Mathematical Principles of Topological and Geometric Data Analysis (Paperback)
暫譯: 拓撲與幾何資料分析的數學原理（平裝）

Joharinad, Parvaneh, Jost, Jürgen

Mathematical Principles of Topological and Geometric Data Analysis (Paperback)

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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)

本書探討並示範如何將幾何工具應用於資料分析。全書首先系統性地介紹所需的數學基礎，涵蓋從範疇論（category theory）、代數拓撲（algebraic topology）、黎曼幾何（Riemannian geometry）、算子理論（operator theory）到網路分析（network analysis）等主題；接著說明並分析一些最重要的降維（dimension reduction）機器學習技術，包括不同類型的流形學習（manifold learning）與核方法（kernel methods）。此外，本書還以超凸性（hyperconvexity）概念為基礎，發展出一種廣義度量空間（generalized metric spaces）的新曲率概念，可用於幾何資訊的拓撲表示。

近年來出現了一項引人入勝的發展：原本在純數學研究，尤其是幾何學研究中所創造的概念與方法，已成為機器學習中分析資料的強大工具。其根本原因在於，資料通常都具有某種距離概念，用來量化資料點之間的差異。當然，若要成功應用，這些幾何工具通常必須經過適當的重新定義、廣義化或延伸。

本書主要面向希望全面了解適用於資料分析之幾何概念與方法的數學家；同時，也適合從事機器學習與資料分析的研究人員閱讀，尤其是想了解其所使用方法之系統化數學基礎者。

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 月取得伊朗德黑蘭 Amirkabir University of Technology（Tehran Polytechnic）的數學博士學位。她曾在伊朗贊詹的 Institute for Advanced Studies in Basic Sciences（IASBS）幾何學組擔任助理教授，任職七年。她對幾何學在資料科學與機器學習中的應用深感興趣，尤其是維度縮減（dimensionality reduction）；這是拓撲與幾何資料分析中的一項基礎問題。

她與 Jürgen Jost 的合作始於 2017 年，當時雙方共同進行一項將截面曲率（sectional curvature）概念推廣至資料集的計畫。2020 年，她獲得 Max-Planck Society 的研究補助，得以在德國萊比錫的 Max-Planck Institute for Mathematics in the Sciences 延續合作。自 2022 年 8 月起，她在 Center for Scalable Data Analytics and Artificial Intelligence 擔任資深博士後研究員（senior postdoc）。

Jürgen Jost 於 1984 年至 1996 年間擔任 Ruhr University Bochum 的數學教授；自 1996 年起，他一直擔任位於萊比錫的 Max Planck Institute for Mathematics in the Sciences 所長及常任成員。1998 年，他成為 University of Leipzig 的名譽教授。他也是位於美國新墨西哥州的 Santa Fe Institute for the Sciences of Complexity 外籍成員。

他的研究涵蓋純數學與理論物理不同領域的主題性研究，包括 Riemann 幾何與代數幾何、幾何分析、變分法、偏微分方程、動力系統，以及圖論與超圖理論；同時也從事複雜系統的跨領域研究，內容包括演化與理論分子生物學、數學與理論神經科學、非線性動力學與統計物理學、經濟學與社會科學、策略科學，以及科學史與科學哲學。他領導一個由約 40 名科學家、博士後研究員與博士生組成的研究團隊，並與許多國際合作夥伴保持合作關係。

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