{"id":"https://openalex.org/W3106033135","doi":"https://doi.org/10.1007/s10994-021-06037-z","title":"Convex optimization with an interpolation-based projection and its application to deep learning","display_name":"Convex optimization with an interpolation-based projection and its application to deep learning","publication_year":2021,"publication_date":"2021-07-19","ids":{"openalex":"https://openalex.org/W3106033135","doi":"https://doi.org/10.1007/s10994-021-06037-z","mag":"3106033135"},"language":"en","primary_location":{"id":"doi:10.1007/s10994-021-06037-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10994-021-06037-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10994-021-06037-z.pdf","source":{"id":"https://openalex.org/S62148650","display_name":"Machine Learning","issn_l":"0885-6125","issn":["0885-6125","1573-0565"],"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":"Machine Learning","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s10994-021-06037-z.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5075701789","display_name":"Riad Akrour","orcid":"https://orcid.org/0000-0002-8735-6960"},"institutions":[{"id":"https://openalex.org/I31512782","display_name":"Technische Universit\u00e4t Darmstadt","ror":"https://ror.org/05n911h24","country_code":"DE","type":"education","lineage":["https://openalex.org/I31512782"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Riad Akrour","raw_affiliation_strings":["TU Darmstadt, Darmstadt, Germany"],"raw_orcid":"https://orcid.org/0000-0002-8735-6960","affiliations":[{"raw_affiliation_string":"TU Darmstadt, Darmstadt, Germany","institution_ids":["https://openalex.org/I31512782"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5010269091","display_name":"Asma Atamna","orcid":"https://orcid.org/0009-0005-9666-7225"},"institutions":[{"id":"https://openalex.org/I12356871","display_name":"T\u00e9l\u00e9com Paris","ror":"https://ror.org/01naq7912","country_code":"FR","type":"education","lineage":["https://openalex.org/I12356871","https://openalex.org/I205703379","https://openalex.org/I4210145102"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Asma Atamna","raw_affiliation_strings":["T\u00e9l\u00e9com Paris, Paris, France","Telecom Paris,Paris,France"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"T\u00e9l\u00e9com Paris, Paris, France","institution_ids":["https://openalex.org/I12356871"]},{"raw_affiliation_string":"Telecom Paris,Paris,France","institution_ids":["https://openalex.org/I12356871"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5071367253","display_name":"Jan Peters","orcid":"https://orcid.org/0000-0002-5266-8091"},"institutions":[{"id":"https://openalex.org/I31512782","display_name":"Technische Universit\u00e4t Darmstadt","ror":"https://ror.org/05n911h24","country_code":"DE","type":"education","lineage":["https://openalex.org/I31512782"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Jan Peters","raw_affiliation_strings":["TU Darmstadt, Darmstadt, Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"TU Darmstadt, Darmstadt, Germany","institution_ids":["https://openalex.org/I31512782"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5075701789"],"corresponding_institution_ids":["https://openalex.org/I31512782"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.01042186,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"110","issue":"8","first_page":"2267","last_page":"2289"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9987999796867371,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.9980000257492065,"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"}},{"id":"https://openalex.org/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.9980000257492065,"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/projection","display_name":"Projection (relational algebra)","score":0.6201683878898621},{"id":"https://openalex.org/keywords/interpolation","display_name":"Interpolation (computer graphics)","score":0.5738591551780701},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5068619847297668},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.4952665865421295},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.48671820759773254},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.4792892336845398},{"id":"https://openalex.org/keywords/proper-convex-function","display_name":"Proper convex function","score":0.4695107638835907},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"Convex optimization","score":0.44604435563087463},{"id":"https://openalex.org/keywords/convex-set","display_name":"Convex set","score":0.44071465730667114},{"id":"https://openalex.org/keywords/lipschitz-continuity","display_name":"Lipschitz