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Distribution-on-distribution regression via optimal transport maps
[Upper and lower risk bounds for estimating the Wasserstein barycenter of random measures on the real line]

Author

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  • Laya Ghodrati
  • Victor M Panaretos

Abstract

SummaryWe present a framework for performing regression when both covariate and response are probability distributions on a compact interval. Our regression model is based on the theory of optimal transportation, and links the conditional Fréchet mean of the response to the covariate via an optimal transport map. We define a Fréchet-least-squares estimator of this regression map, and establish its consistency and rate of convergence to the true map, under both full and partial observations of the regression pairs. Computation of the estimator is shown to reduce to a standard convex optimization problem, and thus our regression model can be implemented with ease. We illustrate our methodology using real and simulated data.

Suggested Citation

  • Laya Ghodrati & Victor M Panaretos, 2022. "Distribution-on-distribution regression via optimal transport maps [Upper and lower risk bounds for estimating the Wasserstein barycenter of random measures on the real line]," Biometrika, Biometrika Trust, vol. 109(4), pages 957-974.
    • Handle: RePEc:oup:biomet:v:109:y:2022:i:4:p:957-974.
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    File URL: http://hdl.handle.net/10.1093/biomet/asac005
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    Cited by:

    1. Ghodrati, Laya & Panaretos, Victor M., 2023. "Minimax rate for optimal transport regression between distributions," Statistics & Probability Letters, Elsevier, vol. 194(C).
    2. Florian Gunsilius & Meng Hsuan Hsieh & Myung Jin Lee, 2022. "Tangential Wasserstein Projections," Papers 2207.14727, arXiv.org, revised Aug 2022.

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