Author: Abigail Hickok
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
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Book Description
The fields of topological data analysis (TDA) and geometric data analysis (GDA) use algebraic topology and differential geometry to capture topological and geometric structural properties of data that are not captured by other methods in data science and machine learning. The primary tool of TDA---and one of the focuses of this dissertation---is persistent homology, which measures the connected components, holes, and higher-dimensional voids of a data set and tracks how those voids emerge and disappear at different scales. The objective of GDA is to extract new insights by considering geometric invariants of a manifold, such as curvature, rather than topological invariants. Previous studies have demonstrated the power of geometry and topology for analyzing data in complex systems, neuroscience, biology, and many other fields. In my thesis, I study both the theory and applications of topological and geometric data analysis. In the first part of the dissertation, I establish and analyze a new construction, called a "persistence diagram (PD) bundle," for doing multiparameter TDA, and I develop an algorithm to compute a certain class of PD bundles. PD bundles generalize several important constructions in TDA: vineyards, the persistent homology transform, and fibered barcodes. In the second part of the dissertation, I apply TDA to several geospatial and geospatiotemporal data sets. In the last part of the dissertation, I introduce a new method for curvature estimation in point-cloud data.
Author: Jean-Daniel Boissonnat
Publisher: Cambridge University Press
ISBN: 1108419399
Category : Computers
Languages : en
Pages : 247
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Book Description
A rigorous introduction to geometric and topological inference, for anyone interested in a geometric approach to data science.
Author: Kisung You
Publisher:
ISBN:
Category :
Languages : en
Pages : 114
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Book Description
Author: Gunnar Carlsson
Publisher: Cambridge University Press
ISBN: 1108838650
Category : Computers
Languages : en
Pages : 233
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Book Description
This timely text introduces topological data analysis from scratch, with detailed case studies.
Author: Robert W. Ghrist
Publisher: Createspace Independent Publishing Platform
ISBN: 9781502880857
Category : Mathematics
Languages : en
Pages : 0
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Book Description
This book gives an introduction to the mathematics and applications comprising the new field of applied topology. The elements of this subject are surveyed in the context of applications drawn from the biological, economic, engineering, physical, and statistical sciences.
Author: Tamal Krishna Dey
Publisher: Cambridge University Press
ISBN: 1009103199
Category : Mathematics
Languages : en
Pages : 456
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Book Description
Topological data analysis (TDA) has emerged recently as a viable tool for analyzing complex data, and the area has grown substantially both in its methodologies and applicability. Providing a computational and algorithmic foundation for techniques in TDA, this comprehensive, self-contained text introduces students and researchers in mathematics and computer science to the current state of the field. The book features a description of mathematical objects and constructs behind recent advances, the algorithms involved, computational considerations, as well as examples of topological structures or ideas that can be used in applications. It provides a thorough treatment of persistent homology together with various extensions – like zigzag persistence and multiparameter persistence – and their applications to different types of data, like point clouds, triangulations, or graph data. Other important topics covered include discrete Morse theory, the Mapper structure, optimal generating cycles, as well as recent advances in embedding TDA within machine learning frameworks.
Author: Jean-Daniel Boissonnat
Publisher: Cambridge University Press
ISBN: 1108317618
Category : Computers
Languages : en
Pages : 247
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Book Description
Geometric and topological inference deals with the retrieval of information about a geometric object using only a finite set of possibly noisy sample points. It has connections to manifold learning and provides the mathematical and algorithmic foundations of the rapidly evolving field of topological data analysis. Building on a rigorous treatment of simplicial complexes and distance functions, this self-contained book covers key aspects of the field, from data representation and combinatorial questions to manifold reconstruction and persistent homology. It can serve as a textbook for graduate students or researchers in mathematics, computer science and engineering interested in a geometric approach to data science.
Author: Parvaneh Joharinad
Publisher: Springer Nature
ISBN: 303133440X
Category : Mathematics
Languages : en
Pages : 287
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Book Description
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.
Author: Raúl Rabadán
Publisher: Cambridge University Press
ISBN: 1108753396
Category : Science
Languages : en
Pages : 521
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Book Description
Biology has entered the age of Big Data. The 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 questions to a comparison of algebraic invariants, such as numbers, which are typically easier to solve. 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 alongside mathematicians interested in applied topology.
Author: Leonid Polterovich
Publisher: American Mathematical Soc.
ISBN: 1470454955
Category : Education
Languages : en
Pages : 128
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Book Description
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.
