{"id":"https://openalex.org/W7160911938","doi":"https://doi.org/10.48550/arxiv.2605.08377","title":"Embedding Dimension Lower Bounds for Universality of Deep Sets and Janossy Pooling","display_name":"Embedding Dimension Lower Bounds for Universality of Deep Sets and Janossy Pooling","publication_year":2026,"publication_date":"2026-05-08","ids":{"openalex":"https://openalex.org/W7160911938","doi":"https://doi.org/10.48550/arxiv.2605.08377"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2605.08377","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.08377","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":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2605.08377","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5126785209","display_name":"Ali Syed","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Syed, Ali","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5073276348","display_name":"Aditya Nambiar","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Nambiar, Aditya","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5135915270","display_name":"Jonathan W. Siegel","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Siegel, Jonathan W.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.46230000257492065,"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":0.46230000257492065,"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/T11273","display_name":"Advanced Graph Neural Networks","score":0.33009999990463257,"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.0430000014603138,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/embedding","display_name":"Embedding","score":0.7732999920845032},{"id":"https://openalex.org/keywords/universality","display_name":"Universality (dynamical systems)","score":0.7366999983787537},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.4790000021457672},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.40709999203681946},{"id":"https://openalex.org/keywords/homogeneous-space","display_name":"Homogeneous space","score":0.40119999647140503},{"id":"https://openalex.org/keywords/effective-dimension","display_name":"Effective dimension","score":0.399399995803833},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.39100000262260437},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.3828999996185303}],"concepts":[{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.7732999920845032},{"id":"https://openalex.org/C183992945","wikidata":"https://www.wikidata.org/wiki/Q2495574","display_name":"Universality (dynamical systems)","level":2,"score":0.7366999983787537},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6700000166893005},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48910000920295715},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.4790000021457672},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.4650999903678894},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.40709999203681946},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.40119999647140503},{"id":"https://openalex.org/C115311070","wikidata":"https://www.wikidata.org/wiki/Q5347255","display_name":"Effective dimension","level":3,"score":0.399399995803833},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.39100000262260437},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.3828999996185303},{"id":"https://openalex.org/C70437156","wikidata":"https://www.wikidata.org/wiki/Q7228652","display_name":"Pooling","level":2,"score":0.3666999936103821},{"id":"https://openalex.org/C2778049214","wikidata":"https://www.wikidata.org/wiki/Q7512234","display_name":"Sigma","level":2,"score":0.35089999437332153},{"id":"https://openalex.org/C119322782","wikidata":"https://www.wikidata.org/wiki/Q2662236","display_name":"VC dimension","level":2,"score":0.30559998750686646},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.3037000000476837},{"id":"https://openalex.org/C2984842247","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep neural networks","level":3,"score":0.27559998631477356},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.2736000120639801},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.27000001072883606},{"id":"https://openalex.org/C108598597","wikidata":"https://www.wikidata.org/wiki/Q124255","display_name":"Conic section","level":2,"score":0.25360000133514404},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.2517000138759613}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2605.08377","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.08377","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":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2605.08377","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2605.08377","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":"Preprint"},"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":{"In":[0,31],"many":[1],"practical":[2],"applications":[3],"it":[4],"is":[5,55],"important":[6,16,75],"to":[7,85,119],"build":[8],"symmetries":[9],"into":[10],"neural":[11],"network":[12,35],"architectures.":[13],"Consider":[14],"the":[15,34,60,67,74,80,102,114,134,140],"case":[17,33],"of":[18,25,42,69,77,88,105],"permutation":[19],"symmetry":[20],"on":[21,39,101,139],"point":[22],"clouds":[23],"consisting":[24],"$n$":[26,43],"points":[27,44],"in":[28,45,72],"$d$":[29],"dimensions.":[30],"this":[32,70,89,106,112],"learns":[36],"a":[37,40,48,93,120],"function":[38],"set":[41],"$\\mathbb{R}^d$,":[46],"and":[47],"natural":[49],"paradigm":[50],"for":[51,123],"constructing":[52],"invariant":[53],"networks":[54],"Janossy":[56,130],"pooling,":[57,131],"which":[58],"generalizes":[59],"popular":[61],"Deep":[62,110],"Sets":[63],"architecture.":[64,90],"We":[65],"study":[66],"universality":[68,87],"approach,":[71],"particular":[73],"question":[76],"how":[78],"large":[79],"embedding":[81,107,142],"dimension":[82,117,143],"must":[83],"be":[84],"guarantee":[86],"Specifically,":[91],"using":[92],"novel":[94],"technique,":[95],"we":[96,132],"prove":[97,133],"new":[98],"lower":[99,137],"bounds":[100],"required":[103,141],"size":[104],"dimension.":[108],"For":[109,128],"Sets,":[111],"gives":[113],"correct":[115],"minimal":[116],"up":[118],"constant":[121],"factor":[122],"all":[124],"$d":[125],"&gt;":[126,146],"1$.":[127,147],"$k$-ary":[129],"first":[135],"non-trivial":[136],"bound":[138],"when":[144],"$k":[145]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-05-13T00:00:00"}
