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Algebraic Foundations for Applied Topology and Data Analysis

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Algebraic Foundations for Applied Topology and Data AnalysisНазвание: Algebraic Foundations for Applied Topology and Data Analysis
Автор: Hal Schenck
Издательство: Springer
Серия: Mathematics of Data
Год: 2022
Страниц: 231
Язык: английский
Формат: pdf (true), epub
Размер: 18.57 MB

This book gives an intuitive and hands-on introduction to Topological Data Analysis (TDA). Covering a wide range of topics at levels of sophistication varying from elementary (matrix algebra) to esoteric (Grothendieck spectral sequence), it offers a mirror of data science aimed at a general mathematical audience.

The required algebraic background is developed in detail. The first third of the book reviews several core areas of mathematics, beginning with basic linear algebra and applications to data fitting and web search algorithms, followed by quick primers on algebra and topology. The middle third introduces algebraic topology, along with applications to sensor networks and voter ranking. The last third covers key contemporary tools in TDA: persistent and multiparameter persistent homology. Also included is a user’s guide to derived functors and spectral sequences (useful but somewhat technical tools which have recently found applications in TDA), and an appendix illustrating a number of software packages used in the field.

How to reveal, characterize, and exploit the structure in data? Meeting this central challenge of modern Data Science requires the development of new mathematical approaches to data analysis, going beyond traditional statistical methods. Fruitful mathematical methods can originate in geometry, topology, algebra, analysis, stochastics, combinatorics, or indeed virtually any field of mathematics. Confronting the challenge of structure in data is already leading to productive new interactions among mathematics, statistics, and Computer Science, notably in Machine Learning. We invite novel contributions (research monographs, advanced textbooks, and lecture notes) presenting substantial mathematics that is relevant for Data Science. Since the methods required to understand data depend on the source and type of the data, we very much welcome contributions comprising significant discussions of the problems presented by particular applications. We also encourage the use of online resources for exercises, software and data sets. Contributions from all mathematical communities that analyze structures in data are welcome. Examples of potential topics include optimization, topological data analysis, compressed sensing, algebraic statistics, information geometry, manifold learning, tensor decomposition, support vector machines, neural networks, and many more.

Based on a course given as part of a masters degree in statistics, the book is appropriate for graduate students.

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