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Data Visualization with Category Theory and Geometry: With a Critical Analysis and Refinement of UMAP (Mathematics of Data, 3) - Softcover
Synopsis
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.
"synopsis" may belong to another edition of this title.
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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.
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Data Visualization with Category Theory and Geometry
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Springer, Berlin, Max Planck Digital Library, Max-Planck-Institut Für Mathematik In Den Naturwissenschaften, Springer, 2026
ISBN 10: 3031979753
ISBN 13: 9783031979750
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Taschenbuch. Condition: Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -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. 272 pp. Englisch. Seller Inventory # 9783031979750
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Data Visualization with Category Theory and Geometry : With a Critical Analysis and Refinement of UMAP
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Taschenbuch. Condition: Neu. Druck auf Anfrage Neuware - Printed after ordering - 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. Seller Inventory # 9783031979750
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Data Visualization with Category Theory and Geometry
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Data Visualization with Category Theory and Geometry
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Taschenbuch. Condition: Neu. This item is printed on demand - Print on Demand Titel. Neuware -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.Springer Nature Customer Service Center GmbH, Europaplatz 3, 69115 Heidelberg 288 pp. Englisch. Seller Inventory # 9783031979750
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Data Visualization with Category Theory and Geometry | With a Critical Analysis and Refinement of UMAP
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Taschenbuch. Condition: Neu. Data Visualization with Category Theory and Geometry | With a Critical Analysis and Refinement of UMAP | Lukas Silvester Barth (u. a.) | Taschenbuch | Mathematics of Data | xiii | Englisch | 2026 | Springer | EAN 9783031979750 | Verantwortliche Person für die EU: Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg, juergen[dot]hartmann[at]springer[dot]com | Anbieter: preigu. Seller Inventory # 136496154