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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.30266 |
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| _version_ | 1866913170794217472 |
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| author | León, Uriel Martínez Niles-Weed, Jonathan |
| author_facet | León, Uriel Martínez Niles-Weed, Jonathan |
| contents | We perform a mathematical and statistical analysis of the Wasserstein least squares problem, a regression method for vector-valued covariates and distribution-valued responses. Our proposal contrasts with other distributional regression methods by having a direct interpretation in terms of random variables, as a nonparametric analogue of the classic random-effects model. On the mathematical side, we use a strategy of Lavenant (2024) to show that Wasserstein least squares is the canonical extension of Euclidean least squares to the space of probability distributions from the perspective of convex analysis; this viewpoint gives rise to multimarginal and dual formulations of the Wasserstein least squares problem, extending a similar theory for Wasserstein barycenters. We perform a statistical analysis of the Wasserstein least squares problem under the template deformation model, showing, surprisingly, that estimation is possible at the n^{-1/2} rate. As a special case, we obtain improved rates of estimation for Wasserstein barycenters, which are an exponential improvement over those established by Ahidar-Coutrix, Le Gouic and Paris (2020). Finally, we propose a heuristic particle method for Wasserstein least squares and use it to conduct a novel analysis of large-scale demographic data from the RAND Health and Retirement Study. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_30266 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Wasserstein Least Squares: A Canonical Regression Method for Probability Distributions León, Uriel Martínez Niles-Weed, Jonathan Statistics Theory 62J05, 49Q22, 60A10 We perform a mathematical and statistical analysis of the Wasserstein least squares problem, a regression method for vector-valued covariates and distribution-valued responses. Our proposal contrasts with other distributional regression methods by having a direct interpretation in terms of random variables, as a nonparametric analogue of the classic random-effects model. On the mathematical side, we use a strategy of Lavenant (2024) to show that Wasserstein least squares is the canonical extension of Euclidean least squares to the space of probability distributions from the perspective of convex analysis; this viewpoint gives rise to multimarginal and dual formulations of the Wasserstein least squares problem, extending a similar theory for Wasserstein barycenters. We perform a statistical analysis of the Wasserstein least squares problem under the template deformation model, showing, surprisingly, that estimation is possible at the n^{-1/2} rate. As a special case, we obtain improved rates of estimation for Wasserstein barycenters, which are an exponential improvement over those established by Ahidar-Coutrix, Le Gouic and Paris (2020). Finally, we propose a heuristic particle method for Wasserstein least squares and use it to conduct a novel analysis of large-scale demographic data from the RAND Health and Retirement Study. |
| title | Wasserstein Least Squares: A Canonical Regression Method for Probability Distributions |
| topic | Statistics Theory 62J05, 49Q22, 60A10 |
| url | https://arxiv.org/abs/2605.30266 |