{"id":"https://openalex.org/W4385891694","doi":"https://doi.org/10.1137/22m1516208","title":"Low-Rank Univariate Sum of Squares Has No Spurious Local Minima","display_name":"Low-Rank Univariate Sum of Squares Has No Spurious Local Minima","publication_year":2023,"publication_date":"2023-08-08","ids":{"openalex":"https://openalex.org/W4385891694","doi":"https://doi.org/10.1137/22m1516208"},"language":"en","primary_location":{"id":"doi:10.1137/22m1516208","is_oa":false,"landing_page_url":"http://dx.doi.org/10.1137/22m1516208","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://hdl.handle.net/2078.1/303599","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029044993","display_name":"Beno\u00eet Legat","orcid":"https://orcid.org/0000-0002-5921-368X"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Beno\u00eet Legat","raw_affiliation_strings":["Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA"],"affiliations":[{"raw_affiliation_string":"Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5055547569","display_name":"Chenyang Yuan","orcid":"https://orcid.org/0000-0001-7912-0328"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Chenyang Yuan","raw_affiliation_strings":["Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA"],"affiliations":[{"raw_affiliation_string":"Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA","institution_ids":["https://openalex.org/I63966007"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5031935416","display_name":"Pablo A. Parrilo","orcid":"https://orcid.org/0000-0003-1132-8477"},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Pablo Parrilo","raw_affiliation_strings":["Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA"],"affiliations":[{"raw_affiliation_string":"Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139 USA","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5029044993"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":1.6476,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.81024717,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"33","issue":"3","first_page":"2041","last_page":"2061"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T11435","display_name":"Polynomial and algebraic computation","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T12069","display_name":"Commutative Algebra and Its Applications","score":0.9962999820709229,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8400958776473999},{"id":"https://openalex.org/keywords/univariate","display_name":"Univariate","score":0.5471382141113281},{"id":"https://openalex.org/keywords/maxima-and-minima","display_name":"Maxima and minima","score":0.5302569270133972},{"id":"https://openalex.org/keywords/rounding","display_name":"Rounding","score":0.5210487246513367},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.4970286190509796},{"id":"https://openalex.org/keywords/spurious-relationship","display_name":"Spurious relationship","score":0.4573518633842468},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.44796091318130493},{"id":"https://openalex.org/keywords/least-squares-function-approximation","display_name":"Least-squares function approximation","score":0.4453321099281311},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.42522597312927246},{"id":"https://openalex.org/keywords/factorization","display_name":"Factorization","score":0.42181625962257385},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3596518039703369},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.33989912271499634},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.27615928649902344},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1575266420841217},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.12907087802886963},{"id":"https://openalex.org/keywords/multivariate-statistics","display_name":"Multivariate statistics","score":0.09737610816955566}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8400958776473999},{"id":"https://openalex.org/C199163554","wikidata":"https://www.wikidata.org/wiki/Q1681619","display_name":"Univariate","level":3,"score":0.5471382141113281},{"id":"https://openalex.org/C186633575","wikidata":"https://www.wikidata.org/wiki/Q845060","display_name":"Maxima and minima","level":2,"score":0.5302569270133972},{"id":"https://openalex.org/C136625980","wikidata":"https://www.wikidata.org/wiki/Q663208","display_name":"Rounding","level":2,"score":0.5210487246513367},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.4970286190509796},{"id":"https://openalex.org/C97256817","wikidata":"https://www.wikidata.org/wiki/Q1462316","display_name":"Spurious relationship","level":2,"score":0.4573518633842468},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.44796091318130493},{"id":"https://openalex.org/C9936470","wikidata":"https://www.wikidata.org/wiki/Q6510405","display_name":"Least-squares function approximation","level":3,"score":0.4453321099281311},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.42522597312927246},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.42181625962257385},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3596518039703369},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.33989912271499634},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.27615928649902344},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1575266420841217},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.12907087802886963},{"id":"https://openalex.org/C161584116","wikidata":"https://www.wikidata.org/wiki/Q1952580","display_name":"Multivariate statistics","level":2,"score":0.09737610816955566},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/22m1516208","is_oa":false,"landing_page_url":"http://dx.doi.org/10.1137/22m1516208","pdf_url":null,"source":{"id":"https://openalex.org/S928796702","display_name":"SIAM Journal on