{"id":"https://openalex.org/W7160873739","doi":"https://doi.org/10.48550/arxiv.2605.07648","title":"Learning Large-Scale Modular Addition with an Auxiliary Modulus","display_name":"Learning Large-Scale Modular Addition with an Auxiliary Modulus","publication_year":2026,"publication_date":"2026-05-08","ids":{"openalex":"https://openalex.org/W7160873739","doi":"https://doi.org/10.48550/arxiv.2605.07648"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.07648","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.07648","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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2605.07648","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5135848763","display_name":"Hanato Kikuchi","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kikuchi, Hanato","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5135838657","display_name":"Ryosuke Masuya","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Masuya, Ryosuke","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5010514962","display_name":"Kazuhiko Kawamoto","orcid":"https://orcid.org/0000-0003-3701-1961"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kawamoto, Kazuhiko","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5055384327","display_name":"Hiroshi Kera","orcid":"https://orcid.org/0000-0002-9830-0436"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Kera, Hiroshi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"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/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.42719998955726624,"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"}},"topics":[{"id":"https://openalex.org/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.42719998955726624,"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/T12535","display_name":"Machine Learning and Data Classification","score":0.2766999900341034,"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/T13702","display_name":"Machine Learning in Healthcare","score":0.02979999966919422,"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/modular-design","display_name":"Modular design","score":0.7677000164985657},{"id":"https://openalex.org/keywords/scalability","display_name":"Scalability","score":0.536300003528595},{"id":"https://openalex.org/keywords/task","display_name":"Task (project management)","score":0.4596000015735626},{"id":"https://openalex.org/keywords/covariate","display_name":"Covariate","score":0.444599986076355},{"id":"https://openalex.org/keywords/training-set","display_name":"Training set","score":0.4334000051021576},{"id":"https://openalex.org/keywords/key","display_name":"Key (lock)","score":0.4284999966621399},{"id":"https://openalex.org/keywords/modulus","display_name":"Modulus","score":0.38420000672340393},{"id":"https://openalex.org/keywords/sample","display_name":"Sample (material)","score":0.3431999981403351}],"concepts":[{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.7677000164985657},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.536300003528595},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5138999819755554},{"id":"https://openalex.org/C2780451532","wikidata":"https://www.wikidata.org/wiki/Q759676","display_name":"Task (project management)","level":2,"score":0.4596000015735626},{"id":"https://openalex.org/C119043178","wikidata":"https://www.wikidata.org/wiki/Q320723","display_name":"Covariate","level":2,"score":0.444599986076355},{"id":"https://openalex.org/C51632099","wikidata":"https://www.wikidata.org/wiki/Q3985153","display_name":"Training set","level":2,"score":0.4334000051021576},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.4284999966621399},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.40230000019073486},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.38440001010894775},{"id":"https://openalex.org/C193867417","wikidata":"https://www.wikidata.org/wiki/Q6889814","display_name":"Modulus","level":2,"score":0.38420000672340393},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.37119999527931213},{"id":"https://openalex.org/C198531522","wikidata":"https://www.wikidata.org/wiki/Q485146","display_name":"Sample (material)","level":2,"score":0.3431999981403351},{"id":"https://openalex.org/C2777151079","wikidata":"https://www.wikidata.org/wiki/Q141160","display_name":"Parity (physics)","level":2,"score":0.3084999918937683},{"id":"https://openalex.org/C119857082","wikidata":"https://www.wikidata.org/wiki/Q2539","display_name":"Machine learning","level":1,"score":0.2955999970436096},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.29339998960494995},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.2825999855995178},{"id":"https://openalex.org/C129848803","wikidata":"https://www.wikidata.org/wiki/Q2564360","display_name":"Sample size determination","level":2,"score":0.28049999475479126},{"id":"https://openalex.org/C77967617","wikidata":"https://www.wikidata.org/wiki/Q4677561","display_name":"Active learning (machine learning)","level":2,"score":0.27619999647140503},{"id":"https://openalex.org/C136389625","wikidata":"https://www.wikidata.org/wiki/Q334384","display_name":"Supervised learning","level":3,"score":0.26489999890327454},{"id":"https://openalex.org/C2777211547","wikidata":"https://www.wikidata.org/wiki/Q17141490","display_name":"Training (meteorology)","level":2,"score":0.2644999921321869},{"id":"https://openalex.org/C87007009","wikidata":"https://www.wikidata.org/wiki/Q210832","display_name":"Statistical hypothesis testing","level":2,"score":0.2612000107765198},{"id":"https://openalex.org/C97931131","wikidata":"https://www.wikidata.org/wiki/Q5282087","display_name":"Discriminative model","level":2,"score":0.2581999897956848},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.25609999895095825},{"id":"https://openalex.org/C16910744","wikidata":"https://www.wikidata.org/wiki/Q7705759","display_name":"Test data","level":2,"score":0.2556000053882599}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.07648","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.07648","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":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.07648","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.07648","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":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Learning":[0],"parity":[1],"functions,":[2],"more":[3],"general":[4],"modular":[5,23,82],"addition,":[6],"is":[7,38],"a":[8,78],"challenging":[9],"machine":[10],"learning":[11,25],"task":[12],"due":[13],"to":[14,39,135,183],"its":[15],"input":[16,65,104,120],"sensitivity.":[17],"A":[18],"recent":[19],"study":[20,68],"substantially":[21],"scaled":[22],"addition":[24],"in":[26,42],"both":[27],"the":[28,33,46,102,131,167,174],"number":[29,48],"of":[30,49],"summands":[31,50],"and":[32,51,63,70,76,97,108,114,126,146,153,178],"modulus.":[34],"Its":[35],"key":[36],"idea":[37],"increase":[40],"zeros":[41],"training":[43,54,62,107],"sequences,":[44],"reducing":[45],"effective":[47],"thus":[52],"controlling":[53],"difficulty;":[55],"however,":[56],"this":[57,73],"induces":[58],"covariate":[59],"shift":[60],"between":[61],"test":[64],"distributions.":[66],"This":[67],"theoretically":[69],"empirically":[71],"analyzes":[72],"side":[74],"effect":[75],"proposes":[77],"covariate-shift-free":[79],"method":[80,133,139,156,169],"for":[81,118],"addition.":[83],"Specifically,":[84],"we":[85],"introduce":[86],"an":[87],"auxiliary":[88],"modulus":[89,124],"$Kq$":[90],"during":[91],"training,":[92],"which":[93],"reduces":[94],"wrap-around":[95],"frequency":[96],"problem":[98],"difficulty":[99],"while":[100,166],"preserving":[101],"same":[103,175],"distribution":[105],"across":[106],"testing.":[109],"Experiments":[110],"show":[111],"strong":[112],"scalability":[113],"sample":[115],"efficiency:":[116],"even":[117,180],"large":[119,123],"length":[121],"$N$,":[122],"$q$,":[125],"small":[127],"datasets":[128],"--":[129,137],"where":[130],"sparse":[132,168],"fails":[134],"learn":[136],"our":[138,155],"achieves":[140,161,170],"equal":[141],"or":[142],"better":[143],"match":[144],"accuracy":[145],"relaxed":[147],"$\u03c4$-accuracy.":[148],"For":[149],"example,":[150],"at":[151,164],"$N=64$":[152],"$q=974269$,":[154],"trained":[157],"on":[158],"100K":[159],"samples":[160],"$97.0\\%$":[162],"$\u03c4$-accuracy":[163],"$\u03c4=0.05$,":[165],"only":[171],"$9.5\\%$":[172],"with":[173],"data":[176],"size":[177],"$93.9\\%$":[179],"when":[181],"extended":[182],"1M":[184],"samples.":[185]},"counts_by_year":[],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2026-05-12T00:00:00"}