continuity","score":0.4250409007072449},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4039672613143921},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.40110278129577637},{"id":"https://openalex.org/keywords/regular-polygon","display_name":"Regular polygon","score":0.3745304346084595},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3031514883041382},{"id":"https://openalex.org/keywords/image","display_name":"Image (mathematics)","score":0.10559439659118652},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.0727643072605133}],"concepts":[{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.6201683878898621},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.5738591551780701},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5068619847297668},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.4952665865421295},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.48671820759773254},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.4792892336845398},{"id":"https://openalex.org/C50862404","wikidata":"https://www.wikidata.org/wiki/Q3075259","display_name":"Proper convex function","level":5,"score":0.4695107638835907},{"id":"https://openalex.org/C157972887","wikidata":"https://www.wikidata.org/wiki/Q463359","display_name":"Convex optimization","level":3,"score":0.44604435563087463},{"id":"https://openalex.org/C49870271","wikidata":"https://www.wikidata.org/wiki/Q193657","display_name":"Convex set","level":4,"score":0.44071465730667114},{"id":"https://openalex.org/C22324862","wikidata":"https://www.wikidata.org/wiki/Q652707","display_name":"Lipschitz continuity","level":2,"score":0.4250409007072449},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4039672613143921},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.40110278129577637},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.3745304346084595},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3031514883041382},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.10559439659118652},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0727643072605133},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":7,"locations":[{"id":"doi:10.1007/s10994-021-06037-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10994-021-06037-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10994-021-06037-z.pdf","source":{"id":"https://openalex.org/S62148650","display_name":"Machine Learning","issn_l":"0885-6125","issn":["0885-6125","1573-0565"],"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":"Machine Learning","raw_type":"journal-article"},{"id":"pmh:oai:tubiblio.ulb.tu-darmstadt.de:150912","is_oa":false,"landing_page_url":"http://tubiblio.ulb.tu-darmstadt.de/view/person/Akrour=3ARiad=3A=3A.html>","pdf_url":null,"source":{"id":"https://openalex.org/S4377196390","display_name":"TUbilio (Technical University of Darmstadt)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I31512782","host_organization_name":"Technische Universit\u00e4t Darmstadt","host_organization_lineage":["https://openalex.org/I31512782"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"NonPeerReviewed"},{"id":"pmh:oai:arXiv.org:2011.07016","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2011.07016","pdf_url":"https://arxiv.org/pdf/2011.07016","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"},{"id":"mag:3106033135","is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/2011.07016","pdf_url":null,"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":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:oai:tuprints.ulb.tu-darmstadt.de:23526","is_oa":true,"landing_page_url":"https://tuprints.ulb.tu-darmstadt.de/23526/1/s10994-021-06037-z.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306401590","display_name":"Technischen Universit\u00e4t Darmstadt","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I31512782","host_organization_name":"Technische Universit\u00e4t Darmstadt","host_organization_lineage":["https://openalex.org/I31512782"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"info:eu-repo/semantics/article"},{"id":"doi:10.48550/arxiv.2011.07016","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2011.07016","pdf_url":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"doi:10.26083/tuprints-00023526","is_oa":true,"landing_page_url":"https://doi.org/10.26083/tuprints-00023526","pdf_url":null,"source":{"id":"https://openalex.org/S7407051655","display_name":"TUprints","issn_l":null,"issn":[],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.1007/s10994-021-06037-z","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10994-021-06037-z","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10994-021-06037-z.pdf","source":{"id":"https://openalex.org/S62148650","display_name":"Machine Learning","issn_l":"0885-6125","issn":["0885-6125","1573-0565"],"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":"Machine