Optimization","issn_l":"1052-6234","issn":["1052-6234","1095-7189"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Optimization","raw_type":"journal-article"},{"id":"pmh:oai:dial.uclouvain.be:boreal:303599","is_oa":true,"landing_page_url":"http://hdl.handle.net/2078.1/303599","pdf_url":null,"source":{"id":"https://openalex.org/S4306400166","display_name":"Digital Access to Libraries","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2800384882","host_organization_name":"Harris County Public Library","host_organization_lineage":["https://openalex.org/I2800384882"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Optimization, Vol. 23, no.3, p. 2041-2061 (2023)","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"pmh:oai:dial.uclouvain.be:boreal:303599","is_oa":true,"landing_page_url":"http://hdl.handle.net/2078.1/303599","pdf_url":null,"source":{"id":"https://openalex.org/S4306400166","display_name":"Digital Access to Libraries","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2800384882","host_organization_name":"Harris County Public Library","host_organization_lineage":["https://openalex.org/I2800384882"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"SIAM Journal on Optimization, Vol. 23, no.3, p. 2041-2061 (2023)","raw_type":"info:eu-repo/semantics/article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3139513672","display_name":null,"funder_award_id":"OAC-1835443","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7625705450","display_name":null,"funder_award_id":"FA8750-19-2-1000","funder_id":"https://openalex.org/F4320338279","funder_display_name":"Air Force Office of Scientific Research"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306621","display_name":"Belgian American Educational Foundation","ror":"https://ror.org/014rsed66"},{"id":"https://openalex.org/F4320338279","display_name":"Air Force Office of Scientific Research","ror":"https://ror.org/011e9bt93"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":26,"referenced_works":["https://openalex.org/W217710249","https://openalex.org/W1591179352","https://openalex.org/W1984149403","https://openalex.org/W2005102197","https://openalex.org/W2008797802","https://openalex.org/W2102182691","https://openalex.org/W2138505091","https://openalex.org/W2143075842","https://openalex.org/W2165281529","https://openalex.org/W2172384618","https://openalex.org/W2283023501","https://openalex.org/W2765336915","https://openalex.org/W2785876459","https://openalex.org/W2789612570","https://openalex.org/W2795405496","https://openalex.org/W2969215180","https://openalex.org/W3015255373","https://openalex.org/W3099705116","https://openalex.org/W3105952791","https://openalex.org/W3106518847","https://openalex.org/W3113425034","https://openalex.org/W3127944417","https://openalex.org/W3136413069","https://openalex.org/W4240837394","https://openalex.org/W4254742099","https://openalex.org/W4402262041"],"related_works":["https://openalex.org/W3092316610","https://openalex.org/W3158001585","https://openalex.org/W3111234197","https://openalex.org/W2108371780","https://openalex.org/W1992924751","https://openalex.org/W4316254163","https://openalex.org/W1524873934","https://openalex.org/W1988639837","https://openalex.org/W2007430971","https://openalex.org/W4213058796"],"abstract_inverted_index":{".We":[0],"study":[1],"the":[2,39,49,105,109,143,164,166],"problem":[3],"of":[4,12,43,51,108,111,115,205],"decomposing":[5],"a":[6,10,17,44,140,155],"polynomial":[7],"\\(p\\)":[8],"into":[9],"sum":[11,50],"\\(r\\)":[13,100],"squares":[14,52],"by":[15,153],"minimizing":[16],"quadratically":[18],"penalized":[19],"objective":[20,32],"\\(f_p(\\boldsymbol{\\mathbf{u}})":[21],"=":[22],"\\left":[23],"\\lVert":[24],"\\sum_{i=1}^r":[25],"u_i^2":[26],"-":[27],"p":[28],"\\right":[29],"\\lVert^2\\).":[30],"This":[31,83],"is":[33,36,84],"nonconvex":[34],"and":[35,134,145],"equivalent":[37],"to":[38,87,97,102,119,201],"rank-\\(r\\)":[40],"Burer\u2013Monteiro":[41],"factorization":[42],"semidefinite":[45],"program":[46],"(SDP)":[47],"encoding":[48],"decomposition.":[53],"We":[54,149],"show":[55,151],"that":[56,75,91,152,184],"for":[57,92,117],"all":[58,76],"univariate":[59],"polynomials":[60],"\\(p\\),":[61],"if":[62],"\\(r":[63],"\\ge":[64],"2\\),":[65],"then":[66],"\\(f_p(\\boldsymbol{\\mathbf{u}})\\)":[67],"has":[68,101,187],"no":[69,121],"spurious":[70,122],"second-order":[71,123,146],"critical":[72,124],"points,":[73],"showing":[74,90],"local":[77],"optima":[78],"are":[79],"also":[80,150],"global":[81],"optima.":[82],"in":[85,95,173],"contrast":[86],"previous":[88],"work":[89],"general":[93],"SDPs,":[94],"addition":[96],"genericity":[98],"conditions,":[99],"be":[103,120,136,171],"roughly":[104],"square":[106],"root":[107],"number":[110],"constraints":[112],"(the":[113],"degree":[114],"\\(p\\))":[116],"there":[118],"points.":[125],"Our":[126],"proof":[127],"uses":[128],"tools":[129],"from":[130],"computational":[131],"algebraic":[132],"geometry":[133],"can":[135,170],"interpreted":[137],"as":[138,196],"constructing":[139],"certificate":[141],"using":[142,177,191],"first-":[144],"necessary":[147],"conditions.":[148],"choosing":[154],"norm":[156],"based":[157],"on":[158,163],"sampling":[159],"equally":[160],"spaced":[161],"points":[162],"circle,":[165],"gradient":[167],"\\(\\nabla":[168],"f_p\\)":[169],"computed":[172],"nearly":[174],"linear":[175],"time":[176],"fast":[178,189],"Fourier":[179],"transforms.":[180],"Experimentally":[181],"we":[182],"demonstrate":[183],"this":[185],"method":[186],"very":[188],"convergence":[190],"first-order":[192],"optimization":[193],"algorithms":[194],"such":[195],"L-BFGS,":[197],"with":[198],"near-linear":[199],"scaling":[200],"million-degree":[202],"polynomials.Keywordsnonconvex":[203],"optimizationsum":[204],"squarestrigonometric":[206],"polynomialsBurer\u2013Monteiro":[207],"methodglobal":[208],"landscapesemidefinite":[209],"programmingMSC":[210],"codes90C2390C2690C22":[211]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1}],"updated_date":"2025-12-24T23:09:58.560324","created_date":"2025-10-10T00:00:00"}
