Mathematical principles of topological and geometric data analysis / Parvaneh Joharinad, Jürgen Jost.
Joharinad, Parvaneh, author.; Jost, Jürgen, 1956- author.
2023
QA611
Available Online

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Title Mathematical principles of topological and geometric data analysis / Parvaneh Joharinad, Jürgen Jost.
Author Joharinad, Parvaneh, author.
ISBN 9783031334405 (electronic bk.)
303133440X (electronic bk.)
9783031334399
3031334396
DOI https://doi.org/10.1007/978-3-031-33440-5
Published Cham : Springer, 2023.
Language English
Description 1 online resource (254 pages) : illustrations (black and white, and color).
Item Number 10.1007/978-3-031-33440-5 doi
Call Number QA611
Dewey Decimal Classification 514
Summary This book explores and demonstrates how geometric tools can be used in data analysis. Beginning with a systematic exposition of the mathematical prerequisites, covering topics ranging from category theory to algebraic topology, Riemannian geometry, operator theory and network analysis, it goes on to describe and analyze some of the most important machine learning techniques for dimension reduction, including the different types of manifold learning and kernel methods. It also develops a new notion of curvature of generalized metric spaces, based on the notion of hyperconvexity, which can be used for the topological representation of geometric information. In recent years there has been a fascinating development: concepts and methods originally created in the context of research in pure mathematics, and in particular in geometry, have become powerful tools in machine learning for the analysis of data. The underlying reason for this is that data are typically equipped with some kind of notion of distance, quantifying the differences between data points. Of course, to be successfully applied, the geometric tools usually need to be redefined, generalized, or extended appropriately. Primarily aimed at mathematicians seeking an overview of the geometric concepts and methods that are useful for data analysis, the book will also be of interest to researchers in machine learning and data analysis who want to see a systematic mathematical foundation of the methods that they use.
Access Note Access limited to authorized users.
Source of Description Description based on print version record.
Added Author Jost, Jürgen, 1956- author.
Series Mathematics of data ; v. 2.
Available in Other Form Mathematical principles of topological and geometric data analysis.
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Introduction
Topological foundations, hypercomplexes and homology
Weighted complexes, cohomology and Laplace operators
The Laplace operator and the geometry of graphs
Metric spaces and manifolds
Linear methods: Kernels, variations, and averaging
Nonlinear schemes: Clustering, feature extraction and dimension reduction
Manifold learning, the scheme of Laplacian eigenmaps
Metrics and curvature.

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