Tropical Fermat-Weber Polytropes
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911671926128640 |
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| author | Sabol, John Barnhill, David Yoshida, Ruriko Miura, Keiji |
| author_facet | Sabol, John Barnhill, David Yoshida, Ruriko Miura, Keiji |
| contents | We study the geometry of tropical Fermat--Weber points, that is, optimal solutions to a location problem over a projective space using a dissimilarity measure derived from the tropical metric. It is well-known that for a given sample, such points are not necessarily unique, and we show that the set of all possible Fermat--Weber points forms a polytrope. This follows from the fact that our location problem turns out to be dual to a particular minimum-cost flow problem, and we describe the polytrope of optimal locations in the terminology of tropical geometry. We also provide a simple gradient descent algorithm that converges to the Fermat--Weber polytrope. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14287 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tropical Fermat-Weber Polytropes Sabol, John Barnhill, David Yoshida, Ruriko Miura, Keiji Combinatorics We study the geometry of tropical Fermat--Weber points, that is, optimal solutions to a location problem over a projective space using a dissimilarity measure derived from the tropical metric. It is well-known that for a given sample, such points are not necessarily unique, and we show that the set of all possible Fermat--Weber points forms a polytrope. This follows from the fact that our location problem turns out to be dual to a particular minimum-cost flow problem, and we describe the polytrope of optimal locations in the terminology of tropical geometry. We also provide a simple gradient descent algorithm that converges to the Fermat--Weber polytrope. |
| title | Tropical Fermat-Weber Polytropes |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.14287 |