The Persistent Topology of Optimal Transport Based Metric Thickenings
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866929288907849728 |
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| author | Adams, Henry Mémoli, Facundo Moy, Michael Wang, Qingsong |
| author_facet | Adams, Henry Mémoli, Facundo Moy, Michael Wang, Qingsong |
| contents | A metric thickening of a given metric space $X$ is any metric space admitting an isometric embedding of $X$. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces. We introduce two new families of metric thickenings, the $p$-Vietoris-Rips and $p$-Čech metric thickenings for all $1\le p\le \infty$, which include all measures on $X$ whose $p$-diameter or $p$-radius is bounded from above, equipped with an optimal transport metric. The $p$-diameter (resp. $p$-radius) of a measure is a certain $\ell_p$ relaxation of the usual notion of diameter (resp. radius) of a subset of a metric space. These families recover the previously studied Vietoris-Rips and Čech metric thickenings when $p=\infty$. As our main contribution, we prove a stability theorem for the persistent homology of $p$-Vietoris-Rips and $p$-Čech metric thickenings, which is novel even in the case $p=\infty$. In the specific case $p=2$, we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the $2$-Vietoris-Rips thickenings of the $n$-sphere as the scale increases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_15061 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The Persistent Topology of Optimal Transport Based Metric Thickenings Adams, Henry Mémoli, Facundo Moy, Michael Wang, Qingsong Metric Geometry Algebraic Topology 55N31, 51F99, 53C23 A metric thickening of a given metric space $X$ is any metric space admitting an isometric embedding of $X$. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces. We introduce two new families of metric thickenings, the $p$-Vietoris-Rips and $p$-Čech metric thickenings for all $1\le p\le \infty$, which include all measures on $X$ whose $p$-diameter or $p$-radius is bounded from above, equipped with an optimal transport metric. The $p$-diameter (resp. $p$-radius) of a measure is a certain $\ell_p$ relaxation of the usual notion of diameter (resp. radius) of a subset of a metric space. These families recover the previously studied Vietoris-Rips and Čech metric thickenings when $p=\infty$. As our main contribution, we prove a stability theorem for the persistent homology of $p$-Vietoris-Rips and $p$-Čech metric thickenings, which is novel even in the case $p=\infty$. In the specific case $p=2$, we prove a Hausmann-type theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the $2$-Vietoris-Rips thickenings of the $n$-sphere as the scale increases. |
| title | The Persistent Topology of Optimal Transport Based Metric Thickenings |
| topic | Metric Geometry Algebraic Topology 55N31, 51F99, 53C23 |
| url | https://arxiv.org/abs/2109.15061 |