Mathematical Principles Of Topological And Geometric Data Analysis

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Mathematical Principles of Topological and Geometric Data Analysis

Parvaneh Joharinad,Jürgen Jost
Mathematical Principles of Topological and Geometric Data Analysis Book PDF

Author : Parvaneh Joharinad,Jürgen Jost
Publisher : Unknown
Page : 0 pages
File Size : 51,5 Mb
Release : 2023
Category : Electronic
ISBN : 3031334418

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Mathematical Principles of Topological and Geometric Data Analysis by Parvaneh Joharinad,Jürgen Jost Pdf

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.

Mathematical Principles of Topological and Geometric Data Analysis

Parvaneh Joharinad,Jürgen Jost
Mathematical Principles of Topological and Geometric Data Analysis Book PDF

Author : Parvaneh Joharinad,Jürgen Jost
Publisher : Springer Nature
Page : 287 pages
File Size : 47,9 Mb
Release : 2023-07-29
Category : Mathematics
ISBN : 9783031334405

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Mathematical Principles of Topological and Geometric Data Analysis by Parvaneh Joharinad,Jürgen Jost Pdf

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.

Computational Topology for Data Analysis

Tamal Krishna Dey,Yusu Wang
Computational Topology for Data Analysis Book PDF

Author : Tamal Krishna Dey,Yusu Wang
Publisher : Cambridge University Press
Page : 455 pages
File Size : 44,5 Mb
Release : 2022-03-10
Category : Computers
ISBN : 9781009098168

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Computational Topology for Data Analysis by Tamal Krishna Dey,Yusu Wang Pdf

This book provides a computational and algorithmic foundation for techniques in topological data analysis, with examples and exercises.

Geometric and Topological Inference

Jean-Daniel Boissonnat,Frédéric Chazal,Mariette Yvinec
Geometric and Topological Inference Book PDF

Author : Jean-Daniel Boissonnat,Frédéric Chazal,Mariette Yvinec
Publisher : Cambridge University Press
Page : 247 pages
File Size : 50,5 Mb
Release : 2018-09-27
Category : Computers
ISBN : 9781108419390

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Geometric and Topological Inference by Jean-Daniel Boissonnat,Frédéric Chazal,Mariette Yvinec Pdf

A rigorous introduction to geometric and topological inference, for anyone interested in a geometric approach to data science.

Topological Data Analysis for Scientific Visualization

Julien Tierny
Topological Data Analysis for Scientific Visualization Book PDF

Author : Julien Tierny
Publisher : Springer
Page : 150 pages
File Size : 42,7 Mb
Release : 2018-01-16
Category : Mathematics
ISBN : 9783319715070

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Topological Data Analysis for Scientific Visualization by Julien Tierny Pdf

Combining theoretical and practical aspects of topology, this book provides a comprehensive and self-contained introduction to topological methods for the analysis and visualization of scientific data. Theoretical concepts are presented in a painstaking but intuitive manner, with numerous high-quality color illustrations. Key algorithms for the computation and simplification of topological data representations are described in detail, and their application is carefully demonstrated in a chapter dedicated to concrete use cases. With its fine balance between theory and practice, "Topological Data Analysis for Scientific Visualization" constitutes an appealing introduction to the increasingly important topic of topological data analysis for lecturers, students and researchers.

Algebraic Foundations for Applied Topology and Data Analysis

Hal Schenck
Algebraic Foundations for Applied Topology and Data Analysis Book PDF

Author : Hal Schenck
Publisher : Springer Nature
Page : 231 pages
File Size : 43,5 Mb
Release : 2022-11-21
Category : Mathematics
ISBN : 9783031066641

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Algebraic Foundations for Applied Topology and Data Analysis by Hal Schenck Pdf

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

Topological Persistence in Geometry and Analysis

Leonid Polterovich,Daniel Rosen,Karina Samvelyan,Jun Zhang
Topological Persistence in Geometry and Analysis Book PDF

Author : Leonid Polterovich,Daniel Rosen,Karina Samvelyan,Jun Zhang
Publisher : American Mathematical Soc.
Page : 128 pages
File Size : 50,8 Mb
Release : 2020-05-11
Category : Education
ISBN : 9781470454951

