基于最优迁移映射的分布-分布回归
Distribution-on-Distribution Regression via Optimal Transport Maps
摘要：
我们提出了一个框架进行回归时，协变量和响应的概率分布在一个紧凑的间隔$Ω\subset\mathbb{R}$。我们的回归模型是基于最优运输理论，并通过最优运输图将响应分布的条件Fréchet均值与协变量分布联系起来。我们定义了这个回归映射的Fréchet最小二乘估计，并建立了它的一致性和收敛到真实映射的速度，在全部和部分观测的回归对。估计的计算减少到保序回归问题，因此我们的回归模型可以很容易地实现。我们说明我们的方法使用真实和模拟数据。
英文摘要：
We present a framework for performing regression when both covariate and response are probability distributions on a compact interval $Ω\subset\mathbb{R}$. Our regression model is based on the theory of optimal transportation and links the conditional Fréchet mean of the response distribution to the covariate distribution 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 observation of the regression pairs. Computation of the estimator is shown to reduce to an isotonic regression problem, and thus our regression model can be implemented with ease. We illustrate our methodology using real and simulated data.
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