{"id":"https://openalex.org/W7140441192","doi":"https://doi.org/10.48550/arxiv.2603.23784","title":"Latent Algorithmic Structure Precedes Grokking: A Mechanistic Study of ReLU MLPs on Modular Arithmetic","display_name":"Latent Algorithmic Structure Precedes Grokking: A Mechanistic Study of ReLU MLPs on Modular Arithmetic","publication_year":2026,"publication_date":"2026-03-24","ids":{"openalex":"https://openalex.org/W7140441192","doi":"https://doi.org/10.48550/arxiv.2603.23784"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2603.23784","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.23784","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","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.2603.23784","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5130234131","display_name":"Anand Swaroop","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Swaroop, Anand","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5130234131"],"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/T11206","display_name":"Model Reduction and Neural Networks","score":0.1598999947309494,"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":0.1598999947309494,"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/T10320","display_name":"Neural Networks and Applications","score":0.13179999589920044,"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/T12611","display_name":"Neural Networks and Reservoir Computing","score":0.10400000214576721,"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/binary-number","display_name":"Binary number","score":0.5809000134468079},{"id":"https://openalex.org/keywords/perceptron","display_name":"Perceptron","score":0.580299973487854},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5687999725341797},{"id":"https://openalex.org/keywords/generalization","display_name":"Generalization","score":0.48559999465942383},{"id":"https://openalex.org/keywords/modular-design","display_name":"Modular design","score":0.4702000021934509},{"id":"https://openalex.org/keywords/amplitude","display_name":"Amplitude","score":0.3517000079154968},{"id":"https://openalex.org/keywords/pattern-recognition","display_name":"Pattern recognition (psychology)","score":0.32989999651908875},{"id":"https://openalex.org/keywords/binary-data","display_name":"Binary data","score":0.32510000467300415},{"id":"https://openalex.org/keywords/fourier-transform","display_name":"Fourier transform","score":0.32499998807907104}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.6452000141143799},{"id":"https://openalex.org/C48372109","wikidata":"https://www.wikidata.org/wiki/Q3913","display_name":"Binary number","level":2,"score":0.5809000134468079},{"id":"https://openalex.org/C60908668","wikidata":"https://www.wikidata.org/wiki/Q690207","display_name":"Perceptron","level":3,"score":0.580299973487854},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5687999725341797},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.48559999465942383},{"id":"https://openalex.org/C101468663","wikidata":"https://www.wikidata.org/wiki/Q1620158","display_name":"Modular design","level":2,"score":0.4702000021934509},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4528999924659729},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4431000053882599},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3637999892234802},{"id":"https://openalex.org/C180205008","wikidata":"https://www.wikidata.org/wiki/Q159190","display_name":"Amplitude","level":2,"score":0.3517000079154968},{"id":"https://openalex.org/C153180895","wikidata":"https://www.wikidata.org/wiki/Q7148389","display_name":"Pattern recognition (psychology)","level":2,"score":0.32989999651908875},{"id":"https://openalex.org/C2779190172","wikidata":"https://www.wikidata.org/wiki/Q4913888","display_name":"Binary data","level":3,"score":0.32510000467300415},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.32499998807907104},{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.3125999867916107},{"id":"https://openalex.org/C66322947","wikidata":"https://www.wikidata.org/wiki/Q11658","display_name":"Transformer","level":3,"score":0.30809998512268066},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.2953999936580658},{"id":"https://openalex.org/C29001434","wikidata":"https://www.wikidata.org/wiki/Q8084","display_name":"Trigonometry","level":2,"score":0.28859999775886536},{"id":"https://openalex.org/C75172450","wikidata":"https://www.wikidata.org/wiki/Q623950","display_name":"Fast