{"id":"https://openalex.org/W4407198000","doi":"https://doi.org/10.1137/23m1568739","title":"The Troublesome Kernel: On Hallucinations, No Free Lunches, and the Accuracy-Stability Tradeoff in Inverse Problems","display_name":"The Troublesome Kernel: On Hallucinations, No Free Lunches, and the Accuracy-Stability Tradeoff in Inverse Problems","publication_year":2025,"publication_date":"2025-02-06","ids":{"openalex":"https://openalex.org/W4407198000","doi":"https://doi.org/10.1137/23m1568739"},"language":"en","primary_location":{"id":"doi:10.1137/23m1568739","is_oa":false,"landing_page_url":"https://doi.org/10.1137/23m1568739","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"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 Review","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5006433290","display_name":"Nina Maria Gottschling","orcid":"https://orcid.org/0009-0004-0275-7522"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Nina M. Gottschling","raw_affiliation_strings":["University of Cambridge, Cambridge CB3 0WA, UK"],"raw_orcid":"https://orcid.org/0009-0004-0275-7522","affiliations":[{"raw_affiliation_string":"University of Cambridge, Cambridge CB3 0WA, UK","institution_ids":["https://openalex.org/I241749"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5060969026","display_name":"Vegard Antun","orcid":"https://orcid.org/0000-0003-4045-9775"},"institutions":[{"id":"https://openalex.org/I184942183","display_name":"University of Oslo","ror":"https://ror.org/01xtthb56","country_code":"NO","type":"education","lineage":["https://openalex.org/I184942183"]}],"countries":["NO"],"is_corresponding":false,"raw_author_name":"Vegard Antun","raw_affiliation_strings":["Department of Mathematics, University of Oslo, Blindern, 0316 Oslo, Norway"],"raw_orcid":"https://orcid.org/0000-0003-4045-9775","affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Oslo, Blindern, 0316 Oslo, Norway","institution_ids":["https://openalex.org/I184942183"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102813441","display_name":"Anders C. Hansen","orcid":"https://orcid.org/0000-0003-2700-9446"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Anders C. Hansen","raw_affiliation_strings":["University of Cambridge, Cambridge CB3 0WA, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Cambridge, Cambridge CB3 0WA, UK","institution_ids":["https://openalex.org/I241749"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5065352983","display_name":"Ben Adcock","orcid":null},"institutions":[{"id":"https://openalex.org/I18014758","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28","country_code":"CA","type":"education","lineage":["https://openalex.org/I18014758"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Ben Adcock","raw_affiliation_strings":["Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Simon Fraser University, Burnaby, BC V5A 1S6, Canada","institution_ids":["https://openalex.org/I18014758"]}]}],"institutions":[],"countries_distinct_count":3,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":14.4214,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.99107259,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"67","issue":"1","first_page":"73","last_page":"104"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9994000196456909,"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"}},"topics":[{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9994000196456909,"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"}},{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11236","display_name":"Control Systems and Identification","score":0.9725000262260437,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.6327143907546997},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5937629342079163},{"id":"https://openalex.org/keywords/inverse","display_name":"Inverse","score":0.5766714215278625},{"id":"https://openalex.org/keywords/kernel","display_name":"Kernel (algebra)","score":0.544044017791748},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5240252614021301},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.4424537122249603},{"id":"https://openalex.org/keywords/econometrics","display_name":"Econometrics","score":0.3758991062641144},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.3514547348022461},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.3435785174369812},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.28574204444885254},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.1598571538925171},{"id":"https://openalex.org/keywords/machine-learning","display_name":"Machine