tex-math></inline-formula> because of the specific asymmetric nature of our problem. Compared to our recent work, we improve the sample complexity of the AltMin iterations by a factor of <inline-formula> <tex-math notation="LaTeX">r^{2} </tex-math></inline-formula>, and that of the initialization by a factor of <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>. We argue, based on comparison with related well-studied problems, why the above sample complexity cannot be improved any further for non-convex solutions to LRPR. We also extend our result to the noisy case; we prove stability to corruption by small additive noise. |
|---|---|
| AbstractList | This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an [Formula Omitted] rank-[Formula Omitted] matrix [Formula Omitted] from [Formula Omitted], [Formula Omitted], when each [Formula Omitted] is an m-length vector containing independent phaseless linear projections of [Formula Omitted]. Here [Formula Omitted] takes element-wise magnitudes of a vector. The different matrices [Formula Omitted] are i.i.d. and each contains i.i.d. standard Gaussian entries. We obtain an improved guarantee for AltMinLowRaP, which is an Alternating Minimization solution to LRPR that was introduced and studied in our recent work. As long as the right singular vectors of [Formula Omitted] satisfy the incoherence assumption, we can show that the AltMinLowRaP estimate converges geometrically to [Formula Omitted] if the total number of measurements [Formula Omitted]. In addition, we also need [Formula Omitted] because of the specific asymmetric nature of our problem. Compared to our recent work, we improve the sample complexity of the AltMin iterations by a factor of [Formula Omitted], and that of the initialization by a factor of [Formula Omitted]. We argue, based on comparison with related well-studied problems, why the above sample complexity cannot be improved any further for non-convex solutions to LRPR. We also extend our result to the noisy case; we prove stability to corruption by small additive noise. This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an <inline-formula> <tex-math notation="LaTeX">n \times q </tex-math></inline-formula> rank-<inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula> matrix <inline-formula> <tex-math notation="LaTeX">{ \boldsymbol {X}^{\ast}} </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">\boldsymbol {y}_{k} = | \boldsymbol {A}_{k}^\top \boldsymbol {x}^{\ast} _{k}| </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">k=1, 2,\ldots, q </tex-math></inline-formula>, when each <inline-formula> <tex-math notation="LaTeX">\boldsymbol {y}_{k} </tex-math></inline-formula> is an m-length vector containing independent phaseless linear projections of <inline-formula> <tex-math notation="LaTeX">\boldsymbol {x}^{\ast}_{k} </tex-math></inline-formula>. Here <inline-formula> <tex-math notation="LaTeX">|.| </tex-math></inline-formula> takes element-wise magnitudes of a vector. The different matrices <inline-formula> <tex-math notation="LaTeX">\boldsymbol {A}_{k} </tex-math></inline-formula> are i.i.d. and each contains i.i.d. standard Gaussian entries. We obtain an improved guarantee for AltMinLowRaP, which is an Alternating Minimization solution to LRPR that was introduced and studied in our recent work. As long as the right singular vectors of <inline-formula> <tex-math notation="LaTeX">{ \boldsymbol {X}^{\ast}} </tex-math></inline-formula> satisfy the incoherence assumption, we can show that the AltMinLowRaP estimate converges geometrically to <inline-formula> <tex-math notation="LaTeX">{ \boldsymbol {X}^{\ast}} </tex-math></inline-formula> if the total number of measurements <inline-formula> <tex-math notation="LaTeX">mq \gtrsim nr^{2} (r + \log (1/\epsilon)) </tex-math></inline-formula>. In addition, we also need <inline-formula> <tex-math notation="LaTeX">m \gtrsim max(r, \log q, \log n) </tex-math></inline-formula> because of the specific asymmetric nature of our problem. Compared to our recent work, we improve the sample complexity of the AltMin iterations by a factor of <inline-formula> <tex-math notation="LaTeX">r^{2} </tex-math></inline-formula>, and that of the initialization by a factor of <inline-formula> <tex-math notation="LaTeX">r </tex-math></inline-formula>. We argue, based on comparison with related well-studied problems, why the above sample complexity cannot be improved any further for non-convex solutions to LRPR. We also extend our result to the noisy case; we prove stability to corruption by small additive noise. |
| Author | Nayer, Seyedehsara Vaswani, Namrata |
| Author_xml | – sequence: 1 givenname: Seyedehsara orcidid: 0000-0002-3042-1186 surname: Nayer fullname: Nayer, Seyedehsara organization: Department of Electrical and Computer Engineeing, Iowa State University, Ames, IA, USA – sequence: 2 givenname: Namrata orcidid: 0000-0003-2774-0650 surname: Vaswani fullname: Vaswani, Namrata email: namrata@iastate.edu organization: Department of Electrical and Computer Engineeing, Iowa State University, Ames, IA, USA |
| BookMark | eNp9kEFLAzEQhYNUsFbvgpcFTx62ZrLJJjlKqVooKLWeQ8xOMHW7W7Op4r93S4sHD56GB-97A98pGTRtg4RcAB0DUH2znC3HjDIYFwBMUXFEhiCEzHUp-IAMKQWVa87VCTntulUfuQA2JNfPdr2pMZ96H1zAJmXz9itb2OY9e3qzHWYLTDHgp63PyLG3dYfnhzsiL3fT5eQhnz_ezya389wxDSmHqvKoSlcCk9JxBopbhii4K8DRV-lkJax1rKqorqhzCrBSjEvtNPfMYzEiV_vdTWw_ttgls2q3selfGia07gep5n2r3LdcbLsuojcuJJtC26RoQ22Amp0W02sxOy3moKUH6R9wE8Paxu__kMs9EhDxt65FIRUtix9w4m29 |
| CODEN | IETTAW |
| CitedBy_id | crossref_primary_10_1109_TIT_2024_3442211 crossref_primary_10_1109_TCI_2023_3263810 crossref_primary_10_1109_TIT_2022_3212374 crossref_primary_10_1109_TIT_2025_3563450 crossref_primary_10_1109_TSP_2022_3208430 |
| Cites_doi | 10.1109/TIT.2020.3016183 10.1109/TSP.2017.2684758 10.1109/TSP.2017.2656844 10.1109/TIT.2010.2046205 10.1109/TIT.2020.2984478 10.1214/16-AOS1443 10.1007/s10208-011-9099-z 10.1109/TIT.2019.2891653 10.1002/cpa.21432 10.1214/10-AOS850 10.1109/TCOMM.2009.04.070065 10.1109/TIT.2019.2902924 10.1137/120893707 10.1017/CBO9780511794308.006 10.1109/TCI.2019.2948758 10.1145/2488608.2488693 10.1109/TIT.2018.2800663 10.1007/s10208-009-9045-5 10.1109/ICIP.2012.6467015 10.1109/TIT.2015.2399924 |
| ContentType | Journal Article |
| Copyright | Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2021 |
| Copyright_xml | – notice: Copyright The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 2021 |
| DBID | 97E RIA RIE AAYXX CITATION 7SC 7SP 8FD JQ2 L7M L~C L~D |
| DOI | 10.1109/TIT.2021.3112805 |
| DatabaseName | IEEE All-Society Periodicals Package (ASPP) 2005–Present IEEE All-Society Periodicals Package (ASPP) 1998–Present IEEE Electronic Library (IEL) CrossRef Computer and Information Systems Abstracts Electronics & Communications Abstracts Technology Research Database ProQuest Computer Science Collection Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Academic Computer and Information Systems Abstracts Professional |