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Topological Persistence in Geometry and Analysis by Leonid Polterovich,Daniel Rosen,Karina Samvelyan,Jun Zhang Pdf

The theory of persistence modules originated in topological data analysis and became an active area of research in algebraic topology. This book provides a concise and self-contained introduction to persistence modules and focuses on their interactions with pure mathematics, bringing the reader to the cutting edge of current research. In particular, the authors present applications of persistence to symplectic topology, including the geometry of symplectomorphism groups and embedding problems. Furthermore, they discuss topological function theory, which provides new insight into oscillation of functions. The book is accessible to readers with a basic background in algebraic and differential topology.

Topics in Mathematical Analysis and Differential Geometry

Nicolas K. Laos
Topics in Mathematical Analysis and Differential Geometry Book PDF

Author : Nicolas K. Laos
Publisher : World Scientific
Page : 580 pages
File Size : 42,5 Mb
Release : 1998
Category : Mathematics
ISBN : 9810231806

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Topics in Mathematical Analysis and Differential Geometry by Nicolas K. Laos Pdf

This book studies the interplay between mathematical analysis and differential geometry as well as the foundations of these two fields. The development of a unified approach to topological vector spaces, differential geometry and algebraic and differential topology of function manifolds led to the broad expansion of global analysis. This book serves as a self-contained reference on both the prerequisites for further study and the recent research results which have played a decisive role in the advancement of global analysis.

Computational Topology

Herbert Edelsbrunner,John L. Harer
Computational Topology Book PDF

Author : Herbert Edelsbrunner,John L. Harer
Publisher : American Mathematical Society
Page : 241 pages
File Size : 51,6 Mb
Release : 2022-01-31
Category : Mathematics
ISBN : 9781470467692

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Computational Topology by Herbert Edelsbrunner,John L. Harer Pdf

Combining concepts from topology and algorithms, this book delivers what its title promises: an introduction to the field of computational topology. Starting with motivating problems in both mathematics and computer science and building up from classic topics in geometric and algebraic topology, the third part of the text advances to persistent homology. This point of view is critically important in turning a mostly theoretical field of mathematics into one that is relevant to a multitude of disciplines in the sciences and engineering. The main approach is the discovery of topology through algorithms. The book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles. Thus the text could serve equally well in a course taught in a mathematics department or computer science department.

First Concepts of Topology

William G. Chinn,Norman Earl Steenrod
First Concepts of Topology Book PDF

Author : William G. Chinn,Norman Earl Steenrod
Publisher : MAA
Page : 170 pages
File Size : 52,8 Mb
Release : 1966
Category : Mathematics
ISBN : 9780883856185

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First Concepts of Topology by William G. Chinn,Norman Earl Steenrod Pdf

Over 150 problems and solutions.

Topological Data Analysis for Genomics and Evolution

Raul Rabadan,Andrew J. Blumberg
Topological Data Analysis for Genomics and Evolution Book PDF

Author : Raul Rabadan,Andrew J. Blumberg
Publisher : Cambridge University Press
Page : 522 pages
File Size : 51,9 Mb
Release : 2019-12-19
Category : Science
ISBN : 9781108757492

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Topological Data Analysis for Genomics and Evolution by Raul Rabadan,Andrew J. Blumberg Pdf

Biology has entered the age of Big Data. A technical revolution has transformed the field, and extracting meaningful information from large biological data sets is now a central methodological challenge. Algebraic topology is a well-established branch of pure mathematics that studies qualitative descriptors of the shape of geometric objects. It aims to reduce comparisons of shape to a comparison of algebraic invariants, such as numbers, which are typically easier to work with. Topological data analysis is a rapidly developing subfield that leverages the tools of algebraic topology to provide robust multiscale analysis of data sets. This book introduces the central ideas and techniques of topological data analysis and its specific applications to biology, including the evolution of viruses, bacteria and humans, genomics of cancer, and single cell characterization of developmental processes. Bridging two disciplines, the book is for researchers and graduate students in genomics and evolutionary biology as well as mathematicians interested in applied topology.