learning","score":0.12098777294158936},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.05874750018119812}],"concepts":[{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.6327143907546997},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5937629342079163},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.5766714215278625},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.544044017791748},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5240252614021301},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.4424537122249603},{"id":"https://openalex.org/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"score":0.3758991062641144},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.3514547348022461},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.3435785174369812},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.28574204444885254},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.1598571538925171},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.12098777294158936},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.05874750018119812},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/23m1568739","is_oa":false,"landing_page_url":"https://doi.org/10.1137/23m1568739","pdf_url":null,"source":{"id":"https://openalex.org/S160107561","display_name":"SIAM Review","issn_l":"0036-1445","issn":["0036-1445","1095-7200"],"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 Review","raw_type":"journal-article"},{"id":"pmh:oai:elib.dlr.de:209444","is_oa":false,"landing_page_url":"https://doi.org/10.1137/23M1568739>.","pdf_url":null,"source":{"id":"https://openalex.org/S4377196266","display_name":"elib (German Aerospace Center)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I2898391981","host_organization_name":"Deutsches Zentrum f\u00fcr Luft- und Raumfahrt e. V. (DLR)","host_organization_lineage":["https://openalex.org/I2898391981"],"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":"PeerReviewed"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/8","display_name":"Decent work and economic growth","score":0.4399999976158142}],"awards":[{"id":"https://openalex.org/G3959550084","display_name":null,"funder_award_id":"R611675","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"}],"funders":[{"id":"https://openalex.org/F4320310984","display_name":"Pacific Institute for the Mathematical Sciences","ror":"https://ror.org/04w49dw51"},{"id":"https://openalex.org/F4320319993","display_name":"Leverhulme Trust","ror":"https://ror.org/012mzw131"},{"id":"https://openalex.org/F4320322551","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28"},{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"},{"id":"https://openalex.org/F4320334627","display_name":"Engineering and Physical Sciences Research Council","ror":"https://ror.org/0439y7842"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":58,"referenced_works":["https://openalex.org/W1543241987","https://openalex.org/W2020390700","https://openalex.org/W2140089781","https://openalex.org/W2493000045","https://openalex.org/W2525748878","https://openalex.org/W2611467245","https://openalex.org/W2762685704","https://openalex.org/W2801215870","https://openalex.org/W2803224943","https://openalex.org/W2883105305","https://openalex.org/W2899283552","https://openalex.org/W2910683834","https://openalex.org/W2911290743","https://openalex.org/W2924551358","https://openalex.org/W2942199805","https://openalex.org/W2952020389","https://openalex.org/W2964301649","https://openalex.org/W2979544394","https://openalex.org/W2988626048","https://openalex.org/W2990138404","https://openalex.org/W2999511788","https://openalex.org/W3001319253","https://openalex.org/W3021094251","https://openalex.org/W3033822002","https://openalex.org/W3047729951","https://openalex.org/W3098020164","https://openalex.org/W3098900881","https://openalex.org/W3103921058","https://openalex.org/W3112965401","https://openalex.org/W3126002913","https://openalex.org/W3126069796","https://openalex.org/W3126432987","https://openalex.org/W3128363678","https://openalex.org/W3133902371","https://openalex.org/W3143784018","https://openalex.org/W3157723514","https://openalex.org/W3158148831","https://openalex.org/W3166995168","https://openalex.org/W3179592584","https://openalex.org/W3195875706","https://openalex.org/W3198619676","https://openalex.org/W3201812816","https://openalex.org/W3204290769","https://openalex.org/W4206305733","https://openalex.org/W4220974656","https://openalex.org/W4221051856","https://openalex.org/W4223430324","https://openalex.org/W4223506650","https://openalex.org/W4224106166","https://openalex.org/W4236362309","https://openalex.org/W4290725208","https://openalex.org/W4296966010","https://openalex.org/W4301168154","https://openalex.org/W4313437003","https://openalex.org/W4377004284","https://openalex.org/W4387500561","https://openalex.org/W4391927908","https://openalex.org/W4398150900"],"related_works":["https://openalex.org/W4317939968","https://openalex.org/W4390144857","https://openalex.org/W1493568480","https://openalex.org/W4385967523","https://openalex.org/W4281647907","https://openalex.org/W2638372604","https://openalex.org/W3114513573","https://openalex.org/W2596081731","https://openalex.org/W2331917905","https://openalex.org/W3155039083"],"abstract_inverted_index":{"Methods":[0],"inspired":[1],"by":[2,178],"Artificial":[3],"Intelligence":[4],"(AI)":[5],"are":[6,170],"starting":[7],"to":[8,61,136,186,199,213,229,235],"fundamentally":[9],"change":[10],"computational":[11],"science":[12],"and":[13,23,66,95,165,174,238],"engineering":[14],"through":[15],"breakthrough":[16],"performances":[17],"on":[18,72,78,125,145,223],"challenging":[19],"problems.":[20,193],"However,":[21],"reliability":[22],"trustworthiness":[24],"of":[25,39,106,112,202],"such":[26,97],"techniques":[27],"is":[28,43,160,208,220],"a":[29,83,126,139],"major":[30],"concern.":[31],"In":[32],"inverse":[33,243],"problems":[34,244],"in":[35,63,100,245],"imaging,":[36],"the":[37,64,110,157,200,203,214,217],"focus":[38],"this":[40],"paper,":[41],"there":[42],"increasing":[44],"empirical":[45],"evidence":[46],"that":[47,120,123,143],"methods":[48,122,142,241],"may":[49,175,183],"suffer":[50],"from":[51,133],"hallucinations,":[52],"i.e.,":[53,59,69],"false,":[54],"but":[55,75,210],"realistic-looking":[56],"artifacts;":[57],"instability,":[58],"sensitivity":[60],"perturbations":[62],"data;":[65],"unpredictable":[67],"generalization,":[68],"excellent":[70],"performance":[71],"some":[73],"images,":[74],"significant":[76],"deterioration":[77],"others.":[79],"This":[80],"paper":[81],"provides":[82],"theoretical":[84],"foundation":[85],"for":[86,93,191,242],"these":[87,197,224],"phenomena.":[88],"We":[89],"give":[90],"mathematical":[91],"explanations":[92],"how":[94],"when":[96,216],"effects":[98,198],"arise":[99],"arbitrary":[101],"reconstruction":[102,189],"methods,":[103],"with":[104],"several":[105],"our":[107,226],"results":[108,195],"taking":[109],"form":[111],"`no":[113],"free":[114],"lunch'":[115],"theorems.":[116],"Specifically,":[117],"we":[118],"show":[119],"(i)":[121],"overperform":[124,144],"single":[127],"image":[128,135],"can":[129,150],"wrongly":[130],"transfer":[131],"details":[132],"one":[134],"another,":[137],"creating":[138],"hallucination,":[140],"(ii)":[141],"two":[146],"or":[147,152],"more":[148],"images":[149],"hallucinate":[151],"be":[153,176,184],"unstable,":[154],"(iii)":[155],"optimizing":[156],"accuracy-stability":[158],"trade-off":[159],"generally":[161],"difficult,":[162],"(iv)":[163],"hallucinations":[164],"instabilities,":[166],"if":[167],"they":[168],"occur,":[169],"not":[171],"rare":[172],"events,":[173],"encouraged":[177],"standard":[179],"training,":[180],"(v)":[181],"it":[182,207],"impossible":[185],"construct":[187],"optimal":[188],"maps":[190],"certain":[192],"Our":[194],"trace":[196],"kernel":[201],"forward":[204,218],"operator":[205,219],"whenever":[206],"nontrivial,":[209],"also":[211],"apply":[212],"case":[215],"ill-conditioned.":[221],"Based":[222],"insights,":[225],"work":[227],"aims":[228],"spur":[230],"research":[231],"into":[232],"new":[233],"ways":[234],"develop":[236],"robust":[237],"reliable":[239],"AI-based":[240],"imaging.":[246]},"counts_by_year":[{"year":2026,"cited_by_count":6},{"year":2025,"cited_by_count":9}],"updated_date":"2026-06-16T09:24:06.705377","created_date":"2025-10-10T00:00:00"}
