{"id":"https://openalex.org/W3042743256","doi":"https://doi.org/10.1007/s10208-021-09537-5","title":"Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports","display_name":"Deep Composition of Tensor-Trains Using Squared Inverse Rosenblatt Transports","publication_year":2021,"publication_date":"2021-09-21","ids":{"openalex":"https://openalex.org/W3042743256","doi":"https://doi.org/10.1007/s10208-021-09537-5","mag":"3042743256"},"language":"en","primary_location":{"id":"doi:10.1007/s10208-021-09537-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09537-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09537-5.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/s10208-021-09537-5.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5067931953","display_name":"Tiangang Cui","orcid":"https://orcid.org/0000-0002-4840-8545"},"institutions":[{"id":"https://openalex.org/I56590836","display_name":"Monash University","ror":"https://ror.org/02bfwt286","country_code":"AU","type":"education","lineage":["https://openalex.org/I56590836"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Tiangang Cui","raw_affiliation_strings":["School of Mathematics, Monash University, VIC 3800, Australia","School of Mathematics, Monash University, VIC, 3800, Australia"],"raw_orcid":"https://orcid.org/0000-0002-4840-8545","affiliations":[{"raw_affiliation_string":"School of Mathematics, Monash University, VIC 3800, Australia","institution_ids":["https://openalex.org/I56590836"]},{"raw_affiliation_string":"School of Mathematics, Monash University, VIC, 3800, Australia","institution_ids":["https://openalex.org/I56590836"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5023624132","display_name":"Sergey Dolgov","orcid":"https://orcid.org/0000-0002-1647-4214"},"institutions":[{"id":"https://openalex.org/I51601045","display_name":"University of Bath","ror":"https://ror.org/002h8g185","country_code":"GB","type":"education","lineage":["https://openalex.org/I51601045"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"Sergey Dolgov","raw_affiliation_strings":["Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK","Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK"],"raw_orcid":"https://orcid.org/0000-0002-1647-4214","affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK","institution_ids":["https://openalex.org/I51601045"]},{"raw_affiliation_string":"Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK","institution_ids":["https://openalex.org/I51601045"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5023624132"],"corresponding_institution_ids":["https://openalex.org/I51601045"],"apc_list":{"value":2390,"currency":"EUR","value_usd":2990},"apc_paid":{"value":2390,"currency":"EUR","value_usd":2990},"fwci":2.6223,"has_fulltext":true,"cited_by_count":25,"citation_normalized_percentile":{"value":0.91252955,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"22","issue":"6","first_page":"1863","last_page":"1922"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.9998999834060669,"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":0.9998999834060669,"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/T11304","display_name":"Advanced Neuroimaging Techniques and Applications","score":0.9950000047683716,"subfield":{"id":"https://openalex.org/subfields/2741","display_name":"Radiology, Nuclear Medicine and Imaging"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9940000176429749,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6777821183204651},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.6025010943412781},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5129746794700623},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.47392305731773376},{"id":"https://openalex.org/keywords/random-variable","display_name":"Random variable","score":0.4659596085548401},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.4434118866920471},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.4266360402107239},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3598117232322693},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2594258189201355},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09363231062889099}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6777821183204651},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.6025010943412781},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5129746794700623},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.47392305731773376},{"id":"https://openalex.org/C122123141","wikidata":"https://www.wikidata.org/wiki/Q176623","display_name":"Random variable","level":2,"score":0.4659596085548401},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.4434118866920471},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.4266360402107239},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3598117232322693},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2594258189201355},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09363231062889099},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1007/s10208-021-09537-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09537-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09537-5.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:2007.06968","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2007.06968","pdf_url":"https://arxiv.org/pdf/2007.06968","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/s10208-021-09537-5","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10208-021-09537-5","pdf_url":"https://link.springer.com/content/pdf/10.1007/s10208-021-09537-5.