持续图空间的度量几何
Metric Geometry of Spaces of Persistence Diagrams
AI总结：
本文构造函子 $\mathcal{D}_p$ 将度量对映射为持续图度量空间，证明其连续性、多种度量性质保持性、Fréchet 均值存在性，并推出欧氏持续图空间具有无限多维数。
摘要：
持续图是在拓扑数据分析中发挥核心作用的对象。在本文中，我们研究持续图空间的局部与整体几何性质。为此，我们构造了一族函子 $\mathcal{D}_p$，其中 $1\leq p\leq\infty$，它为每个度量对 $(X,A)$ 指定一个带基点的度量空间 $\mathcal{D}_p(X,A)$。此外，我们证明 $\mathcal{D}_{\infty}$ 关于度量对的 Gromov–Hausdorff 收敛是序列连续的；并证明 $\mathcal{D}_p$ 保持若干有用的度量性质：当 $p\in[1,\infty)$ 时保持完备性和可分性，当 $p=2$ 时保持测地性以及 Alexandrov 意义下的非负曲率。对于后一种情形，我们描述了空图处方向空间的度量。我们还证明，在 $\mathcal{D}_p(X,A)$（$1\leq p\leq\infty$）上，任何具有有限二阶矩和紧支撑的 Borel 概率测度的 Fréchet 均值集均非空。作为该几何框架的应用，我们证明欧氏持续图空间 $\mathcal{D}_p(\mathbb{R}^{2n},\Delta_n)$（$1\leq n$，$1\leq p<\infty$）具有无限的覆盖维数、Hausdorff 维数、渐近维数、Assouad 维数和 Assouad–Nagata 维数。
英文摘要：
Persistence diagrams are objects that play a central role in topological data analysis. In the present article, we investigate the local and global geometric properties of spaces of persistence diagrams. In order to do this, we construct a family of functors $\mathcal{D}_p$, $1\leq p \leq\infty$, that assign, to each metric pair $(X,A)$, a pointed metric space $\mathcal{D}_p(X,A)$. Moreover, we show that $\mathcal{D}_{\infty}$ is sequentially continuous with respect to the Gromov-Hausdorff convergence of metric pairs, and we prove that $\mathcal{D}_p$ preserves several useful metric properties, such as completeness and separability, for $p \in [1,\infty)$, and geodesicity and non-negative curvature in the sense of Alexandrov, for $p=2$. For the latter case, we describe the metric of the space of directions at the empty diagram. We also show that the Fréchet mean set of a Borel probability measure on $\mathcal{D}_p(X,A)$, $1\leq p \leq\infty$, with finite second moment and compact support is non-empty. As an application of our geometric framework, we prove that the space of Euclidean persistence diagrams, $\mathcal{D}_{p}(\mathbb{R}^{2n},Δ_n)$, $1\leq n$ and $1\leq p<\infty$, has infinite covering, Hausdorff, asymptotic, Assouad, and Assouad-Nagata dimensions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
