Metric Geometry of Spaces of Persistence Diagrams
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866914904404918272 |
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| author | Che, Mauricio Galaz-García, Fernando Guijarro, Luis Solis, Ingrid Amaranta Membrillo |
| author_facet | Che, Mauricio Galaz-García, Fernando Guijarro, Luis Solis, Ingrid Amaranta Membrillo |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2109_14697 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Metric Geometry of Spaces of Persistence Diagrams Che, Mauricio Galaz-García, Fernando Guijarro, Luis Solis, Ingrid Amaranta Membrillo Metric Geometry Algebraic Topology 53C23, 55N31, 54F45 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. |
| title | Metric Geometry of Spaces of Persistence Diagrams |
| topic | Metric Geometry Algebraic Topology 53C23, 55N31, 54F45 |
| url | https://arxiv.org/abs/2109.14697 |