Spherical Coordinates from Persistent Cohomology
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866913725831708672 |
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| author | Schonsheck, Nikolas C. Schonsheck, Stefan C. |
| author_facet | Schonsheck, Nikolas C. Schonsheck, Stefan C. |
| contents | We describe a method to obtain spherical parameterizations of arbitrary data through the use of persistent cohomology and variational optimization. We begin by computing the second-degree persistent cohomology of the filtered Vietoris-Rips (VR) complex of a data set $X$ and extract a cocycle $α$ from any significant feature. From this cocycle, we define an associated map $α: VR(X) \to S^2$ and use this map as an infeasible initialization for a variational model, which we show has a unique solution (up to rigid motion). We then employ an alternating gradient descent/Möbius transformation update method to solve the problem and generate a more suitable, i.e., smoother, representative of the homotopy class of $α$, preserving the relevant topological feature. Finally, we conduct numerical experiments on both synthetic and real-world data sets to show the efficacy of our proposed approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_02791 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Spherical Coordinates from Persistent Cohomology Schonsheck, Nikolas C. Schonsheck, Stefan C. Algebraic Topology Numerical Analysis 55-04, 65-04 We describe a method to obtain spherical parameterizations of arbitrary data through the use of persistent cohomology and variational optimization. We begin by computing the second-degree persistent cohomology of the filtered Vietoris-Rips (VR) complex of a data set $X$ and extract a cocycle $α$ from any significant feature. From this cocycle, we define an associated map $α: VR(X) \to S^2$ and use this map as an infeasible initialization for a variational model, which we show has a unique solution (up to rigid motion). We then employ an alternating gradient descent/Möbius transformation update method to solve the problem and generate a more suitable, i.e., smoother, representative of the homotopy class of $α$, preserving the relevant topological feature. Finally, we conduct numerical experiments on both synthetic and real-world data sets to show the efficacy of our proposed approach. |
| title | Spherical Coordinates from Persistent Cohomology |
| topic | Algebraic Topology Numerical Analysis 55-04, 65-04 |
| url | https://arxiv.org/abs/2209.02791 |