Principal Curves In Metric Spaces And The Space Of Probability Measures
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arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913825190576128 |
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| author | Warren, Andrew Afanassiev, Anton Kobayashi, Forest Kim, Young-Heon Schiebinger, Geoffrey |
| author_facet | Warren, Andrew Afanassiev, Anton Kobayashi, Forest Kim, Young-Heon Schiebinger, Geoffrey |
| contents | We introduce principal curves in Wasserstein space, and in general compact metric spaces. Our motivation for the Wasserstein case comes from optimal-transport-based trajectory inference, where a developing population of cells traces out a curve in Wasserstein space. Our framework enables new experimental procedures for collecting high-density time-courses of developing populations of cells: time-points can be processed in parallel (making it easier to collect more time-points). However, then the time of collection is unknown, and must be recovered by solving a seriation problem (or one-dimensional manifold learning problem).
We propose an estimator based on Wasserstein principal curves, and prove it is consistent for recovering a curve of probability measures in Wasserstein space from empirical samples. This consistency theorem is obtained via a series of results regarding principal curves in compact metric spaces. In particular, we establish the validity of certain numerical discretization schemes for principal curves, which is a new result even in the Euclidean setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04168 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Principal Curves In Metric Spaces And The Space Of Probability Measures Warren, Andrew Afanassiev, Anton Kobayashi, Forest Kim, Young-Heon Schiebinger, Geoffrey Statistics Theory Primary: 62G05, 49Q20, Secondary: 62P10, 62R20 We introduce principal curves in Wasserstein space, and in general compact metric spaces. Our motivation for the Wasserstein case comes from optimal-transport-based trajectory inference, where a developing population of cells traces out a curve in Wasserstein space. Our framework enables new experimental procedures for collecting high-density time-courses of developing populations of cells: time-points can be processed in parallel (making it easier to collect more time-points). However, then the time of collection is unknown, and must be recovered by solving a seriation problem (or one-dimensional manifold learning problem). We propose an estimator based on Wasserstein principal curves, and prove it is consistent for recovering a curve of probability measures in Wasserstein space from empirical samples. This consistency theorem is obtained via a series of results regarding principal curves in compact metric spaces. In particular, we establish the validity of certain numerical discretization schemes for principal curves, which is a new result even in the Euclidean setting. |
| title | Principal Curves In Metric Spaces And The Space Of Probability Measures |
| topic | Statistics Theory Primary: 62G05, 49Q20, Secondary: 62P10, 62R20 |
| url | https://arxiv.org/abs/2505.04168 |