Fourier transform","level":2,"score":0.2851000130176544},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.2833000123500824},{"id":"https://openalex.org/C2780801425","wikidata":"https://www.wikidata.org/wiki/Q5164392","display_name":"Construct (python library)","level":2,"score":0.28040000796318054},{"id":"https://openalex.org/C182365436","wikidata":"https://www.wikidata.org/wiki/Q50701","display_name":"Variable (mathematics)","level":2,"score":0.2784999907016754},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.2685000002384186},{"id":"https://openalex.org/C95623464","wikidata":"https://www.wikidata.org/wiki/Q1096149","display_name":"Classifier (UML)","level":2,"score":0.26179999113082886},{"id":"https://openalex.org/C160920958","wikidata":"https://www.wikidata.org/wiki/Q7662746","display_name":"Synthetic data","level":2,"score":0.25949999690055847},{"id":"https://openalex.org/C44280652","wikidata":"https://www.wikidata.org/wiki/Q104837","display_name":"Phase (matter)","level":2,"score":0.25690001249313354},{"id":"https://openalex.org/C207864730","wikidata":"https://www.wikidata.org/wiki/Q179467","display_name":"Fourier series","level":2,"score":0.25290000438690186}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2603.23784","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.23784","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2603.23784","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.23784","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Grokking-the":[0],"phenomenon":[1],"where":[2,56],"validation":[3],"accuracy":[4,152],"of":[5,11,103,142,163],"neural":[6],"networks":[7],"on":[8,90,167],"modular":[9],"addition":[10],"two":[12],"integers":[13],"rises":[14],"long":[15],"after":[16],"training":[17],"data":[18,92,169],"has":[19],"been":[20,22],"memorized-has":[21],"characterized":[23],"in":[24,33,45],"previous":[25],"works":[26],"as":[27],"producing":[28],"sinusoidal":[29],"input":[30,54,198],"weight":[31,66],"distributions":[32,67],"transformers":[34],"and":[35,93,101,109,123,134,156,204],"multi-layer":[36],"perceptrons":[37],"(MLPs).":[38],"We":[39,97],"find":[40],"empirically":[41],"that":[42,170,178],"ReLU":[43],"MLPs":[44],"our":[46],"experimental":[47],"setting":[48],"instead":[49],"learn":[50],"near-binary":[51],"square":[52,121,202],"wave":[53],"weights,":[55,208],"intermediate-valued":[57],"weights":[58,106,115,125,162,199],"appear":[59],"exclusively":[60],"near":[61],"sign-change":[62],"boundaries,":[63],"alongside":[64],"output":[65,124,207],"whose":[68],"dominant":[69,139],"Fourier":[70,140],"phases":[71,157],"satisfy":[72],"a":[73,164],"phase-sum":[74],"relation":[75,82],"$\u03d5_{\\mathrm{out}}":[76],"=":[77],"\u03d5_a":[78],"+":[79],"\u03d5_b$;":[80],"this":[81],"holds":[83],"even":[84],"when":[85,153],"the":[86,99,131,138,143,154,161,183,197,206],"model":[87,145,149,165],"is":[88],"trained":[89,166],"noisy":[91,168],"fails":[94],"to":[95],"grok.":[96],"extract":[98],"frequency":[100],"phase":[102],"each":[104],"neuron's":[105],"via":[107],"DFT":[108],"construct":[110],"an":[111,189],"idealized":[112,148],"MLP:":[113],"Input":[114],"are":[116,158],"replaced":[117],"by":[118,126,130],"perfect":[119],"binary":[120],"waves":[122,203],"cosines,":[127],"both":[128],"parametrized":[129],"frequencies,":[132],"phases,":[133],"amplitudes":[135],"extracted":[136,159],"from":[137,160],"components":[141],"real":[144],"weights.":[146],"This":[147,176],"achieves":[150,172],"95.5%":[151],"frequencies":[155],"itself":[171],"only":[173],"0.23%":[174],"accuracy.":[175],"suggests":[177],"grokking":[179],"does":[180],"not":[181],"discover":[182],"correct":[184],"algorithm,":[185],"but":[186],"rather":[187],"sharpens":[188],"algorithm":[190],"substantially":[191],"encoded":[192],"during":[193],"memorization,":[194],"progressively":[195],"binarizing":[196],"into":[200],"cleaner":[201],"aligning":[205],"until":[209],"generalization":[210],"becomes":[211],"possible.":[212]},"counts_by_year":[],"updated_date":"2026-03-27T06:05:27.210665","created_date":"2026-03-27T00:00:00"}