Morse theory for chromatic Delaunay triangulations
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866911189640937472 |
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| author | Natarajan, Abhinav Chaplin, Thomas Brown, Adam Jimenez, Maria-Jose |
| author_facet | Natarajan, Abhinav Chaplin, Thomas Brown, Adam Jimenez, Maria-Jose |
| contents | The chromatic alpha filtration is a generalization of the alpha filtration that can encode spatial relationships among classes of labelled point cloud data, and has applications in topological data analysis of multi-species data. In this paper we introduce the chromatic Delaunay-Čech and chromatic Delaunay-Rips filtrations, which are computationally favourable alternatives to the chromatic alpha filtration. We use generalized discrete Morse theory to show that the Čech, chromatic Delaunay-Čech, and chromatic alpha filtrations are related by simplicial collapses. Our result generalizes a result of Bauer and Edelsbrunner from the non-chromatic to the chromatic setting. We also show that the chromatic Delaunay-Rips filtration is locally stable to perturbations of the underlying point cloud. Our results provide theoretical justification for the use of chromatic Delaunay-Čech and chromatic Delaunay-Rips filtrations in applications, and we demonstrate their computational advantage with numerical experiments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_19303 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Morse theory for chromatic Delaunay triangulations Natarajan, Abhinav Chaplin, Thomas Brown, Adam Jimenez, Maria-Jose Algebraic Topology 55N31 (Primary) 55U10, 52-08 (Secondary) The chromatic alpha filtration is a generalization of the alpha filtration that can encode spatial relationships among classes of labelled point cloud data, and has applications in topological data analysis of multi-species data. In this paper we introduce the chromatic Delaunay-Čech and chromatic Delaunay-Rips filtrations, which are computationally favourable alternatives to the chromatic alpha filtration. We use generalized discrete Morse theory to show that the Čech, chromatic Delaunay-Čech, and chromatic alpha filtrations are related by simplicial collapses. Our result generalizes a result of Bauer and Edelsbrunner from the non-chromatic to the chromatic setting. We also show that the chromatic Delaunay-Rips filtration is locally stable to perturbations of the underlying point cloud. Our results provide theoretical justification for the use of chromatic Delaunay-Čech and chromatic Delaunay-Rips filtrations in applications, and we demonstrate their computational advantage with numerical experiments. |
| title | Morse theory for chromatic Delaunay triangulations |
| topic | Algebraic Topology 55N31 (Primary) 55U10, 52-08 (Secondary) |
| url | https://arxiv.org/abs/2405.19303 |