國立高雄大學圖資館 |

登入

回首頁

Data visualization with category theory and geometrywith a critical analysis and refinement of UMAP /

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	Data visualization with category theory and geometryby Lukas Silvester Barth ... [et al.].
其他題名:	with a critical analysis and refinement of UMAP /
其他作者:	Barth, Lukas Silvester.
出版者:	Cham :Springer Nature Switzerland :2025.
面頁冊數:	xiii, 272 p. :ill. (chiefly col.), digital ;24 cm.
Contained By:	Springer Nature eBook
標題:	Information visualization.
電子資源:	https://doi.org/10.1007/978-3-031-97973-6
ISBN:	9783031979736$q(electronic bk.)

Data visualization with category theory and geometrywith a critical analysis and refinement of UMAP /
Data visualization with category theory and geometrywith a critical analysis and refinement of UMAP /[electronic resource] :by Lukas Silvester Barth ... [et al.]. - Cham :Springer Nature Switzerland :2025. - xiii, 272 p. :ill. (chiefly col.), digital ;24 cm. - Mathematics of data,v. 32731-4111 ;. - Mathematics of data ;v. 1..

Chapter 1. Introduction -- Chapter 2. Illustrating UMAP on some simple data sets -- Chapter 3. Metrics and Riemannian manifolds -- Chapter 4. Merging fuzzy simplicial sets and metric spaces: A category theoretical approach -- Chapter 5. UMAP -- Chapter 6. IsUMap: An alternative to the UMAP embedding.

Open access.

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.

ISBN: 9783031979736$q(electronic bk.)

Standard No.: 10.1007/978-3-031-97973-6doiSubjects--Topical Terms:

248886
Information visualization.

LC Class. No.: QA76.9.I52

Dewey Class. No.: 001.4226

Data visualization with category theory and geometrywith a critical analysis and refinement of UMAP /
LDR:02809nmm a2200349 a 4500 001 686031
003 DE-He213
005 20250808130241.0
006 m d
007 cr nn 008maaau
008 260209s2025 sz s 0 eng d
020 $a 9783031979736$q(electronic bk.)
020 $a 9783031979729$q(paper)
024 7 $a 10.1007/978-3-031-97973-6 $2 doi
035 $a 978-3-031-97973-6
040 $a GP $c GP
041 0 $a eng
050 4 $a QA76.9.I52
072 7 $a UYA $2 bicssc
072 7 $a MAT003000 $2 bisacsh
072 7 $a UYA $2 thema
082 0 4 $a 001.4226 $2 23
090 $a QA76.9.I52 $b D232 2025
245 0 0 $a Data visualization with category theory and geometry $h [electronic resource] : $b with a critical analysis and refinement of UMAP / $c by Lukas Silvester Barth ... [et al.].
260 $a Cham : $b Springer Nature Switzerland : $b Imprint: Springer, $c 2025.
300 $a xiii, 272 p. : $b ill. (chiefly col.), digital ; $c 24 cm.
490 1 $a Mathematics of data, $x 2731-4111 ; $v v. 3
505 0 $a Chapter 1. Introduction -- Chapter 2. Illustrating UMAP on some simple data sets -- Chapter 3. Metrics and Riemannian manifolds -- Chapter 4. Merging fuzzy simplicial sets and metric spaces: A category theoretical approach -- Chapter 5. UMAP -- Chapter 6. IsUMap: An alternative to the UMAP embedding.
506 $a Open access.
520 $a 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.
650 0 $a Information visualization. $3 248886
650 0 $a Dimension reduction (Statistics) $3 592293
650 1 4 $a Mathematical Applications in Computer Science. $3 530811
650 2 4 $a Category Theory, Homological Algebra. $3 275954
700 1 $a Barth, Lukas Silvester. $3 1000141
710 2 $a SpringerLink (Online service) $3 273601
773 0 $t Springer Nature eBook
830 0 $a Mathematics of data ; $v v. 1. $3 939468
856 4 0 $u https://doi.org/10.1007/978-3-031-97973-6
950 $a Mathematics and Statistics (SpringerNature-11649)