{"id":"https://openalex.org/W7223850802","doi":"https://doi.org/10.48550/arxiv.2610.35036","title":"Mapping the Solution-Set Structure of Grokking on Latin Squares","display_name":"Mapping the Solution-Set Structure of Grokking on Latin Squares","publication_year":2026,"publication_date":"2026-10-26","ids":{"openalex":"https://openalex.org/W7223850802","doi":"https://doi.org/10.48550/arxiv.2610.35036"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.35036","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.35036","pdf_url":"https://arxiv.org/pdf/2610.35036","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":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.35036","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5140985931","display_name":"Zihan Gu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I19820366","display_name":"Chinese Academy of Sciences","ror":"https://ror.org/034t30j35","country_code":"CN","type":"government","lineage":["https://openalex.org/I19820366"]},{"id":"https://openalex.org/I4210156404","display_name":"Institute of Information Engineering","ror":"https://ror.org/04r53se39","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210156404"]},{"id":"https://openalex.org/I4210165038","display_name":"University of Chinese Academy of Sciences","ror":"https://ror.org/05qbk4x57","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210165038"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Gu, Zihan","raw_affiliation_strings":["Institute of Information Engineering , Chinese Academy of Sciences ;","School of Cyber Security , University of Chinese Academy of Sciences ;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Information Engineering , Chinese Academy of Sciences ;","institution_ids":["https://openalex.org/I19820366","https://openalex.org/I4210156404"]},{"raw_affiliation_string":"School of Cyber Security , University of Chinese Academy of Sciences ;","institution_ids":["https://openalex.org/I4210165038"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5159878875","display_name":"Lulu Zhao","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I45928872","display_name":"Alibaba Group (China)","ror":"https://ror.org/00k642b80","country_code":"CN","type":"company","lineage":["https://openalex.org/I45928872"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhao, Lulu","raw_affiliation_strings":["Alibaba Group ;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Alibaba Group ;","institution_ids":["https://openalex.org/I45928872"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101874980","display_name":"Tianyi Ma","orcid":"https://orcid.org/0009-0005-9123-8827","observed_orcids":["https://orcid.org/0009-0005-9123-8827","https://orcid.org/0000-0001-5132-3756"]},"institutions":[{"id":"https://openalex.org/I183067930","display_name":"Shanghai Jiao Tong University","ror":"https://ror.org/0220qvk04","country_code":"CN","type":"education","lineage":["https://openalex.org/I183067930"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ma, Tianyi","raw_affiliation_strings":["Shanghai Jiaotong University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Jiaotong University","institution_ids":["https://openalex.org/I183067930"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5159941220","display_name":"Hua Zhang","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I19820366","display_name":"Chinese Academy of Sciences","ror":"https://ror.org/034t30j35","country_code":"CN","type":"government","lineage":["https://openalex.org/I19820366"]},{"id":"https://openalex.org/I4210156404","display_name":"Institute of Information Engineering","ror":"https://ror.org/04r53se39","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210156404"]},{"id":"https://openalex.org/I4210165038","display_name":"University of Chinese Academy of Sciences","ror":"https://ror.org/05qbk4x57","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210165038"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhang, Hua","raw_affiliation_strings":["Institute of Information Engineering , Chinese Academy of Sciences ;","School of Cyber Security , University of Chinese Academy of Sciences ;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Information Engineering , Chinese Academy of Sciences ;","institution_ids":["https://openalex.org/I19820366","https://openalex.org/I4210156404"]},{"raw_affiliation_string":"School of Cyber Security , University of Chinese Academy of Sciences ;","institution_ids":["https://openalex.org/I4210165038"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5113392255","display_name":"Wenbo Su","orcid":"https://orcid.org/0009-0009-3800-7543","observed_orcids":["https://orcid.org/0009-0009-3800-7543"]},"institutions":[{"id":"https://openalex.org/I45928872","display_name":"Alibaba Group (China)","ror":"https://ror.org/00k642b80","country_code":"CN","type":"company","lineage":["https://openalex.org/I45928872"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Su, Wenbo","raw_affiliation_strings":["Alibaba Group ;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Alibaba Group ;","institution_ids":["https://openalex.org/I45928872"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5159789573","display_name":"Yanan Cao","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I19820366","display_name":"Chinese Academy of Sciences","ror":"https://ror.org/034t30j35","country_code":"CN","type":"government","lineage":["https://openalex.org/I19820366"]},{"id":"https://openalex.org/I4210156404","display_name":"Institute of Information Engineering","ror":"https://ror.org/04r53se39","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210156404"]},{"id":"https://openalex.org/I4210165038","display_name":"University of Chinese Academy of Sciences","ror":"https://ror.org/05qbk4x57","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210165038"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Cao, Yanan","raw_affiliation_strings":["Institute of Information Engineering , Chinese Academy of Sciences ;","School of Cyber Security , University of Chinese Academy of Sciences ;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Information Engineering , Chinese Academy of Sciences ;","institution_ids":["https://openalex.org/I19820366","https://openalex.org/I4210156404"]},{"raw_affiliation_string":"School of Cyber Security , University of Chinese Academy of Sciences ;","institution_ids":["https://openalex.org/I4210165038"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5159894973","display_name":"Yue Hu","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I19820366","display_name":"Chinese Academy of Sciences","ror":"https://ror.org/034t30j35","country_code":"CN","type":"government","lineage":["https://openalex.org/I19820366"]},{"id":"https://openalex.org/I4210156404","display_name":"Institute of Information Engineering","ror":"https://ror.org/04r53se39","country_code":"CN","type":"facility","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210156404"]},{"id":"https://openalex.org/I4210165038","display_name":"University of Chinese Academy of Sciences","ror":"https://ror.org/05qbk4x57","country_code":"CN","type":"education","lineage":["https://openalex.org/I19820366","https://openalex.org/I4210165038"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Hu, Yue","raw_affiliation_strings":["Institute of Information Engineering , Chinese Academy of Sciences ;","School of Cyber Security , University of Chinese Academy of Sciences ;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Institute of Information Engineering , Chinese Academy of Sciences ;","institution_ids":["https://openalex.org/I19820366","https://openalex.org/I4210156404"]},{"raw_affiliation_string":"School of Cyber Security , University of Chinese Academy of Sciences ;","institution_ids":["https://openalex.org/I4210165038"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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/T10320","display_name":"Neural Networks and Applications","score":0.29030001163482666,"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/T10320","display_name":"Neural Networks and Applications","score":0.29030001163482666,"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/T12072","display_name":"Machine Learning and Algorithms","score":0.274399995803833,"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/T11307","display_name":"Domain Adaptation and Few-Shot Learning","score":0.08410000056028366,"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/grokking","display_name":"grokking","score":0.9739999771118164},{"id":"https://openalex.org/keywords/latin-squares","display_name":"Latin squares","score":0.6919999718666077},{"id":"https://openalex.org/keywords/edge-coloring","display_name":"edge coloring","score":0.6439999938011169},{"id":"https://openalex.org/keywords/bipartite-graphs","display_name":"bipartite graphs","score":0.7089999914169312},{"id":"https://openalex.org/keywords/finite-abelian-groups","display_name":"finite abelian groups","score":0.8709999918937683},{"id":"https://openalex.org/keywords/weight-decay","display_name":"weight decay","score":0.5740000009536743},{"id":"https://openalex.org/keywords/relabeling","display_name":"relabeling","score":0.5519999861717224},{"id":"https://openalex.org/keywords/polynomial-representation","display_name":"polynomial representation","score":0.5099999904632568},{"id":"https://openalex.org/keywords/structure-of-solution-sets","display_name":"solution set structure","score":0.42500001192092896}],"concepts":[{"id":"https://openalex.org/C184898388","wikidata":"https://www.wikidata.org/wiki/Q1435712","display_name":"Pairwise