{"id":"https://openalex.org/W7133719380","doi":"https://doi.org/10.48550/arxiv.2603.03375","title":"The Theory behind UMAP?","display_name":"The Theory behind UMAP?","publication_year":2026,"publication_date":"2026-03-02","ids":{"openalex":"https://openalex.org/W7133719380","doi":"https://doi.org/10.48550/arxiv.2603.03375"},"language":null,"primary_location":{"id":"pmh:doi:10.48550/arxiv.2603.03375","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","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":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5128163872","display_name":"David Wegmann","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Wegmann, David","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5128163872"],"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/T12536","display_name":"Topological and Geometric Data Analysis","score":0.6662999987602234,"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"}},"topics":[{"id":"https://openalex.org/T12536","display_name":"Topological and Geometric Data Analysis","score":0.6662999987602234,"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"}},{"id":"https://openalex.org/T11106","display_name":"Data Management and Algorithms","score":0.051500000059604645,"subfield":{"id":"https://openalex.org/subfields/1711","display_name":"Signal Processing"},"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":0.020500000566244125,"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/metric","display_name":"Metric (unit)","score":0.6225000023841858},{"id":"https://openalex.org/keywords/functor","display_name":"Functor","score":0.5062000155448914},{"id":"https://openalex.org/keywords/realization","display_name":"Realization (probability)","score":0.4465999901294708},{"id":"https://openalex.org/keywords/dimensionality-reduction","display_name":"Dimensionality reduction","score":0.35920000076293945},{"id":"https://openalex.org/keywords/reduction","display_name":"Reduction (mathematics)","score":0.35409998893737793},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.3366999924182892}],"concepts":[{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.6225000023841858},{"id":"https://openalex.org/C156772000","wikidata":"https://www.wikidata.org/wiki/Q864475","display_name":"Functor","level":2,"score":0.5062000155448914},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5059000253677368},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.47119998931884766},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4553000032901764},{"id":"https://openalex.org/C2781089630","wikidata":"https://www.wikidata.org/wiki/Q21856745","display_name":"Realization (probability)","level":2,"score":0.4465999901294708},{"id":"https://openalex.org/C80444323","wikidata":"https://www.wikidata.org/wiki/Q2878974","display_name":"Theoretical computer science","level":1,"score":0.37720000743865967},{"id":"https://openalex.org/C70518039","wikidata":"https://www.wikidata.org/wiki/Q16000077","display_name":"Dimensionality reduction","level":2,"score":0.35920000076293945},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.35409998893737793},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.3366999924182892},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.3278000056743622},{"id":"https://openalex.org/C88626702","wikidata":"https://www.wikidata.org/wiki/Q1128903","display_name":"Continuation","level":2,"score":0.31769999861717224},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3174999952316284},{"id":"https://openalex.org/C26517878","wikidata":"https://www.wikidata.org/wiki/Q228039","display_name":"Key (lock)","level":2,"score":0.3028999865055084},{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.290800005197525},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.27959999442100525},{"id":"https://openalex.org/C136643341","wikidata":"https://www.wikidata.org/wiki/Q1361526","display_name":"Reachability","level":2,"score":0.27720001339912415},{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.2587999999523163},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.2563999891281128}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:doi:10.48550/arxiv.2603.03375","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.48550/arxiv.2603.03375","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.03375","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":"pmh:doi:10.48550/arxiv.2603.03375","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","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":{"In":[0],"2018,":[1],"McInnes":[2,50,76,114],"et":[3,51,77,115],"al.":[4,52],"introduced":[5],"a":[6,22,26,65],"dimensionality":[7],"reduction":[8],"algorithm":[9,109],"called":[10,28],"UMAP,":[11],"which":[12,46],"enjoys":[13],"wide":[14],"popularity":[15],"among":[16],"data":[17],"scientists.":[18],"Their":[19],"work":[20],"introduces":[21],"finite":[23,79,117],"variant":[24,118],"of":[25,45,72,86,107,113],"functor":[27],"the":[29,69,87,94,98,108,111,120],"metric":[30,88],"realization,":[31],"based":[32],"on":[33],"an":[34,83],"unpublished":[35],"draft":[36,40],"by":[37,49],"Spivak.":[38],"This":[39,56],"contains":[41],"many":[42],"errors,":[43],"most":[44],"are":[47],"reproduced":[48],"and":[53,63,75,90,110],"subsequent":[54],"publications.":[55],"article":[57],"aims":[58],"to":[59,119],"repair":[60],"these":[61],"errors":[62],"provide":[64],"self-contained":[66],"document":[67],"with":[68],"full":[70],"derivation":[71],"Spivak's":[73],"functors":[74],"al.'s":[78,116],"variant.":[80],"We":[81],"contribute":[82],"explicit":[84],"description":[85],"realization":[89],"related":[91],"functors.":[92],"At":[93],"end,":[95],"we":[96],"discuss":[97],"UMAP":[99,121],"algorithm,":[100],"as":[101,103],"well":[102],"claims":[104],"about":[105],"properties":[106],"correspondence":[112],"algorithm.":[122]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2026-03-06T00:00:00"}
