Distribution-on-distribution regression via optimal transport maps

• 成果类型: Article
• 署名作者: Ghodrati, Laya; Panaretos, Victor M.
• 署名单位: Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asac005
• 发表日期: 2022
• 页码: 957974
• 关键词: density-functions

摘要

We 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 Frechet mean of the response to the covariate via an optimal transport map. We define a Frechet-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.