| DatabaseTitle | CrossRef Technology Research Database Computer and Information Systems Abstracts – Academic Electronics & Communications Abstracts ProQuest Computer Science Collection Computer and Information Systems Abstracts Advanced Technologies Database with Aerospace Computer and Information Systems Abstracts Professional |
| DatabaseTitleList | Technology Research Database |
| Database_xml | – sequence: 1 dbid: RIE name: IEEE url: https://ieeexplore.ieee.org/ sourceTypes: Publisher |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering Computer Science |
| EISSN | 1557-9654 |
| EndPage | 8206 |
| ExternalDocumentID | 10_1109_TIT_2021_3112805 9537806 |
| Genre | orig-research |
| GrantInformation_xml | – fundername: NSF grantid: CIF-1815101; CIF-2115200 funderid: 10.13039/100000001 |
| GroupedDBID | -~X .DC 0R~ 29I 3EH 4.4 5GY 5VS 6IK 97E AAJGR AARMG AASAJ AAWTH ABAZT ABFSI ABQJQ ABVLG ACGFO ACGFS ACGOD ACIWK AENEX AETEA AETIX AGQYO AGSQL AHBIQ AI. AIBXA AKJIK AKQYR ALLEH ALMA_UNASSIGNED_HOLDINGS ASUFR ATWAV BEFXN BFFAM BGNUA BKEBE BPEOZ CS3 DU5 E.L EBS EJD F5P HZ~ H~9 IAAWW IBMZZ ICLAB IDIHD IFIPE IFJZH IPLJI JAVBF LAI M43 MS~ O9- OCL P2P PQQKQ RIA RIE RNS RXW TAE TN5 VH1 VJK AAYXX CITATION 7SC 7SP 8FD JQ2 L7M L~C L~D |
| ID | FETCH-LOGICAL-c291t-1ddfe86c61277c42184a2ee54c31c0b7c7d5aac2dd09d0cc81ed82479c94f2fe3 |
| IEDL.DBID | RIE |
| ISICitedReferencesCount | 9 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000720518300036&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0018-9448 |
| IngestDate | Sun Nov 09 07:08:00 EST 2025 Sat Nov 29 03:31:46 EST 2025 Tue Nov 18 22:32:50 EST 2025 Wed Aug 27 02:27:03 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 12 |
| Language | English |
| License | https://ieeexplore.ieee.org/Xplorehelp/downloads/license-information/IEEE.html https://doi.org/10.15223/policy-029 https://doi.org/10.15223/policy-037 |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c291t-1ddfe86c61277c42184a2ee54c31c0b7c7d5aac2dd09d0cc81ed82479c94f2fe3 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ORCID | 0000-0002-3042-1186 0000-0003-2774-0650 |
| PQID | 2599218094 |
| PQPubID | 36024 |
| PageCount | 17 |
| ParticipantIDs | crossref_primary_10_1109_TIT_2021_3112805 proquest_journals_2599218094 crossref_citationtrail_10_1109_TIT_2021_3112805 ieee_primary_9537806 |
| PublicationCentury | 2000 |
| PublicationDate | 2021-12-01 |
| PublicationDateYYYYMMDD | 2021-12-01 |
| PublicationDate_xml | – month: 12 year: 2021 text: 2021-12-01 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | New York |
| PublicationPlace_xml | – name: New York |
| PublicationTitle | IEEE transactions on information theory |
| PublicationTitleAbbrev | TIT |
| PublicationYear | 2021 |
| Publisher | IEEE The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| Publisher_xml | – name: IEEE – name: The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
| References | yi (ref34) 2016 ref13 ref12 nayer (ref14) 2019 srinivasa (ref23) 2019 ref37 ref15 ref36 ref31 ref30 ref11 ref1 ref17 wang (ref10) 2016 ref16 ref19 ref18 zheng (ref35) 2016 wang (ref7) 2016 nayer (ref33) 2021 chen (ref4) 2015 cherapanamjeri (ref32) 2016 ref26 ref20 chen (ref28) 2020 vershynin (ref29) 2018; 47 ref27 zhang (ref6) 2017; 18 hardt (ref24) 2014 netrapalli (ref2) 2013 ref8 ref9 ref3 krishnamurthy (ref22) 2014 ref5 anaraki (ref21) 2014 jain (ref25) 2015 |
