Mathematical Principles Of Topological And Geometric Data Analysis
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Mathematical Principles Of Topological And Geometric Data Analysis
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Author : Parvaneh Joharinad
language : en
Publisher: Springer Nature
Release Date : 2023-07-29
Mathematical Principles Of Topological And Geometric Data Analysis written by Parvaneh Joharinad and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-07-29 with Mathematics categories.
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
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Author : Tamal Krishna Dey
language : en
Publisher: Cambridge University Press
Release Date : 2022-03-10
Computational Topology For Data Analysis written by Tamal Krishna Dey and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-03-10 with Computers categories.
This book provides a computational and algorithmic foundation for techniques in topological data analysis, with examples and exercises.
Geometric And Topological Inference
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Author : Jean-Daniel Boissonnat
language : en
Publisher: Cambridge University Press
Release Date : 2018-09-27
Geometric And Topological Inference written by Jean-Daniel Boissonnat and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-09-27 with Computers categories.
A rigorous introduction to geometric and topological inference, for anyone interested in a geometric approach to data science.
Topological Data Analysis With Applications
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Author : Gunnar Carlsson
language : en
Publisher: Cambridge University Press
Release Date : 2021-12-16
Topological Data Analysis With Applications written by Gunnar Carlsson and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-12-16 with Computers categories.
This timely text introduces topological data analysis from scratch, with detailed case studies.
Higher Order Systems
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Author : Federico Battiston
language : en
Publisher: Springer Nature
Release Date : 2022-04-26
Higher Order Systems written by Federico Battiston and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-04-26 with Science categories.
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.
The Evolution Of Chemical Knowledge
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Author : Jürgen Jost
language : en
Publisher: Springer Nature
Release Date : 2022-10-05
The Evolution Of Chemical Knowledge written by Jürgen Jost and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-10-05 with Science categories.
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.
Principles Of Computational Geometry
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Author : Devang Patil
language : en
Publisher: Educohack Press
Release Date : 2025-02-20
Principles Of Computational Geometry written by Devang Patil and has been published by Educohack Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2025-02-20 with Science categories.
"Principles of Computational Geometry" delves into the intersection of mathematics, algorithms, and computer science to solve geometric problems using computational methods. We cover a wide range of topics, from fundamental geometric concepts to advanced algorithmic techniques. Our book explores geometric data structures and algorithms designed to efficiently tackle issues like geometric modeling, spatial analysis, and geometric optimization. We introduce readers to key concepts like convex hulls, Voronoi diagrams, and Delaunay triangulations, which serve as building blocks for solving complex geometric problems. Additionally, we discuss techniques for geometric transformation, intersection detection, and geometric search, providing the tools needed to analyze and manipulate geometric data effectively. Throughout the text, we highlight practical applications of computational geometry, ranging from computer graphics and image processing to robotics and geographic information systems. We also explore the theoretical underpinnings of computational geometry, offering insights into the mathematical foundations of algorithms and their computational complexity. Overall, "Principles of Computational Geometry" serves as a comprehensive guide for students, researchers, and practitioners interested in leveraging computational methods to solve geometric problems efficiently and effectively. With its blend of theory and practical applications, our book offers a valuable resource for anyone exploring the rich and diverse field of computational geometry.
Mathematical Problems In Data Science
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Author : Li M. Chen
language : en
Publisher: Springer
Release Date : 2015-12-15
Mathematical Problems In Data Science written by Li M. Chen and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-12-15 with Computers categories.
