Data Visualization with Category Theory and Geometry: With a Critical Analysis and Refinement of UMAP...

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Lukas Silvester Barth Hannaneh Fahimi

Data Visualization with Category Theory and Geometry

With a Critical Analysis and Refinement of UMAP

Lukas Silvester Barth, Hannaneh Fahimi, Parvaneh Joharinad , Jürgen Jost

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English

Springer International Publishing AG
08 August 2025

Algebra; Geometry; Mathematical theory of computation; Information visualisation

Series: Mathematics of Data

Summary
Details

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.

By: Lukas Silvester Barth, Hannaneh Fahimi, Parvaneh Joharinad, Jürgen Jost, Janis Keck
Imprint: Springer International Publishing AG
Country of Publication: Switzerland
Volume: 3
Dimensions: Height: 235mm, Width: 155mm,
ISBN: 9783031979729
ISBN 10: 3031979729
Series: Mathematics of Data
Pages: 272
Publication Date: 08 August 2025
Audience: Professional and scholarly , College/higher education , Undergraduate , Further / Higher Education
Format: Hardback
Publisher's Status: Active