Data Visualization with Category Theory and Geometry With a Critical Analysis and Refinement of UMAP
This open access book provides a robust exposition of the mathematical foundations of data representation, focusing on two essential pillars of dimensionality reduction methods, namely geometry in general and Riemannian geometry in particular, and category theory. Presenting a list of examples consi...
Saved in:
| Main Authors: | , , , , |
|---|---|
| Format: | eBook |
| Language: | English |
| Published: |
Cham
Springer Nature
2025
|
| Series: | Mathematics of Data |
| Subjects: | |
| ISBN: | 3031979737, 9783031979736, 3031979729, 9783031979729 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | This open access book provides a robust exposition of the mathematical foundations of data representation, focusing on two essential pillars of dimensionality reduction methods, namely geometry in general and Riemannian geometry in particular, and category theory. Presenting a list of examples consisting of both geometric objects and empirical datasets, this book provides insights into the different effects of dimensionality reduction techniques on data representation and visualization, with the aim of guiding the reader in understanding the expected results specific to each method in such scenarios. As a showcase, the dimensionality reduction method of “Uniform Manifold Approximation and Projection” (UMAP) has been used in this book, as it is built on theoretical foundations from all the areas we want to highlight here. Thus, this book also aims to systematically present the details of constructing a metric representation of a locally distorted metric space, which is essentially the problem that UMAP is trying to address, from a more general perspective. Explaining how UMAP fits into this broader framework, while critically evaluating the underlying ideas, this book finally introduces an alternative algorithm to UMAP. This algorithm, called IsUMap, retains many of the positive features of UMAP, while improving on some of its drawbacks. |
|---|---|
| AbstractList | This open access book provides a robust exposition of the mathematical foundations of data representation, focusing on two essential pillars of dimensionality reduction methods, namely geometry in general and Riemannian geometry in particular, and category theory. Presenting a list of examples consisting of both geometric objects and empirical datasets, this book provides insights into the different effects of dimensionality reduction techniques on data representation and visualization, with the aim of guiding the reader in understanding the expected results specific to each method in such scenarios. As a showcase, the dimensionality reduction method of “Uniform Manifold Approximation and Projection” (UMAP) has been used in this book, as it is built on theoretical foundations from all the areas we want to highlight here. Thus, this book also aims to systematically present the details of constructing a metric representation of a locally distorted metric space, which is essentially the problem that UMAP is trying to address, from a more general perspective. Explaining how UMAP fits into this broader framework, while critically evaluating the underlying ideas, this book finally introduces an alternative algorithm to UMAP. This algorithm, called IsUMap, retains many of the positive features of UMAP, while improving on some of its drawbacks. |
| Author | Fahimi, Hannaneh Joharinad, Parvaneh Jost, Jürgen Barth, Lukas Silvester Keck, Janis |
| Author_xml | – sequence: 1 fullname: Barth, Lukas Silvester – sequence: 2 fullname: Fahimi, Hannaneh – sequence: 3 fullname: Joharinad, Parvaneh – sequence: 4 fullname: Jost, Jürgen – sequence: 5 fullname: Keck, Janis |
| BookMark | eNpNjM1KAzEURgMqaGsfQHAxLxC9SSbJZCNI1SoU3JRuw538tMHppMyMyPj0ttaFq_MdPjgTct7mNhByw-COAeh7oysqKAhGjTZaUHVGJuKgv6YvyazvUw2KgRJS8itSPuGAxTr1n9ikbxxSbouvNGyLOQ5hk7uxWG3DEdj6YhHyLgzdeE0uIjZ9mP1xStYvz6v5K12-L97mj0uKTLOSUx0D6LqsHLpaegOx8hVoBxGFcsIrF4M3spYleB3RCA_GoGPRgAPOnBdT8nAKZ9yH1u67tMNutBmTbVLdnfbxyd3GcrASwDKupLYMZKnYIXD7P-Az1jl_9JYpaYCLHzovW5g |
