Mathematical Principles of Topological and Geometric Data Analysis
暫譯: 拓撲與幾何數據分析的數學原理
Joharinad, Parvaneh, Jost, Jürgen
- 出版商: Springer
- 出版日期: 2024-07-31
- 售價: $2,220
- 貴賓價: 9.5 折 $2,109
- 語言: 英文
- 頁數: 281
- 裝訂: Quality Paper - also called trade paper
- ISBN: 3031334426
- ISBN-13: 9783031334429
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相關分類:
Data Science
海外代購書籍(需單獨結帳)
商品描述
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 somekind 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.
商品描述(中文翻譯)
這本書探討並展示幾何工具如何應用於數據分析。首先系統性地介紹數學先備知識，涵蓋從範疇理論到代數拓撲、黎曼幾何、算子理論和網絡分析等主題，接著描述和分析一些最重要的機器學習技術以進行降維，包括不同類型的流形學習和核方法。它還基於超凸性（hyperconvexity）的概念，發展了一種新的廣義度量空間的曲率概念，這可以用於幾何信息的拓撲表示。
近年來出現了一個引人入勝的發展：最初在純數學研究，特別是幾何學的背景下創造的概念和方法，已成為機器學習中分析數據的強大工具。這背後的原因是數據通常配備某種距離的概念，以量化數據點之間的差異。當然，為了成功應用，幾何工具通常需要適當地重新定義、概括或擴展。
本書主要針對尋求幾何概念和方法概覽的數學家，這些概念和方法對數據分析有用，同時也會吸引希望了解其所使用方法的系統數學基礎的機器學習和數據分析研究人員。
作者簡介
Parvaneh Joharinad received her PhD in mathematics from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran in March 2013. She worked as an assistant professor in the geometry group at the Institute for Advanced Studies in Basic Sciences (IASBS) in Zanjan, Iran, for seven years. She is interested in the use of geometry in data science and machine learning, and in particular in dimensionality reduction, a fundamental problem in topological and geometric data analysis.
Her collaboration with Jürgen Jost began in 2017, via a project on a generalization of the concept of sectional curvature to datasets. In 2020, she received a grant from the Max-Planck society to continue her collaboration at the Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany. As of August 2022, she started a new position at the Center for Scalable Data Analytics and Artificial Intelligence, as a senior postdoc.
Jürgen Jost worked as a Professor of Mathematics at Ruhr University Bochum from 1984 to 1996 and since 1996 has been director and a permanent member of the Max Planck Institute for Mathematics in the Sciences, Leipzig. In 1998 he became an Honorary Professor at the University of Leipzig. He is also an external member of the Santa Fe Institute for the Sciences of Complexity, New Mexico.
He pursues both topical research in different fields of pure mathematics and theoretical physics (Riemannian and algebraic geometry, geometric analysis, calculus of variations, partial differential equations, dynamical systems, graph and hypergraph theory) and interdisciplinary research in complex systems, including evolutionary and theoretical molecular biology, mathematical and theoretical neuroscience, nonlinear dynamics and statistical physics, economics and social sciences, strategy science, history and philosophy of science. He directs a group of about 40 scientists, postdocs and PhD students, and has manyinternational cooperation partners.
作者簡介(中文翻譯)
Parvaneh Joharinad於2013年3月在伊朗德黑蘭的阿米爾卡比爾科技大學（Amirkabir University of Technology）獲得數學博士學位。她在伊朗贊詹的基礎科學高等研究所（Institute for Advanced Studies in Basic Sciences, IASBS）的幾何學組擔任助理教授七年。她對幾何在數據科學和機器學習中的應用感興趣，特別是在降維方面，這是拓撲和幾何數據分析中的一個基本問題。
她與Jürgen Jost的合作始於2017年，通過一個關於將截面曲率概念推廣到數據集的項目。2020年，她獲得了馬克斯·普朗克學會的資助，以便在德國萊比錫的馬克斯·普朗克科學數學研究所繼續她的合作。自2022年8月起，她在可擴展數據分析與人工智慧中心擔任高級博士後研究員的新職位。
Jürgen Jost於1984年至1996年擔任魯爾大學波鴻的數學教授，自1996年以來一直擔任德國萊比錫的馬克斯·普朗克科學數學研究所的所長和常任成員。1998年，他成為萊比錫大學的名譽教授。他也是新墨西哥州聖塔菲複雜科學研究所的外部成員。
他在純數學和理論物理的不同領域（黎曼幾何和代數幾何、幾何分析、變分法、偏微分方程、動力系統、圖論和超圖論）進行主題研究，並在複雜系統中進行跨學科研究，包括進化和理論分子生物學、數學和理論神經科學、非線性動力學和統計物理學、經濟學和社會科學、策略科學、科學史和科學哲學。他指導著約40名科學家、博士後和博士生，並擁有許多國際合作夥伴。
