{"id":"https://openalex.org/W3135310021","doi":null,"title":"A Closed Form Solution to Best Rank-1 Tensor Approximation via KL divergence Minimization.","display_name":"A Closed Form Solution to Best Rank-1 Tensor Approximation via KL divergence Minimization.","publication_year":2021,"publication_date":"2021-03-04","ids":{"openalex":"https://openalex.org/W3135310021","mag":"3135310021"},"language":"en","primary_location":{"id":"mag:3135310021","doi":null,"is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/2103.02898","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":[],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://export.arxiv.org/pdf/2103.02898","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5032956935","display_name":"Kazu Ghalamkari","orcid":"https://orcid.org/0000-0002-4779-2856","observed_orcids":["https://orcid.org/0000-0002-4779-2856"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kazu Ghalamkari","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0002-4779-2856","affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5066053285","display_name":"Mahito Sugiyama","orcid":"https://orcid.org/0000-0001-5907-9831","observed_orcids":["https://orcid.org/0000-0001-5907-9831"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mahito Sugiyama","raw_affiliation_strings":[],"raw_orcid":"https://orcid.org/0000-0001-5907-9831","affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"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/T12303","display_name":"Tensor decomposition and applications","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.0,"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.0,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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/tensor-decomposition","display_name":"tensor decomposition","score":0.7310000061988831},{"id":"https://openalex.org/keywords/kl-divergence-minimization","display_name":"KL divergence minimization","score":0.7979999780654907},{"id":"https://openalex.org/keywords/analytical-solution","display_name":"analytical solution","score":0.9300000071525574},{"id":"https://openalex.org/keywords/convex-optimization","display_name":"convex optimization","score":0.9779999852180481},{"id":"https://openalex.org/keywords/computational-efficiency","display_name":"computational efficiency","score":0.6179999709129333}],"concepts":[{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.7551335096359253},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.7547702789306641},{"id":"https://openalex.org/C207390915","wikidata":"https://www.wikidata.org/wiki/Q1230525","display_name":"Divergence (linguistics)","level":2,"score":0.6886064410209656},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6562031507492065},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.6440500020980835},{"id":"https://openalex.org/C90199385","wikidata":"https://www.wikidata.org/wiki/Q6692777","display_name":"Low-rank approximation","level":3,"score":0.5051642060279846},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4360443353652954},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.42864513397216797},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.416659414768219},{"id":"https://openalex.org/C9936470","wikidata":"https://www.wikidata.org/wiki/Q6510405","display_name":"Least-squares function approximation","level":3,"score":0.4154165983200073},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.38359442353248596},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.28349578380584717},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.18668586015701294},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.18320953845977783},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.11816000938415527},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09216240048408508},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.059679627418518066},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"mag:3135310021","doi":null,"is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/2103.02898","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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}],"best_oa_location":{"id":"mag:3135310021","doi":null,"is_oa":true,"landing_page_url":"http://export.arxiv.org/pdf/2103.02898","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":25,"referenced_works":["https://openalex.org/W1532854728","https://openalex.org/W1963826206","https://openalex.org/W1967077133","https://openalex.org/W1968154520","https://openalex.org/W2002598080","https://openalex.org/W2004026774","https://openalex.org/W2013870323","https://openalex.org/W2013912476","https://openalex.org/W2018282388","https://openalex.org/W2024165284","https://openalex.org/W2024356620","https://openalex.org/W2031688197","https://openalex.org/W2040661328","https://openalex.org/W2042492924","https://openalex.org/W2073502026","https://openalex.org/W2106221905","https://openalex.org/W2124812506","https://openalex.org/W2126447985","https://openalex.org/W2168071446","https://openalex.org/W2330159389","https://openalex.org/W2429756633","https://openalex.org/W2507473973","https://openalex.org/W2765321384","https://openalex.org/W2890081349","https://openalex.org/W2907363428"],"related_works":["https://openalex.org/W2962708970","https://openalex.org/W7099049975","https://openalex.org/W2765321384","https://openalex.org/W2262986604","https://openalex.org/W3178913129","https://openalex.org/W2988608730","https://openalex.org/W3111270789","https://openalex.org/W2780343906","https://openalex.org/W2395373520","https://openalex.org/W7043113115"],"abstract_inverted_index":{"Tensor":[0],"decomposition":[1],"is":[2,25,70,112],"a":[3,49,63,73,77,88],"fundamentally":[4],"challenging":[5],"problem.":[6],"Even":[7],"the":[8,14,20,37,42,54,59,82,91,119],"simplest":[9],"case":[10],"of":[11,19,41,84,93,115,128],"tensor":[12,56,75],"decomposition,":[13],"rank-1":[15,55,85,94,101,121],"approximation":[16,86,102,122,127],"in":[17],"terms":[18],"Least":[21],"Squares":[22],"(LS)":[23],"error,":[24,44],"known":[26],"to":[27,71,99],"be":[28],"NP-hard.":[29],"Here,":[30],"we":[31,35,45],"show":[32],"that,":[33],"if":[34],"consider":[36],"KL":[38,60],"divergence":[39,61],"instead":[40],"LS":[43],"can":[46],"analytically":[47],"derive":[48],"closed":[50],"form":[51],"solution":[52],"for":[53],"that":[57,109],"minimizes":[58],"from":[62],"given":[64,129],"positive":[65,74],"tensor.":[66],"Our":[67],"key":[68],"insight":[69],"treat":[72],"as":[76,87],"probability":[78],"distribution":[79],"and":[80,124],"formulate":[81],"process":[83],"projection":[89],"onto":[90],"set":[92],"tensors.":[95],"This":[96],"enables":[97],"us":[98],"solve":[100],"by":[103],"convex":[104],"optimization.":[105],"We":[106],"empirically":[107],"demonstrate":[108],"our":[110,133],"algorithm":[111],"an":[113],"order":[114],"magnitude":[116],"faster":[117],"than":[118],"existing":[120],"methods":[123],"gives":[125],"better":[126],"tensors,":[130],"which":[131],"supports":[132],"theoretical":[134],"finding.":[135]},"counts_by_year":[],"updated_date":"2026-10-16T09:59:22.235173","created_date":"2021-03-15T00:00:00"}
