{"id":"https://openalex.org/W2962322817","doi":"https://doi.org/10.1007/s10208-020-09479-4","title":"Tensor-Free Proximal Methods for Lifted Bilinear/Quadratic Inverse Problems with Applications to Phase Retrieval","display_name":"Tensor-Free Proximal Methods for Lifted Bilinear/Quadratic Inverse Problems with Applications to Phase Retrieval","publication_year":2021,"publication_date":"2021-01-04","ids":{"openalex":"https://openalex.org/W2962322817","doi":"https://doi.org/10.1007/s10208-020-09479-4","mag":"2962322817"},"language":"en","primary_location":{"id":"doi:10.1007/s10208-020-09479-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-020-09479-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-020-09479-4.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10208-020-09479-4.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Robert Beinert","orcid":null},"institutions":[{"id":"https://openalex.org/I4577782","display_name":"Technische Universit\u00e4t Berlin","ror":"https://ror.org/03v4gjf40","country_code":"DE","type":"education","lineage":["https://openalex.org/I4577782"]},{"id":"https://openalex.org/I15766117","display_name":"University of Graz","ror":"https://ror.org/01faaaf77","country_code":"AT","type":"education","lineage":["https://openalex.org/I15766117"]}],"countries":["AT","DE"],"is_corresponding":true,"raw_author_name":"Robert Beinert","raw_affiliation_strings":["Institut f\u00fcr Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit\u00e4t Graz, Heinrichstra\u00dfe\u00a036, 8010, Graz, Austria","Institut f\u00fcr Mathematik, Technische Universit\u00e4t Berlin, Stra\u00dfe des 17. Juni 136, 10623, Berlin, Germany"],"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit\u00e4t Graz, Heinrichstra\u00dfe\u00a036, 8010, Graz, Austria","institution_ids":["https://openalex.org/I15766117"]},{"raw_affiliation_string":"Institut f\u00fcr Mathematik, Technische Universit\u00e4t Berlin, Stra\u00dfe des 17. Juni 136, 10623, Berlin, Germany","institution_ids":["https://openalex.org/I4577782"]}]},{"author_position":"last","author":{"id":null,"display_name":"Kristian Bredies","orcid":null},"institutions":[{"id":"https://openalex.org/I15766117","display_name":"University of Graz","ror":"https://ror.org/01faaaf77","country_code":"AT","type":"education","lineage":["https://openalex.org/I15766117"]}],"countries":["AT"],"is_corresponding":false,"raw_author_name":"Kristian Bredies","raw_affiliation_strings":["Institut f\u00fcr Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit\u00e4t Graz, Heinrichstra\u00dfe\u00a036, 8010, Graz, Austria"],"affiliations":[{"raw_affiliation_string":"Institut f\u00fcr Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universit\u00e4t Graz, Heinrichstra\u00dfe\u00a036, 8010, Graz, Austria","institution_ids":["https://openalex.org/I15766117"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I15766117","https://openalex.org/I4577782"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":1.6426,"has_fulltext":true,"cited_by_count":6,"citation_normalized_percentile":{"value":0.81830707,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":98},"biblio":{"volume":"21","issue":"5","first_page":"1181","last_page":"1232"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11183","display_name":"Advanced X-ray Imaging Techniques","score":0.9254000186920166,"subfield":{"id":"https://openalex.org/subfields/3108","display_name":"Radiation"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11183","display_name":"Advanced X-ray Imaging Techniques","score":0.9254000186920166,"subfield":{"id":"https://openalex.org/subfields/3108","display_name":"Radiation"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.020600000396370888,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.005400000140070915,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/lanczos-resampling","display_name":"Lanczos resampling","score":0.6098999977111816},{"id":"https://openalex.org/keywords/phase-retrieval","display_name":"Phase retrieval","score":0.5648999810218811},{"id":"https://openalex.org/keywords/singular-value","display_name":"Singular value","score":0.5227000117301941},{"id":"https://openalex.org/keywords/thresholding","display_name":"Thresholding","score":0.4975000023841858},{"id":"https://openalex.org/keywords/solver","display_name":"Solver","score":0.47839999198913574},{"id":"https://openalex.org/keywords/lanczos-algorithm","display_name":"Lanczos algorithm","score":0.4684999883174896},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.4683000147342682},{"id":"https://openalex.org/keywords/bilinear-interpolation","display_name":"Bilinear