| References_xml | – ident: ref31 doi: 10.1109/TIT.2020.3016183 – ident: ref13 doi: 10.1109/TSP.2017.2684758 – start-page: 2796 year: 2013 ident: ref2 article-title: Phase retrieval using alternating minimization publication-title: Proc Neural Inf Process Syst (NeurIPS) – start-page: 4152 year: 2016 ident: ref34 article-title: Fast algorithms for robust PCA via gradient descent publication-title: Proc Adv Neural Inf Process Syst (NeurIPS) – start-page: 4762 year: 2019 ident: ref14 article-title: Phaseless PCA: Low-rank matrix recovery from column-wise phaseless measurements publication-title: Proc Int Conf Mach Learn (ICML) – start-page: 1341 year: 2014 ident: ref21 article-title: Memory and computation efficient PCA via very sparse random projections publication-title: Proc Int Conf Mach Learn (ICML) – ident: ref9 doi: 10.1109/TSP.2017.2656844 – ident: ref18 doi: 10.1109/TIT.2010.2046205 – start-page: 739 year: 2015 ident: ref4 article-title: Solving random quadratic systems of equations is nearly as easy as solving linear systems publication-title: Proc Neural Inf Process Syst (NeurIPS) – ident: ref15 doi: 10.1109/TIT.2020.2984478 – year: 2016 ident: ref10 article-title: Sparse phase retrieval via truncated amplitude flow publication-title: arXiv 1611 07641 – ident: ref11 doi: 10.1214/16-AOS1443 – ident: ref27 doi: 10.1007/s10208-011-9099-z – year: 2016 ident: ref35 article-title: Convergence analysis for rectangular matrix completion using Burer-Monteiro factorization and gradient descent publication-title: arXiv 1605 07051 – ident: ref30 doi: 10.1109/TIT.2019.2891653 – volume: 18 start-page: 5164 year: 2017 ident: ref6 article-title: A nonconvex approach for phase retrieval: Reshaped Wirtinger flow and incremental algorithms publication-title: J Mach Learn Res – start-page: 1007 year: 2015 ident: ref25 article-title: Fast exact matrix completion with finite samples publication-title: Proc Conf Learn Theory – year: 2016 ident: ref7 article-title: Solving systems of random quadratic equations via truncated amplitude flow publication-title: arXiv 1605 08285 – year: 2021 ident: ref33 article-title: Fast and sample-efficient federated low rank matrix recovery from column-wise linear and quadratic projections publication-title: arXiv 2102 10217 – volume: 47 year: 2018 ident: ref29 publication-title: High-Dimensional Probability An Introduction with Applications in Data Science – ident: ref1 doi: 10.1002/cpa.21432 – ident: ref26 doi: 10.1214/10-AOS850 – ident: ref37 doi: 10.1109/TCOMM.2009.04.070065 – start-page: 638 year: 2014 ident: ref24 article-title: Fast matrix completion without the condition number publication-title: Proc Conf Learn Theory – ident: ref12 doi: 10.1109/TIT.2019.2902924 – year: 2020 ident: ref28 article-title: Spectral methods for data science: A statistical perspective publication-title: arXiv 2012 08496 – ident: ref8 doi: 10.1137/120893707 – start-page: 797 year: 2016 ident: ref32 article-title: Nearly optimal robust matrix completion publication-title: Proc ICML – ident: ref36 doi: 10.1017/CBO9780511794308.006 – ident: ref16 doi: 10.1109/TCI.2019.2948758 – ident: ref19 doi: 10.1145/2488608.2488693 – ident: ref5 doi: 10.1109/TIT.2018.2800663 – ident: ref17 doi: 10.1007/s10208-009-9045-5 – ident: ref20 doi: 10.1109/ICIP.2012.6467015 – ident: ref3 doi: 10.1109/TIT.2015.2399924 – year: 2014 ident: ref22 article-title: Subspace learning from extremely compressed measurements publication-title: arXiv 1404 0751 – start-page: 10101 year: 2019 ident: ref23 article-title: Decentralized sketching of low rank matrices publication-title: Proc Neural Inf Process Syst (NeurIPS) |
