Data Visualization with Category Theory and Geometry

Lukas Silvester Barth

ISBN 10: 3031979729 / ISBN 13: 9783031979729

Published by Springer, Berlin, Springer Nature Switzerland, Max Planck Digital Library, Max-Planck-Institut Für Mathematik In Den Naturwissenschaften, Springer, 2025

About this Item

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 # 9783031979729

Bibliographic Details

• Title: Data Visualization with Category Theory and Geometry
• Publisher: Springer, Berlin, Springer Nature Switzerland, Max Planck Digital Library, Max-Planck-Institut Für Mathematik In Den Naturwissenschaften, Springer
• Publication Date: 2025
• Language: English
• ISBN 10: 3031979729
• ISBN 13: 9783031979729
• Binding: Buch
• Condition: Neu
• Dimensions: 235W x 155H x 19D millimeters 235 millimeters width by 155 millimeters height by 19 millimeters depth
• Weight: 543 grams

About this title

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.