This book describes current problems in data science and Big Data. Key topics are data classification, Graph Cut, the Laplacian Matrix, Google Page Rank, efficient algorithms, hardness of problems, different types of big data, geometric data structures, topological data processing, and various learning methods. For unsolved problems such as incomplete data relation and reconstruction, the book includes possible solutions and both statistical and computational methods for data analysis. Initial chapters focus on exploring the properties of incomplete data sets and partial-connectedness among data points or data sets. Discussions also cover the completion problem of Netflix matrix; machine learning method on massive data sets; image segmentation and video search. This book introduces software tools for data science and Big Data such MapReduce, Hadoop, and Spark. This book contains three parts. The first part explores the fundamental tools of data science. It includes basic graph theoretical methods, statistical and AI methods for massive data sets. In second part, chapters focus on the procedural treatment of data science problems including machine learning methods, mathematical image and video processing, topological data analysis, and statistical methods. The final section provides case studies on special topics in variational learning, manifold learning, business and financial data rec overy, geometric search, and computing models. Mathematical Problems in Data Science is a valuable resource for researchers and professionals working in data science, information systems and networks. Advanced-level students studying computer science, electrical engineering and mathematics will also find the content helpful.
Stochastic Transport In Upper Ocean Dynamics Iii
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Author : Bertrand Chapron
language : en
Publisher: Springer Nature
Release Date : 2024-11-18
Stochastic Transport In Upper Ocean Dynamics Iii written by Bertrand Chapron and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-11-18 with Mathematics categories.
This open-access proceedings volume brings selected, peer-reviewed contributions presented at the Fourth Stochastic Transport in Upper Ocean Dynamics (STUOD) 2023 Workshop, held at IFREMER in Plouzané, France, September 25–28, 2023. The STUOD project is supported by an ERC Synergy Grant, and led by Imperial College London, the National Institute for Research in Computer Science and Automatic Control (INRIA), and the French Research Institute for Exploitation of the Sea (IFREMER). The project aims to deliver new capabilities for assessing variability and uncertainty in upper ocean dynamics. It will provide decision makers a means of quantifying the effects of local patterns of sea level rise, heat uptake, carbon storage, and change of oxygen content and pH in the ocean. Its multimodal monitoring will enhance the scientific understanding of marine debris transport, tracking of oil spills, and accumulation of plastic in the sea. All topics of these proceedings are essential to the scientific foundations of oceanography which has a vital role in climate science. Studies convened in this volume focus on a range of fundamental areas, including: Observations at a high resolution of upper ocean properties such as temperature, salinity, topography, wind, waves and velocity; Large-scale numerical simulations; Data-based stochastic equations for upper ocean dynamics that quantify simulation error; Stochastic data assimilation to reduce uncertainty. These fundamental subjects in modern science and technology are urgently required in order to meet the challenges of climate change faced today by human society. This proceedings volume represents a lasting legacy of crucial scientific expertise to help meet this ongoing challenge, for the benefit of academics and professionals in pure and applied mathematics, computational science, data analysis, data assimilation, and oceanography.
Geometry And Topology In Hamiltonian Dynamics And Statistical Mechanics
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Author : Marco Pettini
language : en
Publisher: Springer Science & Business Media
Release Date : 2007-06-14
Geometry And Topology In Hamiltonian Dynamics And Statistical Mechanics written by Marco Pettini and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-06-14 with Mathematics categories.
Itisaspecialpleasureformetowritethisforewordforaremarkablebookbya remarkableauthor.MarcoPettiniisadeepthinker,whohasspentmanyyears probing the foundations of Hamiltonian chaos and statistical mechanics, in particular phase transitions, from the point of view of geometry and topology. Itisinparticularthequalityofmindoftheauthorandhisdeepphysical,as well as mathematical insights which make this book so special and inspiring. It is a “must” for those who want to venture into a new approach to old problems or want to use new tools for new problems. Although topology has penetrated a number of ?elds of physics, a broad participationoftopologyintheclari?cationandprogressoffundamentalpr- lems in the above-mentioned ?elds has been lacking. The new perspectives topology gives to the above-mentioned problems are bound to help in their clari?cation and to spread to other ?elds of science. The sparsity of geometric thinking and of its use to solve fundamental problems, when compared with purely analytical methods in physics, could be relieved and made highly productive using the material discussed in this book. It is unavoidable that the physicist reader may have then to learn some new mathematics and be challenged to a new way of thinking, but with the author as a guide, he is assured of the best help in achieving this that is presently available.