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/G4588159538","display_name":null,"funder_award_id":"CE140100049","funder_id":"https://openalex.org/F4320334704","funder_display_name":"Australian Research Council"},{"id":"https://openalex.org/G663801976","display_name":"Tensor decomposition sampling algorithms for Bayesian inverse problems","funder_award_id":"EP/T031255/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"},{"id":"https://openalex.org/G7141960837","display_name":null,"funder_award_id":"EP/T031255/1","funder_id":"https://openalex.org/F4320334627","funder_display_name":"Engineering and Physical Sciences Research Council"}],"funders":[{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"},{"id":"https://openalex.org/F4320334704","display_name":"Australian Research Council","ror":"https://ror.org/05mmh0f86"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3042743256.pdf","grobid_xml":"https://content.openalex.org/works/W3042743256.grobid-xml"},"referenced_works_count":83,"referenced_works":["https://openalex.org/W50417968","https://openalex.org/W253307310","https://openalex.org/W385466589","https://openalex.org/W640428934","https://openalex.org/W1522264804","https://openalex.org/W1534349507","https://openalex.org/W1545319692","https://openalex.org/W1592510033","https://openalex.org/W1805381866","https://openalex.org/W1858056047","https://openalex.org/W1934412359","https://openalex.org/W1973396249","https://openalex.org/W1973594349","https://openalex.org/W1978501336","https://openalex.org/W1985093013","https://openalex.org/W1993482030","https://openalex.org/W1995565517","https://openalex.org/W1996869553","https://openalex.org/W1998037613","https://openalex.org/W2001518794","https://openalex.org/W2002664623","https://openalex.org/W2011732752","https://openalex.org/W2013145865","https://openalex.org/W2013164703","https://openalex.org/W2016407890","https://openalex.org/W2016527001","https://openalex.org/W2023103251","https://openalex.org/W2024165284","https://openalex.org/W2031216664","https://openalex.org/W2043719428","https://openalex.org/W2047591100","https://openalex.org/W2049753327","https://openalex.org/W2069739265","https://openalex.org/W2071544114","https://openalex.org/W2078465266","https://openalex.org/W2089657469","https://openalex.org/W2097175338","https://openalex.org/W2104067967","https://openalex.org/W2129728672","https://openalex.org/W2141696759","https://openalex.org/W2159565862","https://openalex.org/W2169837720","https://openalex.org/W2232528752","https://openalex.org/W2274168723","https://openalex.org/W2279346308","https://openalex.org/W2403013574","https://openalex.org/W2496219407","https://openalex.org/W2559655401","https://openalex.org/W2569729041","https://openalex.org/W2806674516","https://openalex.org/W2848712137","https://openalex.org/W2921490382","https://openalex.org/W2948700496","https://openalex.org/W2962947392","https://openalex.org/W2963009826","https://openalex.org/W2963090522","https://openalex.org/W2963849628","https://openalex.org/W2963956018","https://openalex.org/W2963964397","https://openalex.org/W2970641149","https://openalex.org/W2982295692","https://openalex.org/W2992035660","https://openalex.org/W3002052860","https://openalex.org/W3008180535","https://openalex.org/W3034463452","https://openalex.org/W3041693466","https://openalex.org/W3042770619","https://openalex.org/W3080914972","https://openalex.org/W3098700198","https://openalex.org/W3099467939","https://openalex.org/W3100193990","https://openalex.org/W3101006260","https://openalex.org/W3105604858","https://openalex.org/W3120877973","https://openalex.org/W3121471846","https://openalex.org/W3123981669","https://openalex.org/W3150807214","https://openalex.org/W3175177514","https://openalex.org/W3183200481","https://openalex.org/W4212774754","https://openalex.org/W4253937035","https://openalex.org/W4288347845","https://openalex.org/W6602603507"],"related_works":["https://openalex.org/W2154204858","https://openalex.org/W3209622820","https://openalex.org/W4317939968","https://openalex.org/W2382250648","https://openalex.org/W4390144857","https://openalex.org/W2367363545","https://openalex.org/W2586383879","https://openalex.org/W2067895158","https://openalex.org/W2331917905","https://openalex.org/W3155039083"],"abstract_inverted_index":{"Abstract":[0],"Characterising":[1],"intractable":[2,34],"high-dimensional":[3,69],"random":[4,35,40,158],"variables":[5,36,159],"is":[6],"one":[7],"of":[8,18,49,68,137,151,172,179],"the":[9,45,50,81,87,95,114,120,134,149,170,173],"fundamental":[10],"challenges":[11],"in":[12,181],"stochastic":[13],"computation.":[14],"The":[15,141],"recent":[16],"surge":[17],"transport":[19,53,88,106,125,146,155],"maps":[20,156],"offers":[21],"a":[22,65,108,127,177],"mathematical":[23],"foundation":[24],"and":[25,154,164,184,193],"new":[26],"insights":[27],"for":[28,190],"tackling":[29],"this":[30,105],"challenge":[31],"by":[32,56,133,197],"coupling":[33],"with":[37,160],"tractable":[38],"reference":[39,91],"variables.":[41],"This":[42],"paper":[43],"generalises":[44],"functional":[46,123],"tensor-train":[47,110,124],"approximation":[48],"inverse":[51,82,144,194],"Rosenblatt":[52,83,145],"recently":[54],"developed":[55],"Dolgov":[57],"et":[58],"al.":[59],"(Stat":[60],"Comput":[61],"30:603\u2013625,":[62],"2020)":[63],"to":[64,85,89,103,157],"wide":[66],"class":[67],"non-negative":[70],"functions,":[71],"such":[72],"as":[73],"unnormalised":[74],"probability":[75],"density":[76,166],"functions.":[77,167],"First,":[78],"we":[79,118],"extend":[80],"transform":[84],"enable":[86],"general":[90],"measures":[92],"other":[93],"than":[94],"uniform":[96],"measure.":[97],"We":[98,168],"develop":[99],"an":[100],"efficient":[101],"procedure":[102],"compute":[104],"from":[107],"squared":[109],"decomposition":[111],"which":[112],"preserves":[113],"monotonicity.":[115],"More":[116],"crucially,":[117],"integrate":[119],"proposed":[121,174],"order-preserving":[122],"into":[126],"nested":[128],"variable":[129],"transformation":[130],"framework":[131],"inspired":[132],"layered":[135],"structure":[136],"deep":[138,143],"neural":[139],"networks.":[140],"resulting":[142],"significantly":[147],"expands":[148],"capability":[150],"tensor":[152],"approximations":[153],"complicated":[161],"nonlinear":[162],"interactions":[163],"concentrated":[165],"demonstrate":[169],"efficiency":[171],"approach":[175],"on":[176],"range":[178],"applications":[180],"statistical":[182],"learning":[183],"uncertainty":[185],"quantification,":[186],"including":[187],"parameter":[188],"estimation":[189],"dynamical":[191],"systems":[192],"problems":[195],"constrained":[196],"partial":[198],"differential":[199],"equations.":[200]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":4},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2020-07-23T00:00:00"}
