Data Visualization with Category Theory and Geometry
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...
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| Principais autores: | , , , , |
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| Formato: | Online |
| Idioma: | inglês |
| Publicado em: |
Springer Nature
2025
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| Acesso em linha: | ONIX_20250813T121456_9783031979736_38 |
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| _version_ | 1869514523484880896 |
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| author | Barth, Lukas Silvester Fahimi, Hannaneh Joharinad, Parvaneh Jost, Jürgen Keck, Janis |
| author_browse | Barth, Lukas Silvester Fahimi, Hannaneh Joharinad, Parvaneh Jost, Jürgen Keck, Janis |
| author_facet | Barth, Lukas Silvester Fahimi, Hannaneh Joharinad, Parvaneh Jost, Jürgen Keck, Janis |
| author_sort | Barth, Lukas Silvester |
| collection | Directory of Open Access Books |
| description | 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. |
| format | Online |
| id | doab-20.500.12854ir-165902 |
| institution | Directory of Open Access Books |
| language | eng |
| publishDate | 2025 |
| publishDateRange | 2025 |
| publishDateSort | 2025 |
| publisher | Springer Nature |
| publisherStr | Springer Nature |
| record_format | ojs |
| spelling | doab-20.500.12854ir-1659022026-03-19T14:47:59Z Data Visualization with Category Theory and Geometry Barth, Lukas Silvester Fahimi, Hannaneh Joharinad, Parvaneh Jost, Jürgen Keck, Janis Dimension reduction Merging local metrics Data visualization Riemannian geometry Applied category theory UMAP Simplicial complexes Metric realization thema EDItEUR::U Computing and Information Technology::UY Computer science::UYA Mathematical theory of computation thema EDItEUR::P Mathematics and Science::PB Mathematics::PBF Algebra 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. 2025-08-14T05:08:36Z 2025-08-14T05:08:36Z 2025-08-13T10:19:32Z 2025 book ONIX_20250813T121456_9783031979736_38 https://library.oapen.org/handle/20.500.12657/105461 9783031979736 9783031979729 https://directory.doabooks.org/handle/20.500.12854/165902 eng Mathematics of Data open access image/jpeg image/jpeg n/a n/a https://library.oapen.org/bitstream/20.500.12657/105461/1/9783031979736.pdf https://library.oapen.org/bitstream/20.500.12657/105461/1/9783031979736.pdf Springer Nature Springer Nature Switzerland 10.1007/978-3-031-97973-6 10.1007/978-3-031-97973-6 9fa3421d-f917-4153-b9ab-fc337c396b5a Max-Planck-Institut für Mathematik in den Naturwissenschaften 7d48738a-6759-43c9-974a-c93e736baeb1 3752e4e5-3f4e-4600-9e94-68912e1902ba d880bcc0-ecff-409c-9896-c00d438cc124 9783031979736 9783031979729 Max Planck Society (MPG) Springer Nature Switzerland 272 Cham [...] [...] Max Planck Society (MPG) MPG 04a0tr310 open access |
| spellingShingle | Dimension reduction Merging local metrics Data visualization Riemannian geometry Applied category theory UMAP Simplicial complexes Metric realization thema EDItEUR::U Computing and Information Technology::UY Computer science::UYA Mathematical theory of computation thema EDItEUR::P Mathematics and Science::PB Mathematics::PBF Algebra Barth, Lukas Silvester Fahimi, Hannaneh Joharinad, Parvaneh Jost, Jürgen Keck, Janis Data Visualization with Category Theory and Geometry |
| title | Data Visualization with Category Theory and Geometry |
| title_full | Data Visualization with Category Theory and Geometry |
| title_fullStr | Data Visualization with Category Theory and Geometry |
| title_full_unstemmed | Data Visualization with Category Theory and Geometry |
| title_short | Data Visualization with Category Theory and Geometry |
| title_sort | data visualization with category theory and geometry |
| topic | Dimension reduction Merging local metrics Data visualization Riemannian geometry Applied category theory UMAP Simplicial complexes Metric realization thema EDItEUR::U Computing and Information Technology::UY Computer science::UYA Mathematical theory of computation thema EDItEUR::P Mathematics and Science::PB Mathematics::PBF Algebra |
| topic_facet | Dimension reduction Merging local metrics Data visualization Riemannian geometry Applied category theory UMAP Simplicial complexes Metric realization thema EDItEUR::U Computing and Information Technology::UY Computer science::UYA Mathematical theory of computation thema EDItEUR::P Mathematics and Science::PB Mathematics::PBF Algebra |
| url | ONIX_20250813T121456_9783031979736_38 |
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