{"id":"https://openalex.org/W7164884728","doi":"https://doi.org/10.48550/arxiv.2606.14954","title":"Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks","display_name":"Representation Costs in Data Science: Foundations and the Quasi-Banach Spaces of Deep Neural Networks","publication_year":2026,"publication_date":"2026-06-12","ids":{"openalex":"https://openalex.org/W7164884728","doi":"https://doi.org/10.48550/arxiv.2606.14954"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2606.14954","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.14954","pdf_url":"https://arxiv.org/pdf/2606.14954","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/2606.14954","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5020368316","display_name":"Greg Ongie","orcid":"https://orcid.org/0000-0003-1417-3925","observed_orcids":["https://orcid.org/0000-0003-1417-3925"]},"institutions":[{"id":"https://openalex.org/I102461120","display_name":"Marquette University","ror":"https://ror.org/04gr4te78","country_code":"US","type":"education","lineage":["https://openalex.org/I102461120"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Ongie, Greg","raw_affiliation_strings":["Marquette University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Marquette University","institution_ids":["https://openalex.org/I102461120"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5039057672","display_name":"Rahul Parhi","orcid":"https://orcid.org/0000-0002-1971-7699","observed_orcids":["https://orcid.org/0000-0002-1971-7699"]},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Parhi, Rahul","raw_affiliation_strings":["University of California , San Diego"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of California , San Diego","institution_ids":["https://openalex.org/I36258959"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":3,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.5332000255584717,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.5332000255584717,"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/T10320","display_name":"Neural Networks and Applications","score":0.36010000109672546,"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":0.019099999219179153,"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/representer-theorem","display_name":"representer theorem","score":0.9110000133514404},{"id":"https://openalex.org/keywords/quasi-banach-spaces","display_name":"quasi-Banach spaces","score":0.9580000042915344},{"id":"https://openalex.org/keywords/deep-neural-network","display_name":"deep neural network","score":0.6710000038146973},{"id":"https://openalex.org/keywords/relu-neural-networks","display_name":"ReLU networks","score":0.9039999842643738},{"id":"https://openalex.org/keywords/weight-decay-regularization","display_name":"weight decay regularization","score":0.875},{"id":"https://openalex.org/keywords/kernel-methods","display_name":"kernel methods","score":0.7229999899864197},{"id":"https://openalex.org/keywords/reproducing-kernel-hilbert-space","display_name":"reproducing kernel Hilbert space","score":0.9440000057220459},{"id":"https://openalex.org/keywords/wavelets","display_name":"wavelets","score":0.8619999885559082},{"id":"https://openalex.org/keywords/besov-spaces","display_name":"Besov spaces","score":0.9940000176429749},{"id":"https://openalex.org/keywords/shallow-neural-networks","display_name":"shallow neural networks","score":0.8709999918937683},{"id":"https://openalex.org/keywords/data-fitting","display_name":"data fitting","score":0.4020000100135803}],"concepts":[{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.678600013256073},{"id":"https://openalex.org/C117251300","wikidata":"https://www.wikidata.org/wiki/Q1849855","display_name":"Parametric statistics","level":2,"score":0.6148999929428101},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.57669997215271},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.48669999837875366},{"id":"https://openalex.org/C102366305","wikidata":"https://www.wikidata.org/wiki/Q1097688","display_name":"Nonparametric statistics","level":2,"score":0.461899995803833},{"id":"https://openalex.org/C74193536","wikidata":"https://www.wikidata.org/wiki/Q574844","display_name":"Kernel (algebra)","level":2,"score":0.4510999917984009},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4250999987125397},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.42500001192092896},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3880999982357025},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.3797999918460846},{"id":"https://openalex.org/C24574437","wikidata":"https://www.wikidata.org/wiki/Q7135228","display_name":"Parametric model","level":3,"score":0.3772999942302704},{"id":"https://openalex.org/C2984842247","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep neural networks","level":3,"score":0.35839998722076416},{"id":"https://openalex.org/C47702885","wikidata":"https://www.wikidata.org/wiki/Q5441227","display_name":"Feedforward neural network","level":3,"score":0.3488999903202057},{"id":"https://openalex.org/C172623408","wikidata":"https://www.wikidata.org/wiki/Q7314263","display_name":"Representer theorem","level":5,"score":0.3382999897003174},{"id":"https://openalex.org/C142730499","wikidata":"https://www.wikidata.org/wiki/Q934367","display_name":"Function space","level":2,"score":0.3352999985218048},{"id":"https://openalex.org/C122280245","wikidata":"https://www.wikidata.org/wiki/Q620622","display_name":"Kernel method","level":3,"score":0.33000001311302185},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.32659998536109924},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.3215000033378601},{"id":"https://openalex.org/C80884492","wikidata":"https://www.wikidata.org/wiki/Q3345678","display_name":"Reproducing kernel Hilbert space","level":3,"score":0.3001999855041504},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2919999957084656},{"id":"https://openalex.org/C2778334786","wikidata":"https://www.wikidata.org/wiki/Q1586270","display_name":"Variation (astronomy)","level":2,"score":0.2913999855518341},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.28769999742507935},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.27239999175071716},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.2549999952316284},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.2549999952316284}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2606.14954","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.14954","pdf_url":"https://arxiv.org/pdf/2606.14954","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.2606.14954","doi":"https://doi.org/10.48550/arxiv.2606.14954","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2606.14954","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2606.14954","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2606.14954","pdf_url":"https://arxiv.org/pdf/2606.14954","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":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7164884728.pdf","grobid_xml":"https://content.openalex.org/works/W7164884728.grobid-xml"},"referenced_works_count":6,"referenced_works":["https://openalex.org/W2767434645","https://openalex.org/W4392828334","https://openalex.org/W4406191556","https://openalex.org/W4407091390","https://openalex.org/W4415023072","https://openalex.org/W7162453115"],"related_works":["https://openalex.org/W4392828334","https://openalex.org/W4413387854","https://openalex.org/W3090857165","https://openalex.org/W3199436419","https://openalex.org/W3162775926","https://openalex.org/W7128441712","https://openalex.org/W3127165678","https://openalex.org/W2099720998","https://openalex.org/W4417511765","https://openalex.org/W4412112932"],"abstract_inverted_index":{"We":[0],"develop":[1],"a":[2,98,101,112,126],"general":[3],"framework":[4,51],"for":[5],"analyzing":[6],"representation":[7,24,95],"costs":[8],"induced":[9,131],"by":[10,132],"parameter-space":[11,37],"regularizers":[12],"in":[13],"data-fitting":[14],"methods.":[15],"For":[16,88],"an":[17],"arbitrary":[18],"parametric":[19],"method,":[20],"we":[21,91],"define":[22],"its":[23],"cost":[25,96],"and":[26,32,38,45,55,61,64,67,71,103],"native":[27,109],"function":[28],"space,":[29],"prove":[30,92],"existence,":[31],"identify":[33,125],"conditions":[34],"under":[35,105],"which":[36],"function-space":[39],"problems":[40],"have":[41],"equal":[42],"infimal":[43],"values":[44],"minimizers":[46],"transfer":[47],"between":[48],"them.":[49],"This":[50],"yields":[52],"representer":[53],"theorems":[54],"recovers":[56],"classical":[57],"formulations---including":[58],"kernel":[59],"methods":[60],"RKHSs,":[62],"wavelets":[63],"Besov":[65],"spaces,":[66],"shallow":[68],"neural":[69],"networks":[70,84],"variation":[72],"spaces---as":[73],"special":[74],"cases.":[75],"Our":[76],"main":[77],"new":[78],"results":[79,124],"concern":[80],"depth-$L$":[81],"feedforward":[82],"ReLU":[83],"with":[85,115],"weight-decay":[86],"regularization.":[87],"these":[89],"networks,":[90],"that":[93],"the":[94,108],"is":[97,111],"power":[99],"of":[100],"quasi-seminorm":[102],"that,":[104],"suitable":[106],"hypotheses,":[107],"space":[110,114],"quasi-Banach":[113,129],"nonconvex":[116],"unit":[117],"ball":[118],"when":[119],"$L":[120],">":[121],"2$.":[122],"These":[123],"novel":[127],"depth-dependent":[128],"geometry":[130],"weight":[133],"decay.":[134]},"counts_by_year":[{"year":2026,"cited_by_count":3}],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-06-17T00:00:00"}