用类型化拓扑揭示数据集的轨迹与分支结构，为聚类等任务提供新思路。
Typed Topological Structures Of Datasets
- 引入类型化拓扑，用类型标记开集以分析数据内在结构。
- 数据在 $R^2$ 中被组织成有序组件，形成可编码为整数序列的轨迹。
- 发现分支关系可用一种叫 typed-II 的伪树表示，适合异常检测等应用。
一个位于 $R^2$ 上的数据集 $X$ 是一个有限拓扑空间。当前研究多聚焦于统计方法与代数拓扑方法。文献 \\[1\\] 引入了类型化拓扑空间的概念，展现出研究有限拓扑空间（如数据集）的潜力，是一种从一般拓扑视角出发的新方法。类型化拓扑空间是其开集被赋予类型的拓扑空间，可基于特定类型重新定义拓扑概念与方法。本文构建了一套特殊的类型及其相关的类型化拓扑结构用于数据集 $X$，从而揭示其内部结构。特别地，$R^2$ 存在一个自然商空间，其中 $X$ 被组织为若干轨迹，每条轨迹又被划分为有序组件，这些组件可表示为整数序列。跨越轨迹的组件构成分支，其关系可由一类称为 typed-II 伪树的结构良好刻画。此类结构为凸包计算、孔洞检测、聚类与异常检测等问题提供了新算法平台。
原文摘要 · Abstract (English)
A datatset $X$ on $R^2$ is a finite topological space. Current research of a dataset focuses on statistical methods and the algebraic topological method \cite{carlsson}. In \cite{hu}, the concept of typed topological space was introduced and showed to have the potential for studying finite topological spaces, such as a dataset. It is a new method from the general topology perspective. A typed topological space is a topological space whose open sets are assigned types. Topological concepts and methods can be redefined using open sets of certain types. In this article, we develop a special set of types and its related typed topology on a dataset $X$. Using it, we can investigate the inner structure of $X$. In particular, $R^2$ has a natural quotient space, in which $X$ is organized into tracks, and each track is split into components. Those components are in a order. Further, they can be represented by an integer sequence. Components crossing tracks form branches, and the relationship can be well represented by a type of pseudotree (called typed-II pseudotree). Such structures provide a platform for new algorithms for problems such as calculating convex hull, holes, clustering and anomaly detection.
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