论文标题
入门拓扑数据分析
Introductory Topological Data Analysis
论文作者
论文摘要
本文介绍了拓扑数据分析。从度量空间的概念和一些基本图理论开始，我们以数据集并演示了它们的一些拓扑特性。我们讨论简单的复合物以及它们如何与称为神经定理的事物相关。为此，我们介绍了拓扑领域的概念，例如开放式封面，同构和同型等效。这导致我们从基础数据集讨论过滤数据并从拓扑上不变的简单复合物。然后，对于持续的同源性和贝蒂数字有一个小简介，因为这些是TDA的有用分析工具。本文中的大部分数字的代码随附的在线附录可从bit.ly/tda_2020获得。
This paper introduces topological data analysis. Starting from notions of a metric space and some elementary graph theory, we take example sets of data and demonstrate some of their topological properties. We discuss simplicial complexes and how they relate to something called the Nerve Theorem. For this we introduce notions from the field of topology such as open covers, homeomorphism and homotopy equivalences. This leads us into discussing filtering data and deriving topologically invariant simplicial complexes from the underlying data set. There is then a small introduction to persistent homology and Betti numbers as these are useful analytical tools for TDA. An accompanying online appendix for the code producing the bulk of the figures in this paper is available at bit.ly/TDA_2020.
