Given a map $f:X \to M$ from a topological space $X$ to a metric space $M$, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when $M=\mathbb{R}$ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of $f$. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.

翻译：给定从拓扑空间$X$到度量空间$M$的映射$f:X \to M$，装饰化Reeb空间由Reeb空间及其属性函数构成，其中属性函数的值可恢复在构建Reeb空间过程中丢失的几何信息。例如，当$M=\mathbb{R}$为实直线时，Reeb空间即为著名的Reeb图，其属性可包含用于概括$f$水平集拓扑的持续同调图。本文引入多种形式的装饰化Reeb空间，并证明我们的构造具有Gromov-Hausdorff稳定性。我们同时给出了从有限样本逼近装饰化Reeb空间的结果，并利用这些结果开发了将这些构造应用于点云数据的计算框架。