论文标题
越野$ \ unicode {x2013} $ rips rips complex complexteres a resces a rips complex a
Vietoris$\unicode{x2013}$Rips Complexes of Metric Spaces Near a Metric Graph
论文作者
论文摘要
对于足够小的规模$β> 0 $，越野$ \ unicode {x2013} $ rips complex $ \ mathcal {r}_β（s s） 类型。虽然定性结果是显着的，并且自然而然地概括为恢复Riemannian歧管以外的空间$ \ unicode {x2014} $，例如地理公制空间，具有正凸radius $ \ unicode {x2014} $，一般性是为代价的。尽管已知比例参数$β$仅取决于地球空间的几何特性，但如何定量选择为给定的地理空间选择这种$β$仍然难以捉摸。在这项工作中，我们专注于一种特殊类型的大地测量空间的拓扑恢复，称为度量图。对于抽象度量图$ \ MATHCAL {g} $和A（示例）度量空间$ s $，带有一个小的gromov $ \ unicode {x2013} $ hausdorff的距离，我们将根据$ \ nathcal {g} $ for $ \ math的convexity radius of convexity radius of $β$的描述，以$ \ maths}（相当于$ \ Mathcal {g} $。我们的调查还扩展到对越野的研究{x2013} $撕裂的欧几里得子集$ s \ subset \ subset \ subset \ subset \ mathbb {r}^d $，带有小的hausdorff距离嵌入式度量图$ \ mathcal {g} g} {g} \ subseet \ subseet \ subbb {从$ s $点的成对欧几里得距离中，我们介绍了基于路径的vietoris $ \ unicode {x2013} $ rips compleactes $ \ mathcal {r}^\varepsilon_β（s）的家庭（由$ \ varepsilon $）介绍一个家庭（$ \ varepsilon $）。根据$ \ Mathcal {G} $嵌入的凸度半径和失真，我们展示了如何选择合适的参数$ \ varepsilon $和一个比例$β$，使得$ \ nathcal {r}^\varepsilon_β（s）$是同质$ \ nathcal的同性恋。
For a sufficiently small scale $β>0$, the Vietoris$\unicode{x2013}$Rips complex $\mathcal{R}_β(S)$ of a metric space $S$ with a small Gromov$\unicode{x2013}$Hausdorff distance to a closed Riemannian manifold $M$ has been already known to recover $M$ up to homotopy type. While the qualitative result is remarkable and generalizes naturally to the recovery of spaces beyond Riemannian manifolds$\unicode{x2014}$such as geodesic metric spaces with a positive convexity radius$\unicode{x2014}$the generality comes at a cost. Although the scale parameter $β$ is known to depend only on the geometric properties of the geodesic space, how to quantitatively choose such a $β$ for a given geodesic space is still elusive. In this work, we focus on the topological recovery of a special type of geodesic space, called a metric graph. For an abstract metric graph $\mathcal{G}$ and a (sample) metric space $S$ with a small Gromov$\unicode{x2013}$Hausdorff distance to it, we provide a description of $β$ based on the convexity radius of $\mathcal{G}$ in order for $\mathcal{R}_β(S)$ to be homotopy equivalent to $\mathcal{G}$. Our investigation also extends to the study of the Vietoris$\unicode{x2013}$Rips complexes of a Euclidean subset $S\subset\mathbb{R}^d$ with a small Hausdorff distance to an embedded metric graph $\mathcal{G}\subset\mathbb{R}^d$. From the pairwise Euclidean distances of points of $S$, we introduce a family (parametrized by $\varepsilon$) of path-based Vietoris$\unicode{x2013}$Rips complexes $\mathcal{R}^\varepsilon_β(S)$ for a scale $β>0$. Based on the convexity radius and distortion of the embedding of $\mathcal{G}$, we show how to choose a suitable parameter $\varepsilon$ and a scale $β$ such that $\mathcal{R}^\varepsilon_β(S)$ is homotopy equivalent to $\mathcal{G}$.