| SSID | ssj0014512 |
| Score | 2.4833703 |
| Snippet | This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an <inline-formula> <tex-math notation="LaTeX">n \times q </tex-math></inline-formula>... This work studies the Low Rank Phase Retrieval (LRPR) problem: recover an [Formula Omitted] rank-[Formula Omitted] matrix [Formula Omitted] from [Formula... |
| SourceID | proquest crossref ieee |
| SourceType | Aggregation Database Enrichment Source Index Database Publisher |
| StartPage | 8190 |
| SubjectTerms | Complexity Complexity theory compressive sensing Extraterrestrial measurements Incoherence low rank matrix recovery Low-rank Mathematical analysis Matrix algebra Noise measurement Phase measurement Phase retrieval phase retrieval (PR) Principal component analysis Stability analysis |
| Title | Sample-Efficient Low Rank Phase Retrieval |
| URI | https://ieeexplore.ieee.org/document/9537806 https://www.proquest.com/docview/2599218094 |
| Volume | 67 |
| WOSCitedRecordID | wos000720518300036&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVIEE databaseName: IEEE customDbUrl: eissn: 1557-9654 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0014512 issn: 0018-9448 databaseCode: RIE dateStart: 19630101 isFulltext: true titleUrlDefault: https://ieeexplore.ieee.org/ providerName: IEEE |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV3PS8MwGP2Yw4MenG6K0yk9eBlYl2RpkxxFNhRkjDlht5ImKYrSyX7ov2-SdmWgCN56SEJ4X369fsl7AFcmdg4WwrKT2FiCoixhTbFSIVcx5twZirDUm02w0YjPZmJcg-vqLYwxxl8-Mzfu0-fy9Vyt3a-ynoj6jDt97R3G4uKtVpUxoBEulMGxncCWc2xSkkj0pg9TSwQJtvzUrsbOqG5rC_KeKj8WYr-7DBv_69chHJSnyOC2CPsR1EzehMbGoSEoJ2wT9rfkBlvQfZJOCzgceNkI22TwOP8KJjJ_C8YvdjcLJt5ey469Y3geDqZ392FplRAqIvAqxFpnhscWYMKYoo63SWJMRFUfK5QyxXQkpSJaI6GRUhwbzQllQgmakcz0T6Cez3NzCoGMjDCaSoZoRqUNcMxQhlLMUGoPAxlqQ2-DXqJKHXFnZ_GeeD6BRGLxThzeSYl3G7pVjY9CQ-OPsi2Hb1WuhLYNnU2AknKSLRPL3ARx-mP07Pda57Dn2i5un3SgvlqszQXsqs_V63Jx6cfPN5oxwHU |
| linkProvider | IEEE |
| linkToHtml | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwlV3PS8MwGP0YU1APTjfF6dQevAysS7K0aY4ijg3nGHPCbqVNUhSlk_3Qf98k_cFAEbz1kKTlfUm-vCZ5D-BK-cbBgmt24itNUIQmrDEWwg2Ej4PAGIqw2JpNsNEomM34uALX5V0YpZQ9fKZuzKPdy5dzsTa_yjrc67LA6GtveZQSlN3WKvcMqIczbXCsh7BmHcWmJOKd6WCqqSDBmqHq-dhY1W0kIeuq8mMqtvmlV_vflx3Afr6OdG6zwB9CRaV1qBUeDU4-ZOuwtyE42ID2U2TUgN17Kxyhm3SG8y9nEqVvzvhF5zNnYg22dO87gufe_fSu7-ZmCa4gHK9cLGWiAl9DTBgT1DC3iCjlUdHFAsVMMOlFkSBSIi6REAFWMiCUccFpQhLVPYZqOk_VCTiRp7iSNGKIJjTSIfYZSlCMGYr1ciBBTegU6IUiVxI3hhbvoWUUiIca79DgHeZ4N6Fd1vjIVDT-KNsw-Jblcmib0CoCFObDbBlq7saJUSCjp7_XuoSd_vRxGA4Ho4cz2DXvyc6itKC6WqzVOWyLz9XrcnFh-9I3ypHDvA |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Sample-Efficient+Low+Rank+Phase+Retrieval&rft.jtitle=IEEE+transactions+on+information+theory&rft.au=Nayer%2C+Seyedehsara&rft.au=Vaswani%2C+Namrata&rft.date=2021-12-01&rft.issn=0018-9448&rft.eissn=1557-9654&rft.volume=67&rft.issue=12&rft.spage=8190&rft.epage=8206&rft_id=info:doi/10.1109%2FTIT.2021.3112805&rft.externalDBID=n%2Fa&rft.externalDocID=10_1109_TIT_2021_3112805 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0018-9448&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0018-9448&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0018-9448&client=summon |