interpolation","score":0.4553999900817871},{"id":"https://openalex.org/keywords/smoothness","display_name":"Smoothness","score":0.4521999955177307},{"id":"https://openalex.org/keywords/quadratic-equation","display_name":"Quadratic equation","score":0.43880000710487366}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6840000152587891},{"id":"https://openalex.org/C119256216","wikidata":"https://www.wikidata.org/wiki/Q913012","display_name":"Lanczos resampling","level":3,"score":0.6098999977111816},{"id":"https://openalex.org/C81793267","wikidata":"https://www.wikidata.org/wiki/Q7180962","display_name":"Phase retrieval","level":3,"score":0.5648999810218811},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5436000227928162},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.5227000117301941},{"id":"https://openalex.org/C191178318","wikidata":"https://www.wikidata.org/wiki/Q2256906","display_name":"Thresholding","level":3,"score":0.4975000023841858},{"id":"https://openalex.org/C2778770139","wikidata":"https://www.wikidata.org/wiki/Q1966904","display_name":"Solver","level":2,"score":0.47839999198913574},{"id":"https://openalex.org/C20501136","wikidata":"https://www.wikidata.org/wiki/Q366640","display_name":"Lanczos algorithm","level":4,"score":0.4684999883174896},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.4683000147342682},{"id":"https://openalex.org/C205203396","wikidata":"https://www.wikidata.org/wiki/Q612143","display_name":"Bilinear interpolation","level":2,"score":0.4553999900817871},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.4521999955177307},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.43880000710487366},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.43070000410079956},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.43059998750686646},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4066999852657318},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.39739999175071716},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.39309999346733093},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3903000056743622},{"id":"https://openalex.org/C75553542","wikidata":"https://www.wikidata.org/wiki/Q178161","display_name":"A priori and a posteriori","level":2,"score":0.38999998569488525},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.38690000772476196},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.3716999888420105},{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.3709999918937683},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.36970001459121704},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.33880001306533813},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.3188999891281128},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.310699999332428},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.2856999933719635},{"id":"https://openalex.org/C90199385","wikidata":"https://www.wikidata.org/wiki/Q6692777","display_name":"Low-rank approximation","level":3,"score":0.2842000126838684},{"id":"https://openalex.org/C106430172","wikidata":"https://www.wikidata.org/wiki/Q6002272","display_name":"Image restoration","level":4,"score":0.27559998631477356},{"id":"https://openalex.org/C2781089630","wikidata":"https://www.wikidata.org/wiki/Q21856745","display_name":"Realization (probability)","level":2,"score":0.2736000120639801},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.2709999978542328},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.26669999957084656},{"id":"https://openalex.org/C99498987","wikidata":"https://www.wikidata.org/wiki/Q2210247","display_name":"Noise (video)","level":3,"score":0.2628999948501587},{"id":"https://openalex.org/C72169020","wikidata":"https://www.wikidata.org/wiki/Q194404","display_name":"Monotonic function","level":2,"score":0.26159998774528503},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.2581000030040741},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.25769999623298645},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.25360000133514404}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s10208-020-09479-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-020-09479-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-020-09479-4.