| ContentType | eBook |
| DBID | V1H A7I |
| DOI | 10.1007/978-3-031-97973-6 |
| DatabaseName | DOAB: Directory of Open Access Books OAPEN |
| DatabaseTitleList | |
| Database_xml | – sequence: 1 dbid: V1H name: DOAB: Directory of Open Access Books url: https://directory.doabooks.org/ sourceTypes: Publisher |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics Computer Science |
| ExternalDocumentID | oai_library_oapen_org_20_500_12657_105461 165902 |
| GrantInformation_xml | – fundername: Max-Planck-Institut für Mathematik in den Naturwissenschaften |
| GroupedDBID | ALMA_UNASSIGNED_HOLDINGS IEZ V1H A7I |
| ID | FETCH-LOGICAL-a17142-7fe07b48cacb5d90f8d807c0fa36c3d6cfed95b540d7fa93d099ac1f90c021cd3 |
| IEDL.DBID | A7I |
| ISBN | 3031979737 9783031979736 3031979729 9783031979729 |
| IngestDate | Mon Dec 01 21:23:13 EST 2025 Mon Nov 03 04:46:22 EST 2025 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | false |
| IsScholarly | false |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-a17142-7fe07b48cacb5d90f8d807c0fa36c3d6cfed95b540d7fa93d099ac1f90c021cd3 |
| OpenAccessLink | https://library.oapen.org/handle/20.500.12657/105461 |
| PageCount | 272 |
| ParticipantIDs | oapen_primary_oai_library_oapen_org_20_500_12657_105461 oapen_doabooks_165902 |
| PublicationCentury | 2000 |
| PublicationDate | 2025 |
| PublicationDateYYYYMMDD | 2025-01-01 |
| PublicationDate_xml | – year: 2025 text: 2025 |
| PublicationDecade | 2020 |
| PublicationPlace | Cham |
| PublicationPlace_xml | – name: Cham |
| PublicationSeriesTitle | Mathematics of Data |
| PublicationYear | 2025 |
| Publisher | Springer Nature |
| Publisher_xml | – name: Springer Nature |
| SSID | ssib061063552 ssib061768739 ssib061134969 |
| Score | 2.4441876 |
| Snippet | This open access book provides a robust exposition of the mathematical foundations of data representation, focusing on two essential pillars of dimensionality... |
| SourceID | oapen |
| SourceType | Publisher |
| SubjectTerms | Algebra Applied category theory Computer science Computing and Information Technology Data visualization Dimension reduction Mathematical theory of computation Mathematics Mathematics and Science Merging local metrics Metric realization Riemannian geometry Simplicial complexes UMAP |
| Subtitle | With a Critical Analysis and Refinement of UMAP |
| Title | Data Visualization with Category Theory and Geometry |
| URI | https://directory.doabooks.org/handle/20.500.12854/165902 https://library.oapen.org/handle/20.500.12657/105461 |
| Volume | 3 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV09T8MwELVQywBLoRRRvpSBNZUTJ3Y8okLpABUDirpFju1IHWhQEpD677lz0o-5i6XLKUp8OsvvbL13hDzlOrFcJ7C-FS18QOCRn0SKwxCFOQ1zrl0fsvRdLBbJcik_O1JYvT-7mJQKqnl3k9-qDUCRPokpiiHwGMnicYQ1T5-HocS8fha7ogfwAG6i4d5GBT4u9zYgbMEkEj2QwiMk4MsDg4lWl-fQeWgzvr0c7fRpmQ8e37l81zcJf_1gk5oNjpveBelbJDxckhO7HpLBts2D1636ITn_2Em71lckelGN8tJVjWzMlsPp4WGuN0XNibLaeC3d34Mvem-2_LZNtRmRdPb6NZ37XesFX2FHdADdhaUijxKtdB4bSYvEJFRoWijGNTNcF9bIOAe8Z0ShJDOANJUOCkk1oAZt2DXprcu1vSGe0owJLa0BsAgZEOcFQ1BgglglQlo1JiMXjsyUCiuLOgs4isqMiWif_7S6GxkqYXcBzFoPBDALaQaRy1zksjZyt0e_eUfOQmzw685Y7kmvqX7tAznVf82qrh5dnsGYBvN_3XLG6A |
| linkProvider | Open Access Publishing in European Networks |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.title=Data+Visualization+with+Category+Theory+and+Geometry&rft.au=Barth%2C+Lukas+Silvester&rft.au=Fahimi%2C+Hannaneh&rft.au=Joharinad%2C+Parvaneh&rft.au=Jost%2C+J%C3%BCrgen&rft.series=Mathematics+of+Data&rft.date=2025-01-01&rft.pub=Springer+Nature&rft.isbn=9783031979729&rft.volume=3&rft_id=info:doi/10.1007%2F978-3-031-97973-6&rft.externalDBID=A7I&rft.externalDocID=oai_library_oapen_org_20_500_12657_105461 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=9783031979736/lc.gif&client=summon&freeimage=true |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=9783031979736/mc.gif&client=summon&freeimage=true |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=9783031979736/sc.gif&client=summon&freeimage=true |