Learning","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320323384","display_name":"Technische Universit\u00e4t Darmstadt","ror":"https://ror.org/05n911h24"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3106033135.pdf","grobid_xml":"https://content.openalex.org/works/W3106033135.grobid-xml"},"referenced_works_count":54,"referenced_works":["https://openalex.org/W1409984952","https://openalex.org/W1632601927","https://openalex.org/W1971290991","https://openalex.org/W2012587148","https://openalex.org/W2046866204","https://openalex.org/W2076680677","https://openalex.org/W2113642685","https://openalex.org/W2119579400","https://openalex.org/W2124477018","https://openalex.org/W2136885855","https://openalex.org/W2143101939","https://openalex.org/W2144037063","https://openalex.org/W2172968643","https://openalex.org/W2178935672","https://openalex.org/W2290452516","https://openalex.org/W2296319761","https://openalex.org/W2592457170","https://openalex.org/W2593658277","https://openalex.org/W2607928667","https://openalex.org/W2611147814","https://openalex.org/W2785315711","https://openalex.org/W2798766386","https://openalex.org/W2803895400","https://openalex.org/W2891122218","https://openalex.org/W2946148574","https://openalex.org/W2949608212","https://openalex.org/W2950583227","https://openalex.org/W2962749646","https://openalex.org/W2963070905","https://openalex.org/W2963326510","https://openalex.org/W2963414638","https://openalex.org/W2963970238","https://openalex.org/W2964121744","https://openalex.org/W2964206659","https://openalex.org/W2968585341","https://openalex.org/W2970791107","https://openalex.org/W2981021779","https://openalex.org/W2985010153","https://openalex.org/W3029645440","https://openalex.org/W3046091798","https://openalex.org/W3105047877","https://openalex.org/W3209581048","https://openalex.org/W4211092921","https://openalex.org/W4230471307","https://openalex.org/W4250589301","https://openalex.org/W4396738453","https://openalex.org/W6628116069","https://openalex.org/W6631190155","https://openalex.org/W6638018090","https://openalex.org/W6653548176","https://openalex.org/W6754302822","https://openalex.org/W6767977373","https://openalex.org/W6780559895","https://openalex.org/W6842097421"],"related_works":["https://openalex.org/W2769430593","https://openalex.org/W2327959774","https://openalex.org/W2029020144","https://openalex.org/W2890123607","https://openalex.org/W2608511404","https://openalex.org/W2984209057","https://openalex.org/W3095525892","https://openalex.org/W2951878358","https://openalex.org/W2755875357","https://openalex.org/W2770728682","https://openalex.org/W2518134372","https://openalex.org/W2558001808","https://openalex.org/W155983061","https://openalex.org/W2053216996","https://openalex.org/W1523958148","https://openalex.org/W2275633728","https://openalex.org/W2170388968","https://openalex.org/W2444301330","https://openalex.org/W2920370333","https://openalex.org/W2998247079"],"abstract_inverted_index":{"Abstract":[0],"Convex":[1],"optimizers":[2],"have":[3],"known":[4],"many":[5],"applications":[6],"as":[7],"differentiable":[8],"layers":[9,19,37],"within":[10],"deep":[11],"neural":[12,49,146],"architectures.":[13],"One":[14],"application":[15],"of":[16,34,46,100,103,137],"these":[17,35],"convex":[18,26,36,118],"is":[20,77],"to":[21,42,67,82,127],"project":[22],"points":[23],"into":[24],"a":[25,47,64,85,149,153],"set.":[27],"However,":[28],"both":[29],"forward":[30],"and":[31,80,106,109,116,119,152],"backward":[32],"passes":[33],"are":[38],"significantly":[39],"more":[40],"expensive":[41],"compute":[43,83],"than":[44],"those":[45],"typical":[48],"network.":[50],"We":[51,90],"investigate":[52],"in":[53,143,148],"this":[54],"paper":[55],"whether":[56],"an":[57,68,73,93],"inexact,":[58],"but":[59],"cheaper":[60],"projection,":[61],"can":[62],"drive":[63],"descent":[65],"algorithm":[66,95],"optimum.":[69],"Specifically,":[70],"we":[71,131],"propose":[72,92],"interpolation-based":[74],"projection":[75,108,140],"that":[76,96],"computationally":[78],"cheap":[79],"easy":[81],"given":[84],"convex,":[86],"domain":[87,121],"defining,":[88],"function.":[89],"then":[91],"optimization":[94],"follows":[97],"the":[98,101,104,107,128,134,138],"gradient":[99],"composition":[102],"objective":[105],"prove":[110],"its":[111],"convergence":[112],"for":[113],"linear":[114],"objectives":[115],"arbitrary":[117],"Lipschitz":[120],"defining":[122],"inequality":[123],"constraints.":[124],"In":[125],"addition":[126],"theoretical":[129],"contributions,":[130],"demonstrate":[132],"empirically":[133],"practical":[135],"interest":[136],"interpolation":[139],"when":[141],"used":[142],"conjunction":[144],"with":[145],"networks":[147],"reinforcement":[150],"learning":[151,155],"supervised":[154],"setting.":[156]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}