Geometric Discrepancy

Jiri Matousek
Geometric Discrepancy Book PDF

Author : Jiri Matousek
Publisher : Springer Science & Business Media
Page : 293 pages
File Size : 46,8 Mb
Release : 2009-12-02
Category : Mathematics
ISBN : 9783642039423

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Geometric Discrepancy by Jiri Matousek Pdf

What is the "most uniform" way of distributing n points in the unit square? How big is the "irregularity" necessarily present in any such distribution? This book is an accessible and lively introduction to the area of geometric discrepancy theory, with numerous exercises and illustrations. In separate, more specialized parts, it also provides a comprehensive guide to recent research.

Subdivision Surfaces

Jörg Peters,Ulrich Reif
Subdivision Surfaces Book PDF

Author : Jörg Peters,Ulrich Reif
Publisher : Springer Science & Business Media
Page : 212 pages
File Size : 42,6 Mb
Release : 2008-08-24
Category : Mathematics
ISBN : 9783540764069

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Subdivision Surfaces by Jörg Peters,Ulrich Reif Pdf

Since their first appearance in 1974, subdivision algorithms for generating surfaces of arbitrary topology have gained widespread popularity in computer graphics and are being evaluated in engineering applications. This development was complemented by ongoing efforts to develop appropriate mathematical tools for a thorough analysis, and today, many of the fascinating properties of subdivision are well understood. This book summarizes the current knowledge on the subject. It contains both meanwhile classical results as well as brand-new, unpublished material, such as a new framework for constructing C^2-algorithms. The focus of the book is on the development of a comprehensive mathematical theory, and less on algorithmic aspects. It is intended to serve researchers and engineers - both new to the beauty of the subject - as well as experts, academic teachers and graduate students or, in short, anybody who is interested in the foundations of this flourishing branch of applied geometry.

The Evolution of Chemical Knowledge

Jürgen Jost,Guillermo Restrepo
The Evolution of Chemical Knowledge Book PDF

Author : Jürgen Jost,Guillermo Restrepo
Publisher : Springer Nature
Page : 130 pages
File Size : 41,8 Mb
Release : 2022-10-05
Category : Science
ISBN : 9783031100949

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The Evolution of Chemical Knowledge by Jürgen Jost,Guillermo Restrepo Pdf

Chemistry shapes and creates the disposition of the world's resources and provides novel substances for the welfare and hazard of our civilisation at an exponential rate. Can we model the evolution of chemical knowledge? This book not only provides a positive answer to the question, it provides the formal models and available data to model chemical knowledge as a complex dynamical system based on the mutual interaction of the social, semiotic and material systems of chemistry. These systems, which have evolved over the history, include the scientists and institutions supporting chemical knowledge (social system); theories, concepts and forms of communication (semiotic system) and the substances, reactions and technologies (material system) central for the chemical practice. These three systems, which have traditionally been mostly studied in isolation, are brought together in this book in a grand historical narrative, on the basis of comprehensive data sets and supplemented by appropriate tools for their formal analysis. We thereby develop a comprehensive picture of the evolution of chemistry, needed for better understanding the past, present and future of chemistry as a discipline. The interdisciplinary character of this book and its non-technical language make it an ideal complement to more traditional material in undergraduate and graduate courses in chemistry, history of science and digital humanities.

Higher-Order Systems

Federico Battiston,Giovanni Petri
Higher Order Systems Book PDF

Author : Federico Battiston,Giovanni Petri
Publisher : Springer Nature
Page : 436 pages
File Size : 49,6 Mb
Release : 2022-04-26
Category : Science
ISBN : 9783030913748

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Higher-Order Systems by Federico Battiston,Giovanni Petri Pdf

The book discusses the potential of higher-order interactions to model real-world relational systems. Over the last decade, networks have emerged as the paradigmatic framework to model complex systems. Yet, as simple collections of nodes and links, they are intrinsically limited to pairwise interactions, limiting our ability to describe, understand, and predict complex phenomena which arise from higher-order interactions. Here we introduce the new modeling framework of higher-order systems, where hypergraphs and simplicial complexes are used to describe complex patterns of interactions among any number of agents. This book is intended both as a first introduction and an overview of the state of the art of this rapidly emerging field, serving as a reference for network scientists interested in better modeling the interconnected world we live in.