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1907.04875","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1907.04875","pdf_url":"https://arxiv.org/pdf/1907.04875","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"}],"best_oa_location":{"id":"doi:10.1007/s10208-020-09479-4","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-020-09479-4","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-020-09479-4.pdf","source":{"id":"https://openalex.org/S151639445","display_name":"Foundations of Computational Mathematics","issn_l":"1615-3375","issn":["1615-3375","1615-3383"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Foundations of Computational Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5721630101","display_name":null,"funder_award_id":"P28858","funder_id":"https://openalex.org/F4320321181","funder_display_name":"Austrian Science Fund"}],"funders":[{"id":"https://openalex.org/F4320321181","display_name":"Austrian Science Fund","ror":"https://ror.org/013tf3c58"},{"id":"https://openalex.org/F4320324094","display_name":"Technische Universit\u00e4t Berlin","ror":"https://ror.org/03v4gjf40"},{"id":"https://openalex.org/F4320325582","display_name":"Karl-Franzens-Universit\u00e4t Graz","ror":"https://ror.org/01faaaf77"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2962322817.pdf","grobid_xml":"https://content.openalex.org/works/W2962322817.grobid-xml"},"referenced_works_count":50,"referenced_works":["https://openalex.org/W79676478","https://openalex.org/W613352991","https://openalex.org/W1488291192","https://openalex.org/W1514248469","https://openalex.org/W1542689491","https://openalex.org/W1570089119","https://openalex.org/W1590432342","https://openalex.org/W1968843188","https://openalex.org/W1977361006","https://openalex.org/W1991596255","https://openalex.org/W1991922259","https://openalex.org/W1994217658","https://openalex.org/W2005423095","https://openalex.org/W2006262045","https://openalex.org/W2008232487","https://openalex.org/W2011359124","https://openalex.org/W2014372188","https://openalex.org/W2019569173","https://openalex.org/W2030823150","https://openalex.org/W2037363603","https://openalex.org/W2044613770","https://openalex.org/W2072208551","https://openalex.org/W2078397124","https://openalex.org/W2092663520","https://openalex.org/W2100556411","https://openalex.org/W2103972604","https://openalex.org/W2113972737","https://openalex.org/W2117062784","https://openalex.org/W2118550318","https://openalex.org/W2119058682","https://openalex.org/W2128382396","https://openalex.org/W2130320505","https://openalex.org/W2136328862","https://openalex.org/W2149216076","https://openalex.org/W2151211297","https://openalex.org/W2236527604","https://openalex.org/W2305216423","https://openalex.org/W2405803404","https://openalex.org/W2465002984","https://openalex.org/W2495048266","https://openalex.org/W2611328865","https://openalex.org/W2617270682","https://openalex.org/W2798477839","https://openalex.org/W2912522929","https://openalex.org/W2963841451","https://openalex.org/W2963863416","https://openalex.org/W3014655645","https://openalex.org/W3148325197","https://openalex.org/W4292363360","https://openalex.org/W6605217672"],"related_works":[],"abstract_inverted_index":{"Abstract":[0],"We":[1],"propose":[2],"and":[3,15,27,91,108],"study":[4],"a":[5,20,44,98,141],"class":[6],"of":[7,47,81,111],"novel":[8,99],"algorithms":[9,49,135],"that":[10,145],"aim":[11],"at":[12],"solving":[13],"bilinear":[14],"quadratic":[16],"inverse":[17],"problems.":[18],"Using":[19,97],"convex":[21],"relaxation":[22],"based":[23],"on":[24,64],"tensorial":[25],"lifting,":[26],"applying":[28],"first-order":[29],"proximal":[30],"algorithms,":[31],"these":[32,48],"problems":[33,54],"could":[34],"be":[35,62],"solved":[36],"numerically":[37],"by":[38,87],"singular":[39,83],"value":[40,84],"thresholding":[41,85],"methods.":[42],"However,":[43],"direct":[45],"realization":[46],"for,":[50],"e.g.,":[51],"image":[52],"recovery":[53,130,149],"is":[55,70],"often":[56],"impracticable,":[57],"since":[58],"computations":[59],"have":[60],"to":[61,118,139],"performed":[63],"the":[65,105,112,116,119,133,148],"tensor-product":[66],"space,":[67],"whose":[68],"dimension":[69],"usually":[71],"tremendous.":[72],"To":[73],"overcome":[74],"this":[75],"limitation,":[76],"we":[77,102,126],"derive":[78],"tensor-free":[79,134],"versions":[80],"common":[82],"methods":[86],"exploiting":[88],"low-rank":[89],"representations":[90],"incorporating":[92],"an":[93,128],"augmented":[94],"Lanczos":[95],"process.":[96],"reweighting":[100],"technique,":[101],"further":[103],"improve":[104,147],"convergence":[106],"behavior":[107],"rank":[109],"evolution":[110],"iterative":[113],"algorithms.":[114],"Applying":[115],"method":[117],"two-dimensional":[120],"masked":[121],"Fourier":[122],"phase":[123],"retrieval":[124],"problem,":[125],"obtain":[127],"efficient":[129],"method.":[131],"Moreover,":[132],"are":[136],"flexible":[137],"enough":[138],"incorporate":[140],"priori":[142],"smoothness":[143],"constraints":[144],"greatly":[146],"results.":[150]},"counts_by_year":[{"year":2023,"cited_by_count":4},{"year":2022,"cited_by_count":2}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2019-07-23T00:00:00"}