comparison","level":2,"score":0.7283999919891357},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6809999942779541},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.48489999771118164},{"id":"https://openalex.org/C158887663","wikidata":"https://www.wikidata.org/wiki/Q679367","display_name":"Latin square","level":4,"score":0.4602999985218048},{"id":"https://openalex.org/C181789720","wikidata":"https://www.wikidata.org/wiki/Q4812191","display_name":"Asymptotically optimal algorithm","level":2,"score":0.45419999957084656},{"id":"https://openalex.org/C162307627","wikidata":"https://www.wikidata.org/wiki/Q204833","display_name":"Enhanced Data Rates for GSM Evolution","level":2,"score":0.4514000117778778},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42890000343322754},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.349700003862381},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3400000035762787},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.337799996137619},{"id":"https://openalex.org/C311688","wikidata":"https://www.wikidata.org/wiki/Q2393193","display_name":"Time complexity","level":2,"score":0.3122999966144562},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.28139999508857727},{"id":"https://openalex.org/C28719098","wikidata":"https://www.wikidata.org/wiki/Q44946","display_name":"Point (geometry)","level":2,"score":0.27950000762939453}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.35036","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.35036","pdf_url":"https://arxiv.org/pdf/2610.35036","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":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.35036","doi":"https://doi.org/10.48550/arxiv.2610.35036","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.35036","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.35036","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.35036","pdf_url":"https://arxiv.org/pdf/2610.35036","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":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7223850802.pdf","grobid_xml":"https://content.openalex.org/works/W7223850802.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7166883831","https://openalex.org/W7168651567","https://openalex.org/W4386559757","https://openalex.org/W7130703375","https://openalex.org/W7133330574","https://openalex.org/W7137943497","https://openalex.org/W4403884091","https://openalex.org/W7083679675","https://openalex.org/W7124269709","https://openalex.org/W7169474067"],"abstract_inverted_index":{"Grokking":[0],"is":[1],"a":[2,119,162],"delayed":[3],"transition":[4],"from":[5,67,114],"memorisation":[6],"to":[7,50],"generalisation.":[8],"A":[9],"network":[10],"first":[11],"fits":[12],"observed":[13,91],"examples":[14],"while":[15],"remaining":[16],"near":[17],"chance":[18],"on":[19,30,151],"held-out":[20],"inputs,":[21],"then":[22],"generalises":[23],"abruptly.":[24],"Prior":[25],"work":[26],"has":[27],"focused":[28],"mainly":[29],"this":[31,51],"delay,":[32],"finding":[33],"that":[34,140,159],"the":[35,43,48,55,166,169],"network's":[36],"main":[37],"loss":[38],"becomes":[39],"weight":[40],"decay":[41],"after":[42],"fitting":[44],"phase.":[45],"We":[46],"name":[47],"solution":[49],"optimization":[52],"problem":[53],"as":[54,76,84,93],"minimum-norm":[56,64],"endpoint,":[57],"and":[58,88],"study":[59],"which":[60],"tasks":[61,75,125,152],"make":[62],"all":[63],"endpoints":[65],"generalise":[66,144],"partial":[68,115],"observations.":[69],"Specifically,":[70],"we":[71,103],"model":[72],"pairwise":[73],"input":[74],"edge":[77],"colourings":[78,83],"of":[79,142,153,165],"$K_{q,q}$,":[80],"with":[81],"Latin":[82,122],"our":[85],"structured":[86],"family,":[87],"classify":[89],"each":[90],"instance":[92],"all-good,":[94],"mixed,":[95],"or":[96,134],"no-good":[97,135],"by":[98],"its":[99],"endpoint":[100],"set.":[101],"Furthermore,":[102],"prove":[104],"two":[105],"contrasting":[106],"results:":[107],"(1)":[108],"Finite-abelian-group":[109],"families":[110],"are":[111],"asymptotically":[112,127],"all-good":[113,124],"observations;":[116],"(2)":[117],"Under":[118],"uniform":[120],"random":[121],"square,":[123],"become":[126],"rare":[128],"for":[129],"observation-based":[130],"learners,":[131],"leaving":[132],"mixed":[133],"behaviour.":[136],"Moreover,":[137],"experiments":[138],"show":[139],"relabellings":[141],"addition":[143],"despite":[145],"their":[146],"dense":[147],"polynomial":[148],"representations.":[149],"Finally,":[150],"composite":[154],"order,":[155],"finite-training":[156],"results":[157],"suggest":[158],"predictors":[160],"learn":[161],"coarse":[163],"part":[164],"output":[167],"before":[168],"full":[170],"label.":[171]},"counts_by_year":[],"updated_date":"2026-10-30T12:39:25.321706","created_date":"2026-10-28T00:00:00"}
